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期权期货与其他衍生产品第九版课后习题与答案Chapter

期权期货与其他衍生产品第九版课后习题与答案Chapter

CHAPTER 31Interest Rate Derivatives: Models of the Short Rate Practice QuestionsProblem 31.1.What is the difference between an equilibrium model and a no-arbitrage model?Equilibrium models usually start with assumptions about economic variables and derive the behavior of interest rates. The initial term structure is an output from the model. In ano-arbitrage model the initial term structure is an input. The behavior of interest rates in ano-arbitrage model is designed to be consistent with the initial term structure.Problem 31.2.Suppose that the short rate is currently 4% and its standard deviation is 1% per annum. What happens to the standard deviation when the short rate increases to 8% in (a) Vasicek’s model;(b) Rendleman and Bartter’s mod el; and (c) the Cox, Ingersoll, and Ross model?In Vasicek’s model the standard deviation stays at 1%. In the Rendleman and Bartter model the standard deviation is proportional to the level of the short rate. When the short rate increases from 4% to 8% the standard deviation increases from 1% to 2%. In the Cox, Ingersoll, and Ross model the standard deviation of the short rate is proportional to the square root of the short rate. When the short rate increases from 4% to 8% the standard deviation increases from 1% to 1.414%.Problem 31.3.If a stock price were mean reverting or followed a path-dependent process there would be market inefficiency. Why is there not market inefficiency when the short-term interest rate does so?If the price of a traded security followed a mean-reverting or path-dependent process there would be market inefficiency. The short-term interest rate is not the price of a traded security. In other words we cannot trade something whose price is always the short-term interest rate. There is therefore no market inefficiency when the short-term interest rate follows amean-reverting or path-dependent process. We can trade bonds and other instruments whose prices do depend on the short rate. The prices of these instruments do not followmean-reverting or path-dependent processes.Problem 31.4.Explain the difference between a one-factor and a two-factor interest rate model.In a one-factor model there is one source of uncertainty driving all rates. This usually means that in any short period of time all rates move in the same direction (but not necessarily by the same amount). In a two-factor model, there are two sources of uncertainty driving all rates. The first source of uncertainty usually gives rise to a roughly parallel shift in rates. The second gives rise to a twist where long and short rates moves in opposite directions.Problem 31.5.Can the approach described in Section 31.4 for decomposing an option on a coupon-bearing bond into a portfolio of options on zero-coupon bonds be used in conjunction with a two-factor model? Explain your answer.No. The approach in Section 31.4 relies on the argument that, at any given time, all bond prices are moving in the same direction. This is not true when there is more than one factor.Problem 31.6.Suppose that 01a =. and 01b =. in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is 10% and the initial standard deviation of the short rate change in a short time t ∆is 0.zero-coupon bond that matures in year 10.In Vasicek’s model, 01a =., 01b =., and 002σ=. so that01101(10)(1)63212101B t t e -.⨯,+=-=..22(63212110)(010100002)00004632121(10)exp 00104A t t ⎡⎤.-.⨯.-..⨯.,+=-⎢⎥..⎣⎦071587=. The bond price is therefore 63212101071587038046e -.⨯..=.In the Cox, Ingersoll, and Ross model, 01a =., 01b =.and 00200632σ=.=.. Also013416γ==. Define10()(1)2092992a e γβγγ=+-+=.102(1)(10)607650e B t t γβ-,+==.225()2(10)069746ab a e A t t σγγβ/+⎛⎫,+==. ⎪⎝⎭The bond price is therefore 60765001069746037986e -.⨯..=.Problem 31.7.Suppose that 01a =., 008b =., and 0015σ=. in Vasicek’s model with the initial value of the short rate being 5%. Calculate the price of a one-year European call option on azero-coupon bond with a principal of $100 that matures in three years when the strike price is $87.Using the notation in the text, 3s =, 1T =, 100L =, 87K =, and2010015(1002588601P e σ-⨯..=-=.. From equation (31.6), (01)094988P ,=., (03)085092P ,=., and 114277h =. so thatequation (31.20) gives the call price as call price is 100085092(114277)87094988(111688)259N N ⨯.⨯.-⨯.⨯.=. or $2.59.Problem 31.8.Repeat Problem 31.7 valuing a European put option with a strike of $87. What is the put –call parity relationship between the prices of European call and put options? Show that the put and call option prices satisfy put –call parity in this case.As mentioned in the text, equation (31.20) for a call option is essentially the same as Bl ack’s model. By analogy with Black’s formulas corresponding expression for a put option is (0)()(0)()P KP T N h LP s N h σ,-+-,- In this case the put price is 87094988(111688)100085092(114277)014N N ⨯.⨯-.-⨯.⨯-.=.Since the underlying bond pays no coupon, put –call parity states that the put price plus the bond price should equal the call price plus the present value of the strike price. The bond price is 85.09 and the present value of the strike price is 870949888264⨯.=.. Put –call parity is therefore satisfied:82642598509014.+.=.+.Problem 31.9.Suppose that 005a =., 008b =., and 0015σ=. in Vasicek’s model with the initialshort-term interest rate being 6%. Calculate the price of a 2.1-year European call option on a bond that will mature in three years. Suppose that the bond pays a coupon of 5% semiannually. The principal of the bond is 100 and the strike price of the option is 99. The strike price is the cash price (not the quoted price) that will be paid for the bond.As explained in Section 31.4, the first stage is to calculate the value of r at time 2.1 years which is such that the value of the bond at that time is 99. Denoting this value of r by r *, we must solve(2125)(2130)25(2125)1025(2130)99B r B r A e A e **-.,.-.,...,.+..,.=where the A and B functions are given by equations (31.7) and (31.8). In this case A (2.1, 2.5)=0.999685, A (2.1,3.0)=0.998432, B(2.1,2.5)=0.396027, and B (2.1, 3.0)= 0.88005. and Solver shows that 065989.0*=r . Since434745.2)5.2,1.2(5.2*)5.2,1.2(=⨯-r B e Aand56535.96)0.3,1.2(5.102*)0.3,1.2(=⨯-r B e Athe call option on the coupon-bearing bond can be decomposed into a call option with a strike price of 2.434745 on a bond that pays off 2.5 at time 2.5 years and a call option with a strike price of 96.56535 on a bond that pays off 102.5 at time 3.0 years. The options are valued using equation (31.20).For the first option L =2.5, K = 2.434745, T = 2.1, and s =2.5. Also, A (0,T )=0.991836, B (0,T ) = 1.99351, P (0,T )=0.880022 while A (0,s )=0.988604, B (0,s )=2.350062, andP (0,s )=0.858589. Furthermore σP = 0.008176 and h = 0.223351. so that the option price is 0.009084.For the second option L =102.5, K = 96.56535, T = 2.1, and s =3.0. Also, A (0,T )=0.991836, B (0,T ) = 1.99351, P (0,T )=0.880022 while A (0,s )=0.983904, B (0,s )=2.78584, andP (0,s )=0.832454. Furthermore σP = 0.018168 and h = 0.233343. so that the option price is0.806105.The total value of the option is therefore 0.0090084+0.806105=0.815189.Problem 31.10.Use the answer to Problem 31.9 and put –call parity arguments to calculate the price of a put option that has the same terms as the call option in Problem 31.9.Put-call parity shows that: 0()c I PV K p B ++=+ or 0()()p c PV K B I =+--where c is the call price, K is the strike price, I is the present value of the coupons, and 0B is the bond price. In this case 08152c =., ()99(021)871222PV K P =⨯,.=., 025(025)1025(03)874730B I P P -=.⨯,.+.⨯,=. so that the put price is0815287122287473004644.+.-.=.Problem 31.11.In the Hull –White model, 008a =. and 001σ=.. Calculate the price of a one-year European call option on a zero-coupon bond that will mature in five years when the term structure is flat at 10%, the principal of the bond is $100, and the strike price is $68.Using the notation in the text 011(0)09048P T e -.⨯,==. and 015(0)06065P s e -.⨯,==.. Also4008001(100329008P e σ-⨯..=-=.. and 04192h =-. so that the call price is10006065()6809048()0439P N h N h σ⨯.-⨯.-=.Problem 31.12.Suppose that 005a =. and 0015σ=. in the Hull –White model with the initial term structure being flat at 6% with semiannual compounding. Calculate the price of a 2.1-year European call option on a bond that will mature in three years. Suppose that the bond pays a coupon of 5% per annum semiannually. The principal of the bond is 100 and the strike price of the option is 99. The strike price is the cash price (not the quoted price) that will be paid for the bond.This problem is similar to Problem 31.9. The difference is that the Hull –White model, which fits an initial term structure, is used instead of Vasicek’s model where the initial term structure is determined by the model.The yield curve is flat with a continuously compounded rate of 5.9118%.As explained in Section 31.4, the first stage is to calculate the value of r at time 2.1 years which is such that the value of the bond at that time is 99. Denoting this value of r by r *, we must solve(2125)(2130)25(2125)1025(2130)99B r B r A e A e **-.,.-.,...,.+..,.=where the A and B functions are given by equations (31.16) and (31.17). In this case A (2.1, 2.5)=0.999732, A (2.1,3.0)=0.998656, B(2.1,2.5)=0.396027, and B (2.1, 3.0)= 0.88005. and Solver shows that 066244.0*=r . Since434614.2)5.2,1.2(5.2*)5.2,1.2(=⨯-r B e Aand56539.96)0.3,1.2(5.102*)0.3,1.2(=⨯-r B e Athe call option on the coupon-bearing bond can be decomposed into a call option with a strike price of 2.434614 on a bond that pays off 2.5 at time 2.5 years and a call option with a strike price of 96.56539 on a bond that pays off 102.5 at time 3.0 years. The options are valued using equation (31.20).For the first option L =2.5, K = 2.434614, T = 2.1, and s =2.5. Also, P (0,T )=exp(-0.059118×2.1)=0.88325 and P (0,s )= exp(-0.059118×2.5)=0.862609. Furthermore σP = 0.008176 and h = 0.353374. so that the option price is 0.010523. For the second option L =102.5, K = 96.56539, T = 2.1, and s =3.0. Also, P (0,T )=exp(-0.059118×2.1)=0.88325 and P (0,s )= exp(-0.059118×3.0)=0.837484. Furthermore σP = 0.018168 and h = 0.363366. so that the option price is 0.934074.The total value of the option is therefore 0.010523+0.934074=0.944596.Problem 31.13.Observations spaced at intervals ∆t are taken on the short rate. The ith observation is r i (1 ≤ i ≤ m). Show that the maximum likelihood estimates of a, b, and σ in Vasicek’s model are given by maximizing[]∑=--⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-m i i i i t t r b a r r t 122112)()ln(What is the corresponding result for the CIR model?The change r i –r i -1 is normally distributed with mean a (b − r i -1) and variance σ2∆t. The probability density of the observation is⎪⎭⎫⎝⎛∆σ---∆πσ--t r b a r r ti i i 21122)(exp 21We wish to maximize∏=--⎪⎭⎫⎝⎛∆σ---∆πσmi i i i t r b a r r t121122)(exp 21Taking logarithms, this is the same as maximizing[]∑=--⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-m i i i i t t r b a r r t 122112)()ln(In the case of the CIR model, the change r i –r i -1 is normally distributed with mean a (b − r i -1) and variance t r i ∆σ-12and the maximum likelihood function becomes[]∑=----⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-mi i i i i i t r t r b a r r t r 11221112)()ln(Problem 31.14.Suppose 005a =., 0015σ=., and the term structure is flat at 10%. Construct a trinomial tree for the Hull –White model where there are two time steps, each one year in length.The time step, t ∆, is 1 so that 0002598r ∆=.=.. Also max 4j = showing that the branching method should change four steps from the center of the tree. With only three steps we never reach the point where the branching changes. The tree is shown in Figure S31.1.Figure S31.1: Tree for Problem 31.14Problem 31.15.Calculate the price of a two-year zero-coupon bond from the tree in Figure 31.6.A two-year zero-coupon bond pays off $100 at the ends of the final branches. At nodeB it is worth 01211008869e -.⨯=.. At nodeC it is worth 010********e -.⨯=.. At nodeD it is worth 00811009231e -.⨯=.. It follows that at node A the bond is worth 011(88690259048059231025)8188e -.⨯.⨯.+.⨯.+.⨯.=. or $81.88Problem 31.16.Calculate the price of a two-year zero-coupon bond from the tree in Figure 31.9 and verify that it agrees with the initial term structure.A two-year zero-coupon bond pays off $100 at time two years. At nodeB it is worth 0069311009330e -.⨯=.. At nodeC it is worth 0052011009493e -.⨯=.. At nodeD it is worth 0034711009659e -.⨯=.. It follows that at node A the bond is worth 003821(933001679493066696590167)9137e -.⨯.⨯.+.⨯.+.⨯.=.or $91.37. Because 00451229137100e -.⨯.=, the price of the two-year bond agrees with the initial term structure.Problem 31.17.Calculate the price of an 18-month zero-coupon bond from the tree in Figure 31.10 and verify that it agrees with the initial term structure.An 18-month zero-coupon bond pays off $100 at the final nodes of the tree. At node E it is worth 0088051009570e -.⨯.=.. At node F it is worth 00648051009681e -.⨯.=.. At node G it is worth 00477051009764e -.⨯.=.. At node H it is worth 00351051009826e -.⨯.=.. At node I it is worth 00259051009871e .⨯.=.. At node B it is worth 0056405(011895700654968102289764)9417e -.⨯..⨯.+.⨯.+.⨯.=.Similarly at nodes C and D it is worth 95.60 and 96.68. The value at node A is therefore 0034305(016794170666956001679668)9392e -.⨯..⨯.+.⨯.+.⨯.=.The 18-month zero rate is 0181500800500418e -.⨯..-.=.. This gives the price of the 18-month zero-coupon bond as 00418151009392e -.⨯.=. showing that the tree agrees with the initial term structure.Problem 31.18.What does the calibration of a one-factor term structure model involve?The calibration of a one-factor interest rate model involves determining its volatility parameters so that it matches the market prices of actively traded interest rate options as closely as possible.Problem 31.19.Use the DerivaGem software to value 14⨯, 23⨯, 32⨯, and 41⨯ European swap options to receive fixed and pay floating. Assume that the one, two, three, four, and five year interest rates are 6%, 5.5%, 6%, 6.5%, and 7%, respectively. The payment frequency on the swap is semiannual and the fixed rate is 6% per annum with semiannual compounding. Use the Hull –White model with 3a %= and 1%σ=. Calcula te the volatility implied by Black’s model for each option.The option prices are 0.1302, 0.0814, 0.0580, and 0.0274. The implied Black volatilities are 14.28%, 13.64%, 13.24%, and 12.81%Problem 31.20.Prove equations (31.25), (31.26), and (31.27).From equation (31.15) ()()()()r t B t t t P t t t A t t t e -,+∆,+∆=,+∆ Also ()()R t t P t t t e -∆,+∆= so that()()()()R t t r t B t t t e A t t t e -∆-,+∆=,+∆or()()()()()()()()R t B t T t B t t t r t B t T B t T B t t t e eA t t t -,∆/,+∆-,,/,+∆=,+∆ Hence equation (31.25) is true with()ˆ()(B t T t Bt T B t t t ,∆,=,+∆ and()()()ˆ()()B t T B t t t A t T At T A t t t ,/,+∆,,=,+∆ or()ˆln ()ln ()ln ()()B t T At T A t T A t t t B t t t ,,=,-,+∆,+∆Problem 31.21.(a) What is the second partial derivative of P(t,T) with respect to r in the Vasicek and CIR models?(b) In Section 31.2, ˆDis presented as an alternative to the standard duration measure D. What is a similar alternative ˆCto the convexity measure in Section 4.9? (c) What is ˆCfor P(t,T)? How would you calculate ˆC for a coupon-bearing bond? (d) Give a Taylor Series expansion for ∆P(t,T) in terms of ∆r and (∆r)2 for Vasicek and CIR.(a) ),(),(),(),(),(2),(222T t P T t B e T t A T t B rT t P r T t B ==∂∂- (b) A corresponding definition for Cˆis 221r QQ ∂∂(c) When Q =P (t ,T ), 2),(ˆT t B C=For a coupon-bearing bond C ˆis a weighted average of the Cˆ’s for the constituent zero -coupon bonds where weights are proportional to bond prices. (d)+∆+∆-=+∆∂∂+∆∂∂=∆22222),(),(21),(),(),(21),(),(r T t P T t B r T t P T t B r rT t P r r T t P T t PProblem 31.22.Suppose that the short rate r is 4% and its real-world process is0.1[0.05]0.01dr r dt dz =-+while the risk-neutral process is0.1[0.11]0.01dr r dt dz =-+(a) What is the market price of interest rate risk?(b) What is the expected return and volatility for a 5-year zero-coupon bond in the risk-neutral world?(c) What is the expected return and volatility for a 5-year zero-coupon bond in the real world?(a) The risk neutral process for r has a drift rate which is 0.006/r higher than the real world process. The volatility is 0.01/r . This means that the market price of interest rate risk is −0.006/0.01 or −0.6.(b) The expected return on the bond in the risk-neutral world is the risk free rate of 4%. The volatility is 0.01×B (0,5) where935.31.01)5,0(51.0=-=⨯-e Bi.e., the volatility is 3.935%.(c) The process followed by the bond price in a risk-neutral world isPdz Pdt dP 03935.004.0-=Note that the coefficient of dz is negative because bond prices are negatively correlated with interest rates. When we move to the real world the return increases by the product of the market price of dz risk and −0.03935. The bond price process becomes:Pdz Pdt dP 03935.0)]03935.06.0(04.0[--⨯-+=orPdz Pdt dP 03935.006361.0-=The expected return on the bond increases from 4% to 6.361% as we move from the risk-neutral world to the real world.Further QuestionsProblem 31.23.Construct a trinomial tree for the Ho and Lee model where 002σ=.. Suppose that the initial zero-coupon interest rate for a maturities of 0.5, 1.0, and 1.5 years are 7.5%, 8%, and 8.5%. Use two time steps, each six months long. Calculate the value of a zero-coupon bond with a face value of $100 and a remaining life of six months at the ends of the final nodes of the tree. Use the tree to value a one-year European put option with a strike price of 95 on the bond. Compare the price given by your tree with the analytic price given by DerivaGem.The tree is shown in Figure S31.2. The probability on each upper branch is 1/6; the probability on each middle branch is 2/3; the probability on each lower branch is 1/6. The six month bond prices nodes E, F, G, H, I are 0144205100e -.⨯., 0119705100e -.⨯., 0095205100e -.⨯., 0070705100e -.⨯., and 0046205100e -.⨯., respectively. These are 93.04, 94.19, 95.35, 96.53, and 97.72. The payoffs from the option at nodes E, F, G, H, and I are therefore 1.96, 0.81, 0, 0, and 0. The value at node B is 0109505(0166719606667081)08192e -.⨯..⨯.+.⨯.=.. The value at node C is00851050166708101292e -.⨯..⨯.⨯=.. The value at node D is zero. The value at node A is 0075005(01667081920666701292)0215e -.⨯..⨯.+.⨯.=. The answer given by DerivaGem using the analytic approach is 0.209.Figure S31.2: Tree for Problem 31.23Problem 31.24.A trader wishes to compute the price of a one-year American call option on a five-year bond with a face value of 100. The bond pays a coupon of 6% semiannually and the (quoted) strike price of the option is $100. The continuously compounded zero rates for maturities of sixmonths, one year, two years, three years, four years, and five years are 4.5%, 5%, 5.5%, 5.8%, 6.1%, and 6.3%. The best fit reversion rate for either the normal or the lognormal model has been estimated as 5%.A one year European call option with a (quoted) strike price of 100 on the bond is actively traded. Its market price is $0.50. The trader decides to use this option for calibration. Use the DerivaGem software with ten time steps to answer the following questions.(a) Assuming a normal model, imply the σ parameter from the price of the European option.(b) Use the σ parameter to calculate the price of the option when it is American. (c) Repeat (a) and (b) for the lognormal model. Show that the model used does notsignificantly affect the price obtained providing it is calibrated to the known European price.(d) Display the tree for the normal model and calculate the probability of a negative interest rate occurring.(e) Display the tree for the lognormal model and verify that the option price is correctly calculated at the node where, with the notation of Section 31.7, 9i = and 1j =-.Using 10 time steps:(a) The implied value of σ is 1.12%.(b) The value of the American option is 0.595(c) The implied value of σ is 18.45% and the value of the American option is 0.595. Thetwo models give the same answer providing they are both calibrated to the same European price.(d) We get a negative interest rate if there are 10 down moves. The probability of this is0.16667×0.16418×0.16172×0.15928×0.15687×0.15448×0.15212×0.14978×0.14747 ×0.14518=8.3×10-9 (e) The calculation is0052880101641791707502789e -.⨯..⨯.⨯=.Problem 31.25.Use the DerivaGem software to value 14⨯, 23⨯, 32⨯, and 41⨯ European swap options to receive floating and pay fixed. Assume that the one, two, three, four, and five year interest rates are 3%, 3.5%, 3.8%, 4.0%, and 4.1%, respectively. The payment frequency on the swap is semiannual and the fixed rate is 4% per annum with semiannual compounding. Use thelognormal model with 5a %=, 15%σ=, and 50 time steps. Calculate the volatility implied by Black’s model for each option.The values of the four European swap options are 1.72, 1.73, 1.30, and 0.65, respectively. The implied Black volatilities are 13.37%, 13.41%, 13.43%, and 13.42%, respectively.Problem 31.26.Verify that the DerivaGem software gives Figure 31.11 for the example considered. Use the software to calculate the price of the American bond option for the lognormal and normalmodels when the strike price is 95, 100, and 105. In the case of the normal model, assume that a = 5% and σ = 1%. Discuss the results in the context of the heavy-tails arguments of Chapter 20.With 100 time steps the lognormal model gives prices of 5.585, 2.443, and 0.703 for strike prices of 95, 100, and 105. With 100 time steps the normal model gives prices of 5.508, 2.522, and 0.895 for the three strike prices respectively. The normal model gives a heavier left tail and a less heavy right tail than the lognormal model for interest rates. This translates into a less heavy left tail and a heavier right tail for bond prices. The arguments in Chapter 20 show that we expect the normal model to give higher option prices for high strike prices and lower option prices for low strike. This is indeed what we find.Problem 31.27.Modify Sample Application G in the DerivaGem Application Builder software to test theconvergence of the price of the trinomial tree when it is used to price a two-year call option on a five-year bond with a face value of 100. Suppose that the strike price (quoted) is 100, the coupon rate is 7% with coupons being paid twice a year. Assume that the zero curve is as in Table 31.2. Compare results for the following cases:(a) Option is European; normal model with 001σ=. and 005a =..(b) Option is European; lognormal model with 015σ=. and 005a =..(c) Option is American; normal model with 001σ=. and 005a =..(d) Option is American; lognormal model with 015σ=. and 005a =..The results are shown in Figure S31.3.Figure S31.3: Tree for Problem 31.27Problem 31.28.Suppose that the (CIR) process for short-rate movements in the (traditional) risk-neutral world is()dr a b r dt =-+and the market price of interest rate risk is λ(a) What is the real world process for r?(b) What is the expected return and volatility for a 10-year bond in the risk-neutral world? (c) What is the expected return and volatility for a 10-year bond in the real world?(a) The volatility of r (i.e., the coefficient of rdz in the process for r ) is real world process for r is therefore increased by r r λσ⨯ so that the process isdz r dt r r b a dr σ+λσ+-=])([(b) The expected return is r and the volatility is (,B t T σin the risk-neutral world.(c) The expected return is r T t B r ),(λσ+ and the volatility is as in (b) in the real world.。

期权基础知识

期权基础知识





5 履约日 是指期权权利可以执行的日期。可以分为欧式期权与 美式期权。 6 到期日 是期权合约双方当事人预先设立的期权买方可以行使 期权的最终有效期限。 7 停板额 又称每日交易限价,即涨跌停板额,是期权合约每日 价格的波动幅度高于或低于上一交易日结算价的幅度。


8 合约月份 即买卖双方交付、接受实货以履行职权合约的 月份。 9 交易时间 即各相关期权市场规定的每个交易日及其进场 交易的具体时间。
一份美国芝加哥期货交易所的玉米期 权合约格式。

交易单位 最小变动价位 每日价格最大波 动限制 执行价格 合约月份 交易时间 最后交易日 合约到期日
一个CBOT期货合约交易单位(5000蒲式 耳)
每蒲式耳1/8美分 每蒲式耳不高于或低于上一交 易日结算权利金各10美分 每蒲式耳10美分的整数倍 1、2、3、5、7、9 同期货
一、期权市场概述

(一) 金融期权合约的定义与种类

(二) 金融期权的交易
(三) 期权交易与期货交易的区别

(一) 金融期权合约的定义与种类

金融期权 (Option),是指它的持有者有权在规 定期限内按双方约定的价格购买或出售一定数 量某种金融资产(称为标的金融资产Underlying Financial Assets) 的合约。
(六) 障碍期权

障碍期权 (Barrier Option) 是指其收益依赖于标 的资产价格在一段特定时期内是否达到了一个 特定水平的期权。常见的障碍期权有两种,一 是封顶期权 (Caps),一是失效期权 (Knockout Option)。
(七) 两值期权

两值期权 (Binary Option) 是具有不连续收益的期 权,当到期日标的资产价格低于协议价格时, 该期权作废,而当到期日标的资产价格高于协 议价格时,期权持有者将得到一个固定的金额。

看涨看跌平价定理

看涨看跌平价定理

看涨看跌平价定理(一)欧式看涨期权与看跌期权之间的平价关系1.无收益资产的欧式期权在标的资产没有收益的情况下,为了推导c和p之间的关系,我们考虑如下两个组合:组合A:一份欧式看涨期权加上金额为Xe-r(T-t) 的现金组合B:一份有效期和协议价格与看涨期权相同的欧式看跌期权加上一单位标的资产在期权到期时,两个组合的价值均为max(ST,X)。

由于欧式期权不能提前执行,因此两组合在时刻t必须具有相等的价值,即:c+Xe-r(T-t)=p+S(1.1)这就是无收益资产欧式看涨期权与看跌期权之间的平价关系(Parity)。

它表明欧式看涨期权的价值可根据相同协议价格和到期日的欧式看跌期权的价值推导出来,反之亦然。

如果式(1.1)不成立,则存在无风险套利机会。

套利活动将最终促使式(1.1)成立。

2.有收益资产欧式期权在标的资产有收益的情况下,我们只要把前面的组合A中的现金改为D+Xe-r(T-t) ,我们就可推导有收益资产欧式看涨期权和看跌期权的平价关系:c+D+Xe-r(T-t)=p+S(1.2)(二)美式看涨期权和看跌期权之间的关系1.无收益资产美式期权由于P>p,从式(1.1)中我们可得:P>c+Xe-r(T-t)-S对于无收益资产看涨期权来说,由于c=C,因此:P>C+Xe-r(T-t)-SC-P<S-Xe-r(T-t)(1.3)为了推导出C和P的更严密的关系,我们考虑以下两个组合:组合A:一份欧式看涨期权加上金额为X的现金组合B:一份美式看跌期权加上一单位标的资产如果美式期权没有提前执行,则在T时刻组合B的价值为max(ST,X),而此时组合A 的价值为max(ST,X)+ Xe-r(T-t)-X。

因此组合A的价值大于组合B。

如果美式期权在T-t 时刻提前执行,则在T-t 时刻,组合B的价值为X,而此时组合A的价值大于等于Xe-r(T-t)。

因此组合A的价值也大于组合B。

北美精算师 fm教材 《衍生品市场》课后答案m48-ch03 shrunk v02

北美精算师 fm教材 《衍生品市场》课后答案m48-ch03 shrunk v02

Chapter3Insurance,Collars,and Other StrategiesQuestion3.1.This question is a direct application of the Put-Call-Parity(equation(3.1))of the textbook.Mim-icking Table3.1.,we have:S&R Index S&R Put Loan Payoff−(Cost+Interest)Profit900.00100.00−1000.000.00−95.68−95.68950.0050.00−1000.000.00−95.68−95.681000.000.00−1000.000.00−95.68−95.681050.000.00−1000.0050.00−95.68−45.681100.000.00−1000.00100.00−95.68 4.321150.000.00−1000.00150.00−95.6854.321200.000.00−1000.00200.00−95.68104.32The payoff diagram looks as follows:We can see from the table and from the payoff diagram that we have in fact reproduced a call with the instruments given in the exercise.The profit diagram on the next page confirms this hypothesis.Part1Insurance,Hedging,and Simple StrategiesQuestion3.2.This question constructs a position that is the opposite to the position of Table3.1.Therefore,we should get the exact opposite results from Table3.1.and the associatedfigures.Mimicking Table 3.1.,we indeed have:S&R Index S&R Put Payoff−(Cost+Interest)Profit−900.00−100.00−1000.001095.6895.68−950.00−50.00−1000.001095.6895.68−1000.000.00−1000.001095.6895.68−1050.000.00−1050.001095.6845.68−1100.000.00−1100.001095.68−4.32−1150.000.00−1150.001095.68−54.32−1200.000.00−1200.001095.68−104.32On the next page,we see the corresponding payoff and profit diagrams.Please note that they match the combined payoff and profit diagrams of Figure3.5.Only the axes have different scales.Chapter3Insurance,Collars,and Other Strategies Payoff-diagram:Profit diagram:Part1Insurance,Hedging,and Simple StrategiesQuestion3.3.In order to be able to draw profit diagrams,we need tofind the future value of the put premium,the call premium and the investment in zero-coupon bonds.We have for:the put premium:$51.777×(1+0.02)=$52.81,the call premium:$120.405×(1+0.02)=$122.81andthe zero-coupon bond:$931.37×(1+0.02)=$950.00Now,we can construct the payoff and profit diagrams of the aggregate position:Payoff diagram:From thisfigure,we can already see that the combination of a long put and the long index looks exactly like a certain payoff of$950,plus a call with a strike price of950.But this is the alternative given to us in the question.We have thus confirmed the equivalence of the two combined positions for the payoff diagrams.The profit diagrams on the next page confirm the equivalence of the two positions(which is again an application of the Put-Call-Parity).Chapter 3Insurance,Collars,and Other StrategiesProfit Diagram for a long 950-strike put and a long index combined:Question 3.4.This question is another application of Put-Call-Parity.Initially,we have the following cost to enter into the combined position:We receive $1,000from the short sale of the index,and we have to pay the call premium.Therefore,the future value of our cost is: $120.405−$1,000 ×(1+0.02)=−$897.19.Note that a negative cost means that we initially have an inflow of money.Now,we can directly proceed to draw the payoff diagram:Part 1Insurance,Hedging,and Simple StrategiesWe can clearly see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $950,which is alleviated by a long put position with a strike price of 950.The following profit diagram,including the cost for the combined position,confirms this:Question 3.5.This question is another application of Put-Call-Parity.Initially,we have the following cost to enter into the combined position:We receive $1,000from the short sale of the index,and we have to pay the call premium.Therefore,the future value of our cost is: $71.802−$1,000 ×(1+0.02)=−$946.76.Note that a negative cost means that we initially have an inflow of money.Chapter3Insurance,Collars,and Other Strategies Now,we can directly proceed to draw the payoff diagram:In order to be able to compare this position to the other suggested position of the exercise,we need tofind the future value of the borrowed$1,029.41.We have:$1,029.41×(1+0.02)=$1,050. We can now see from thefigure that the payoff graph of the short index and the long call looks like afixed obligation of$1,050,which is exactly the future value of the borrowed amount,and a long put position with a strike price of1,050.The following profit diagram,including the cost for the combined position we calculated above,confirms this.The profit diagram is the same as the profit diagram for a loan and a long1,050-strike put with an initial premium of$101.214.Part 1Insurance,Hedging,and Simple StrategiesProfit Diagram of going short the index and buying a 1,050-strike call:Question 3.6.We now move from a graphical representation and verification of the Put-Call-Parity to a mathe-matical representation.Let us first consider the payoff of (a).If we buy the index (let us name it S),we receive at the time of expiration T of the options simply S T .The payoffs of part (b)are a little bit more complicated.If we deal with options and the maximum function,it is convenient to split the future values of the index into two regions:one where S T <K and another one where S T ≥K .We then look at each region separately,and hope to be able to draw a conclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentS T <K =950S T ≥K =950Get repayment of loan $931.37×1.02=$950$931.37×1.02=$950Long Call Option max (S T−950,0)=0S T −950Short Put Option −max $950−S T ,0 0=S T −$950TotalS TS TChapter 3Insurance,Collars,and Other StrategiesWe now see that the total aggregate position only gives us S T ,no matter what the final index value is—but this is the same payoff as in part (a).Our proof for the payoff equivalence is complete.Now let us turn to the profits.If we buy the index today,we need to finance it.Therefore,we borrow $1,000,and have to repay $1,020after one year.The profit for part (a)is thus:S T −$1,020.The profits of the aggregate position in part (b)are the payoffs,less the future value of the call premium plus the future value of the put premium (because we have sold the put),and less the future value of the loan we gave initially.Note that we already know that a risk-less bond is canceling out of the profit calculations.We can again tabulate:InstrumentS T <KS T ≥KFuture value of given loan −$950−$950Long Call Optionmax (S T −950,0)=0S T −950Future value call premium −$120.405×1.02=−$122.81−$120.405×1.02=−$122.81Short Put Option −max$950−S T ,0 0=S T −$950Future value put premium $51.777×1.02=$52.81$51.777×1.02=$52.81TotalS T −1020S T −1020Indeed,we see that the profits for part (a)and part (b)are identical as well.Question 3.7.Let us first consider the payoff of (a).If we short the index (let us name it S),we have to pay at the time of expiration T of the options:−S T .The payoffs of part (b)are more complicated.Let us look again at each region separately,and hope to be able to draw a conclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentS T <K S T ≥KMake repayment of loan −$1029.41×1.02=−$1050−$1029.41×1.02=−$1050Short Call Option −max (S T −1050,0)=0−max (S T −1050,0)=1050−S TLong Put Option max$1050−S T ,00=$1050−S TTotal−S T−S TPart 1Insurance,Hedging,and Simple StrategiesWe see that the total aggregate position gives us −S T ,no matter what the final index value is—but this is the same payoff as part (a).Our proof for the payoff equivalence is complete.Now let us turn to the profits.If we sell the index today,we receive money that we can lend out.Therefore,we can lend $1,000,and be repaid $1,020after one year.The profit for part (a)is thus:$1,020−S T .The profits of the aggregate position in part (b)are the payoffs,less the future value of the put premium plus the future value of the call premium (because we sold the call),and less the future value of the loan we gave initially.Note that we know already that a risk-less bond is canceling out of the profit calculations.We can again tabulate:InstrumentS T <KS T ≥KMake repayment of loan −$1029.41×1.02=−$1050−$1029.41×1.02=−$1050Future value of borrowed $1050$1050moneyShort Call Option −max (S T −1050,0)=0−max (S T −1050,0)=1050−S TFuture value of premium $71.802×1.02=$73.24$71.802×1.02=$73.24Long Put Option max $1050−S T ,0 =0$1050−S TFuture value of premium −$101.214×1.02=−$103.24−$101.214×1.02=−$103.24Total$1,020−S T $1,020−S TIndeed,we see that the profits for part (a)and part (b)are identical as well.Question 3.8.This question is a direct application of the Put-Call-Parity.We will use equation (3.1)in the follow-ing,and input the given variables:Call (K,t)−P ut (K,t)=P V F 0,t −K⇔Call (K,t)−P ut (K,t)−P V F 0,t=−P V (K)⇔Call (K,t)−P ut (K,t)−S 0=−P V (K)⇔$109.20−$60.18−$1,000=−K 1.02⇔K =$970.00Question 3.9.The strategy of buying a call (or put)and selling a call (or put)at a higher strike is called call (put)bull spread.In order to draw the profit diagrams,we need to find the future value of the cost of entering in the bull spread positions.We have:Cost of call bull spread:$120.405−$93.809×1.02=$27.13Cost of put bull spread:$51.777−$74.201×1.02=−$22.87The payoff diagram shows that the payoffs to the put bull spread are uniformly less than the payoffs to the call bull spread.There is a difference,because the put bull spread has a negative initial cost, i.e.,we are receiving money if we enter into it.The difference is exactly$50,the value of the difference between the two strike prices.It is equivalent as well to the value of the difference of the future values of the initial premia.Yet,the higher initial cost for the call bull spread is exactly offset by the higher payoff so that the profits of both strategies are the same.It is easy to show this with equation(3.1),the put-call-parity. Payoff-Diagram:Profit diagram:Question 3.10.The strategy of selling a call (or put)and buying a call (or put)at a higher strike is called call (put)bear spread.In order to draw the profit diagrams,we need to find the future value of the cost of entering in the bull spread positions.We have:Cost of call bear spread: $71.802−$120.405 ×1.02=−$49.575Cost of put bear spread: $101.214−$51.777 ×1.02=$50.426The payoff diagram shows that the payoff to the call bear spread is uniformly less than the payoffs to the put bear spread.The difference is exactly $100,equal to the difference in strikes and as well equal to the difference in the future value of the costs of the spreads.There is a difference,because the call bear spread has a negative initial cost,i.e.,we are receiving money if we enter into it.The higher initial cost for the put bear spread is exactly offset by the higher payoff so that the profits of both strategies turn out to be the same.It is easy to show this with equation (3.1),the put-call-parity.Payoff-Diagram:Profit Diagram:Question 3.11.In order to be able to draw the profit diagram,we need to find the future value of the costs of establishing the suggested position.We need to finance the index purchase,buy the 950-strike put and we receive the premium of the sold call.Therefore,the future value of our cost is: $1,000−$71.802+$51.777 ×1.02=$999.57.Now we can draw the profit diagram:The net option premium cost today is:−$71.802+$51.777=−$20.025.We receive about $20if we enter into this collar.If we want to construct a zero-cost collar and keep the 950-strike put,we would need to increase the strike price of the call.By increasing the strike price of the call,the buyer of the call must wait for larger increases in the underlying index before the option pays off.This makes the call option less attractive,and the buyer of the option is only willing to pay a smaller premium.We receive less money,thus pushing the net option premium towards zero.Question 3.12.Our initial cash required to put on the collar,i.e.the net option premium,is as follows:−$51.873+$51.777=−$0.096.Therefore,we receive only 10cents if we enter into this collar.The position is very close to a zero-cost collar.The profit diagram looks as follows:If we compare this profit diagram with the profit diagram of the previous exercise(3.11.),we see that we traded in the additional call premium(that limited our losses after index decreases)against more participation on the upside.Please note that both maximum loss and gain are higher than in question3.11.Question3.13.The followingfigure depicts the requested profit diagrams.We can see that the aggregation of the bought and sold straddle resembles a bear spread.It is bearish,because we sold the straddle with the smaller strike price.a)b)c)Question3.14.a)This question deals with the option trading strategy known as Box spread.We saw earlier that if we deal with options and the maximum function,it is convenient to split the future values of the index into different regions.Let us name thefinal value of the S&R index S T.We have two strike prices,therefore we will use three regions:One in which S T<950,one in which950≤S T<1,000 and another one in which S T≥1,000.We then look at each region separately,and hope to be ableto see that indeed when we aggregate,there is no S&R risk when we look at the aggregate position.Instrument S T<950950≤S T<1,000S T≥1,000long950call0S T−$950S T−$950short1000call00$1,000−S Tshort950put S T−$95000long1000put$1,000−S T$1,000−S T0Total$50$50$50We see that there is no occurrence of thefinal index value in the row labeled total.The option position does not contain S&R price risk.b)The initial cost is the sum of the long option premia less the premia we receive for the sold options.We have:Cost $120.405−$93.809−$51.77+$74.201=$49.027c)The payoff of the position after 6months is $50,as we can see from the above table.d)The implicit interest rate of the cash flows is:$50.00÷$49.027=1.019∼=1.02.The im-plicit interest rate is indeed 2percent.Question 3.15.a)Profit diagram of the 1:2950-,1050-strike ratio call spread (the future value of the initial cost of which is calculated as: $120.405−2×$71.802 ×1.02=−$23.66):b)Profit diagram of the 2:3950-,1050-strike ratio call spread (the future value of the initial cost of which is calculated as: 2×$120.405−3×$71.802 ×1.02=$25.91.c)We saw that in part a),we were receiving money from our position,and in part b),we had to pay a net option premium to establish the position.This suggests that the true ratio n/m lies between 1:2and2:3.Indeed,we can calculate the ratio n/m as:n×$120.405−m×$71.802=0⇔n×$120.405=m×$71.802⇔n/m=$71.802/$120.405⇔n/m=0.596which is approximately3:5.Question3.16.A bull spread or a bear spread can never have an initial premium of zero,because we are buying the same number of calls(or puts)that we are selling and the two legs of the bull and bear spreads have different strikes.A zero initial premium would mean that two calls(or puts)with different strikes have the same price—and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity.A symmetric butterfly spread cannot have a premium of zero because it would violate the convexitycondition of options.Question3.17.From the textbook we learn how to calculate the right ratioλ.It is equal to:λ=K3−K2K3−K1=1050−10201050−950=0.3In order to construct the asymmetric butterfly,for every1020-strike call we write,we buyλ950-strike calls and1−λ1050-strike calls.Since we can only buy whole units of calls,we will in this example buy three950-strike and seven1050-strike calls,and sell ten1020-strike calls.The profit diagram looks as follows:Question3.18.The following threefigures show the individual legs of each of the three suggested strategies.The last subplot shows the aggregate position.It is evident from thefigures that you can indeed use all the suggested strategies to construct the same butterfly spread.Another method to show the claim of3.18.mathematically would be to establish the equivalence by using the Put-Call-Parity on b) and c)and showing that you can write it in terms of the instruments of a).profit diagram part a)profit diagram part b)Chapter3Insurance,Collars,and Other Strategies profit diagram part c)Question3.19.a)We know from the Put-Call-Parity that if we buy a call and sell a put that are at the money (i.e.,S(0)=K),then the call option is slightly more expensive than the put option,the difference being the value of the stock minus the present value of the strike.Therefore,we can tell that the strike price must be a bit higher than the current stock price,and more precisely,it should be equal to the forward price.b)We sold a collar with no difference in strike prices.The profit diagram will be a straight line,which means that we effectively created a long forward contract.c)Remember that you are buying at the ask and selling at the bid,and that the bid price is always smaller than the ask.Suppose we had established a zero-cost synthetic at the forward price, and now we introduce the bid-ask spread.This means that we have to pay a little more for the call, and receive a little less for the put.We are paying money for the position,and in order to correct it,we must make the put a bit more attractive,and the call less attractive.We do so by shifting the strike price to the right of the forward:Now the buyer of the call must wait a little bit longer before his call pays off,and he is only willing to buy it for less.As the opposite is true for the put,we have established that the strike must be to the right of the forward.Part1Insurance,Hedging,and Simple Strategiesd)If we are creating a synthetic short stock,we buy the put option and sell the call option. We are buying at the ask and selling at the bid,and the bid price is always smaller than the ask. Suppose we had established a zero-cost synthetic short at the forward price,and now we introduce the bid-ask spread.This means that we have to pay a little more for the put,and receive a little less for the call.We are paying money for the position,and in order to correct it,we must make the call a bit more attractive.The call gets more attractive if the strike price decreases,because thefinal payoff is max(S−K,0).Therefore,we have to shift the strike price to the left of the forward price.e)No,transaction fees are not a wash,because we are paying implicitly the bid-ask spread:If we bought a stock today and held it until the expiration of the options,we would get the future stock price less the forward price(which is equivalent to the loan we got tofinance the stock purchase). Now,we established in c)that the strike price is to the right of the forward price.Therefore,we will receive from the collar of part c)the stock price less something that is larger than the forward price: We make a loss compared to the self-financed outright purchase of the stock.These considerations do not yet take into account that we incur transaction costs on two instruments,compared to only one time brokerage fees if we buy the stock directly.It is thus very important to be aware of transaction costs when comparing different investment strategies.Question3.20.Use separate cells for the strike price and the quantities you buy and sell for each strike(i.e.,make use of the plus or minus sign).Then,use the maximum function to calculate payoffs and profits. The best way to solve this problem is probably to have the calculations necessary for the payoff and profit diagrams run in the background,e.g.,in another auxiliary table that you are referencing to.Define the boundaries for the calculations dynamically and symmetrically around the current stock price.Then use the diagram function with the line style to draw the diagrams.。

2011-2012FRM备考详尽学习笔记

2011-2012FRM备考详尽学习笔记

2007年note Topic 35波动率微笑1. 理解为什么在计算看涨期权和看跌期权价格时,所使用的隐含波动率(implied volatility)是相同的:这是根据put-call parity得出的,并不受关于股票价格分布的限制。

2. 实证研究表明:外汇期权(foreign currency option)隐含波动率呈现出微笑的形态(当strike较大或较小的时候,波动率会变大)——因为参与外汇期权的人们预期汇率发生大幅波动的可能性要大于由对数正态分布模型中所预测出的概率。

3. 股票期权(equity option)隐含波动率呈现出半笑的形态(向左偏:当strike较小时,波动率会变大)——人们认为股票价格下跌的可能性要大于其上涨的可能性。

原因:leverage效应:标的资产的价值与其波动性成反比;crashophobia效应:人们恐惧股灾的出现,因此对股票价格下跌提供保护的期权定以较高的价格(只是表明在实际中,标的资产价格下降的概率要大于BS 公式中所假设的那样,但并不意味着随着股票价格降低波动率会提高)。

————这是探究波动率与施权价的关系。

4. 波动率期限结构(volatility term structure):平价期权的波动率与期权到期期限之间的关系——当短期波动率非常低时,波动率函数是距离到期期限时间的增函数;当短期波动率非常高时,波动率函数是距离到期期限时间的减函数(与波动率的均值回归性质有关)。

————这是探究波动率与到期期限的关系。

5. Volatility surface:同时考虑了期限结构和微笑效应,即同时考虑了strike和maturity term。

因此交易者可以使用volatility term structure和surfaces来判断一个期权定价模型得出结论的准确性和一致性。

6. 在计算期权的Greek的时候所遵守的两大原则:i. sticky strike rule(期权的隐含波动率在短期内应当是保持不变的);ii. sticky delta rule(期权价格和标的价格与施权价格的比率之间的关系应当是保持不变的,根据这个假设得出的delta要大于由BS公式得到的delta)。

Option

Option

Option1. Put-call parity(1) Fiduciary Call = C(X) + B(X)B: zero-coupon bond that pays X in T yearsC(X) : X is the exercise priceS : buying a stockPayoff to it0f00P0 = C0 - S0 + [X* exp (- R c f * T)](4) Arbitrage strategy2. Binomial Option Pricing Model(1) Basic Model(2) Arbitrage OpportunityOption – fractional share of the stockFractional share of stock in the arbitrage trade for each option traded (hedge ratio)= Delta = (C+1– C-1) / (S+1– S -1)(3) Two-period Binomial Model3. Options on a fixed-income instrument using a binomial tree(1) Basic SituationStep 1: Price the bond at each node using the projected interest ratesStep 2: Calculate the intrinsic value of the option at each node at the maturity of the option Step 3: Bring the terminal option values determined in Step 2 back to todayRisk-neutral probability of an up- and down- move here is always 50%.If we need to value an American option, the value at any node will be equal to the greater of the PV of the future payoffs or the current intrinsic value (S T– X for call or X - S T for put).(2) Caps and FloorsCaps and Floors are just bundles of European-style options on interest rates, called caplets and floorlets. The value of a cap or floor is the sum of the values of the individual caplets or floorlets.e.g. value of 2-year cap = value of a 1-year caplet + value of a 2-year caplet- expiration value of caplet = max {0, [(one-year rate – cap rate)* notional principal]} / (1+ one-year rate)- expiration value of floorlet = max {0, [floor rate –one-year rate]*notional principal}/(1+ one-year rate)4. Black-Scholes-Merton (BSM) Model(1) continuous time and no-arbitrage assumption. To derive the BSM model, an“instantaneously” riskless portfolio (one that is riskless over the next instant) is used to solve for the option price.(2) BSM assumptions and limitations - P 265Exception: the put value may increase as the option approaches maturity if the option is deep in the money and close to maturity.(4) DeltaDelta call = (C1– C0) / (S1– S0)0 = < Delta call < = 1- 1 = < Delta put < = 0C - Price of the callS - Price of the underlying stockFrom BSM:N(d1) *[N(d1) - 1] *The approximation gets worse when the△S gets larger.Delta is the slope of the prior-to-expiration curve, which evaluates the time value.- intrinsic value0, when call option is out of moneyS T– X, when call option is at the money- time valueThe prior-to-expiration curve lies above the at-expiration diagram by the amount of the time value.5. Dynamic Hedging(1) goal of delta-neutral portfolio (delta-neutral hedge) is to combine a long position in astock with a short position in a call option so that the value of the portfolio doesn’t change when the value of the stock changes.(2) Number of options needed to delta hedge = number of shares hedged / delta of calloption(3) The delta-neutral portfolio must be continually rebalanced to maintain the hedge –dynamic hedge.6. Gamma(1) It is used to measure the rate of change in delta as the underlying stock pricechanges.(2) Long positions in calls and puts have positive gammas. And, call and put on the sameunderlying assets with the same exercise price and time to maturity will have equal gammas.(3) Gamma is the largest when a call or put is at the money and close to expiration. If theoption is either deep in or deep out of the money, gamma approaches zero.(4) Gamma can be used to measure how poorly a dynamic hedge will perform when it isnot rebalanced in response to a change in the asset price.7. Effect of underlying asset cash flowExistence of cash flows on the underlying asset:Decrease the value of a call and increase the value of a putC0(X) + [X/(1 + R f)T] = P0 + (S0– PVCF)8. Options on ForwardsPut-call parity for options on forwardsPortfolio IC0 + (X - F T )/(1+R f)TA call on the forward contract with an exercise price of X that matures at time T on a forward contract at F TPortfolio IIP0A put on the forward contract with an exercise price of XA long position in the forward contractC0 + (X - F T )/(1+R f)T = P09. Options on futures & Options on forwardsAmerican options on futures are more valuable than comparable European options because there is mark to market on futures, early exercise can accelerate the payment of any gains.Since there is no mark to market on forwards, early exercise doesn’t accelerate the payment of any gains. So the value of American and European options on forwards are the same.。

期权定价的有限差分法

期权定价的有限差分法

Liaoning Normal University教师指导本科生科研训练项目研究论文题目:期权定价的有限差分法学院:数学学院专业:数学与应用数学(金融数学)班级序号:6班号学号:**************学生姓名:***指导教师:***2011年12月期权定价的有限差分法学生:刘小芹指导教师:包振华数学学院数学与应用数学(金融数学)专业2008级摘要:期权定价是所有金融应用领域数学上最复杂的问题之一。

第一个完整的期权定价模型由Fisher Black和Myron Scholes创立并于1973年公之于世。

B—S期权定价模型发表的时间和芝加哥期权交易所正式挂牌交易标准化期权合约几乎是同时。

不久,德克萨斯仪器公司就推出了装有根据这一模型计算期权价值程序的计算器。

现在,几乎所有从事期权交易的经纪人都持有各家公司出品的此类计算机,利用按照这一模型开发的程序对交易估价。

这项工作对金融创新和各种新兴金融产品的面世起到了重大的推动作用。

有限差分方法(FDM)是计算机数值模拟最早采用的方法,至今仍被广泛运用,该方法是一种直接将微分问题变为代数问题的近似数值解法,数学概念直观,表达简单,是发展较早且比较成熟的数值方法。

关键词:期权定价;有限差分方法;Abstract: Hedging is one of the important reasons for the development of the financial derivatives. When we use derivatives to hedge the risk for other asset or derivatives, it is the nature method to compute some common insensitive index for hedging tool and the underlying asset,then we can construct the portfolio. In this paper we take into account the options, which is called the dynamic hedging. Most investors use some complex hedging strategies in order to reduce the risk they faced. They try to make their portfolio immune to the small changes in the underlying asset in the short future, which is the so called Delta hedging.Keywords: Hedging; Delta; risk neutral hedging1、引言期权,也即期货合约的选择权,指的是其购买者在交付一定数量的权利金之后,所拥有的在未来一定时间内以一定价格买进或卖出一定数量相关商品合约(不论是实物商品,金融证券或期货)的权利,但不负有必须买进或卖出的义务。

公司理财精要版原书第12版教师手册RWJ_Fund_12e_IM_Chapter24

公司理财精要版原书第12版教师手册RWJ_Fund_12e_IM_Chapter24

CHAPTER 24OPTION VALUATIONSLIDES24.1 Chapter 2424.2 Key Concepts and Skills24.3 Chapter Outline24.4 Protective Put24.5 An Alternative strategy24.6 Comparing the Strategies24.7 Put-Call Parity24.8 Example: Finding the Call Price24.9 Continuous Compounding24.10 Example: Continuous Compounding24.11 Example: PCP with Continuous Compounding24.12 Black-Scholes Option Pricing Model24.13 Example: Black-Scholes OPM24.14 Example: Black-Scholes OPM in a Spreadsheet24.15 Put Values24.16 European vs. American Options24.17 Table 24.424.18 Varying Stock Price and Delta24.19 Work the Web Example24.20 Figure 24.1 Option Prices vs. Stock Price24.21 Example: Delta24.22 Varying Time to Expiration and Theta24.23 Figure 24.2 Option Prices vs. Time to Expiration24.24 Example: Time Premiums24.25 Varying Standard Deviation and Vega24.26 Figure 24.3 Option Prices vs. Return Standard Deviation 24.27 Varying the Risk-free Rate and Rho24.28 Figure 24.4 Option Prices vs. Risk-free Interest Rate 24.29 Implied Standard Deviations24.30 Work the Web Example – 224.31 Equity as a Call Option24.32 Valuing Equity and Changes in Assets24.33 Put-Call Parity and the Balance Sheet Identity24.34 Mergers and Diversification24.35 Extended Example – Part I24.36 Extended Example – Part II24.37 Extended Example – Part III24.38 M&A Conclusions24.39 Extended Example: Low NPV – Part I24.40 Extended Example: Low NPV – Part II24.41 Extended Example: Low NPV – Part III24.42 Extended Example: Negative NPV – Part I24.43 Extended Example: Negative NPV – Part II24.44 Extended Example: Negative NPV – Part III24.45 Extended Example: Negative NPV – Part IV24.46 Conclusions24.47 Quick Quiz24.48 Ethics Issues24.49 Comprehensive Problem24.50 End of ChapterCHAPTER WEB SITESCHAPTER ORGANIZATION24.1 Put-Call ParityProtective PutsAn Alternative StrategyThe ResultContinuous Compounding: A Refresher Course 24.2 The Black-Scholes Option Pricing ModelThe Call Option Pricing FormulaPut Option ValuationA Cautionary Note24.3 More about Black-ScholesVarying the Stock PriceVarying the Time to ExpirationVarying the Standard DeviationVarying the Risk-Free RateImplied Standard Deviations24.4 Valuation of Equity and Debt in a Leveraged FirmValuing the Equity in a Leveraged FirmOptions and the Valuation of Risky Bonds 24.5 Options and Corporate Decisions: Some ApplicationsMergers and DiversificationOptions and Capital Budgeting24.6 Summary and ConclusionsANNOTATED CHAPTER OULTINESlide 1: Chapter 24Slide 2: Key Concepts and SkillsSlide 3: Chapter Outline25.1 Put-Call ParityTerminology Review:Call – right, but not the obligation, to buy the underlying asset at thespecified price on or before a specified datePut – right, but not the obligation, to sell the underlying asset at the specifiedprice on or before a specified dateExercise or strike price – price specified in the option contractA.Protective PutsSlide 4: Protective PutThe strategy:Buy one share of stock at price, S.Buy one put option with strike price, E, and put premium, P.Example: Suppose you buy Citigroup stock for $45, and at thesame time, you purchase a put option with a strike price of $40.You pay $1.80 for the option, and it expires in one year. You planto sell the stock in one year.Consider the following possible payoffs:Stock Price Put Value Combined Value Total Gain or Loss25 15 40 −5.8030 10 40 −5.8035 5 40 −5.8040 0 40 −5.8045 0 45 −1.80Stock Price Put Value Combined Value Total Gain or Loss50 0 50 +3.2055 0 55 +8.2060 0 60 +13.2065 0 65 +18.20The maximum loss has been limited to $5.80.B.An Alternative StrategySlide 5: An Alternative StrategySuppose, instead, you buy a call option with a strike price of E anda call price of C. You invest the remainder in a Treasury Bill.Example: A $40 call option on Citigroup stock is selling for $7.78,and the T-bill has an interest rate of 2.5%. We want to look at thesame investment as in part A, so you invest a total of $46.80. So,you invest 46.80 − 7.78 = 39.02 in T-bills. Consider the payoffs.Combined Value Total Gain or Loss Stock Price Call Value T-bill39.02(1.025)25 0 40 40 −5.8030 0 40 40 −5.8035 0 40 40 −5.8040 0 40 40 −5.8045 5 40 45 −1.8050 10 40 50 +3.2055 15 40 55 +8.2060 20 40 60 +13.2065 25 40 65 +18.20The payoffs are the same with both strategies.C.The ResultSlide 6: Comparing the StrategiesIf the combined value is the same at the end, under all situations,then the cost today must be the same.Slide 7: Put-Call ParityThis leads to the famous put-call parity (PCP) condition:S + P = C + PV(E)where the present value is computed using the risk-free rate. Slide 8: Example: Finding the Call PriceThe PCP condition can be rearranged to solve for any of thecomponents.D.Continuous Compounding: A Refresher CourseSlide 9: Continuous CompoundingEffective annual rate with continuous compounding:EAR = e q– 1where q is the quoted rate.Suppose you have a quoted rate of 5% per year with continuouscompounding:EAR = e.05− 1 = .05127 or 5.127%Time value of money calculations with continuous compounding:FV = PVe RtPV = FVe-Rtwhere R = continuously compounded rate, and t = number ofperiods in terms of yearsSlide 10: Example: Continuous CompoundingExample: What is the present value of $1000 to be received inthree months if the annual continuously compounded rate is 8%?PV = 1000e-.08(3/12) = 980.20Slide 11: Example: PCP with Continuous CompoundingPCP with continuous compoundingS + P = C + Ee−RtExample: Given the following, what does the call have to sell forto prevent arbitrage?S = 80; P = 6; E = 85; R = 10% with continuous compounding; t =9 months (9/12)80 + 6 = C + 85e−.1(9/12)C = 86 − 78.86 = 7.1424.2The Black-Scholes Option Pricing ModelA. The Call Option Pricing Formula Slide 12:Black-Scholes Option Pricing ModelThe Formula: C = SN(d 1) − Ee −Rt N(d 2) where N(d 1) and N(d 2) are probabilities that we compute using the following formulas and then look the numbers up in the standard normal tables. where σ is the standard deviation (or volatility) of the underlying asset returns. Slide 13:Example: Black-Scholes OPMExample: Consider a stock that is currently selling for $35. You are looking at a call option that has an exercise price of $30 and expires in 6 months. The risk-free rate is 4%, compounded continuously. The volatility of stock returns is .25. What is the call price? ()89675.12625.07353.1d 07353.112625.126225.04.3035ln d 221=-==⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛= From Table 24.3 N(d 1) = N(1.07) = (.8554 + .8599)/2 = .8577 N(d 2) = N(.90) = .8159 C = 35(.8577) – 30e -.04(6/12)(.8159) = $6.03t σd d t σt 2σR E S ln d 1221-=⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛=Slide 14: Example: Black-Scholes OPM in a SpreadsheetB.Put Option ValuationSlide 15: Put ValuesLecture Tip: The Black-Scholes model can also be adjusted tosolve for the value of a put option directly.P = Ee−Rt N(−d2) − SN(−d1)where d1 and d2 are computed as before. You just change the signbefore looking it up in the table.As an example, consider the previous information but find thevalue of a put instead of a call.N(−d1) = N(−1.07) = (.1401 + .1446) ⁄ 2 = .1424N(−d2) = N(−.9) = .1841P = 30e−.04(6/12)(.1841) − 35(.1424) = $0.43A slightly different answer will be found below using the PCP. Thedifference is due to rounding.Example: Consider the previous call option example and use put-call parity to find the value of the put.S + P = C + PV(E)35 + P = 6.03 + 30e−.04(6/12)P = $0.44C. A Cautionary NoteSlide 16: European vs. American OptionsBoth PCP and the Black-Scholes model are strictly for Europeanoptions that can be exercised only at expiration. There are timeswhen it would be optimal to exercise a put option early, but thesemodels will not capture that additional value, called the “earlyexercise premium.”24.3 More about Black-ScholesSlide 17: Table 24.4Table 24.4 illustrates the relationship between option values andthe five major inputs.A.Varying the Stock PriceSlide 18: Varying Stock Price and DeltaSlide 19: Work the Web ExampleSlide 20: Figure 24.1 Option Prices vs. Stock PriceSlide 21: Example: DeltaCall prices have a direct relationship with the stock price, while putprices have an inverse relationship. This relationship is calleddelta.For European options:Call delta = N(d1)Put delta = N(d1) − 1You can use delta to estimate the new option value given a smallchange in the stock price.Lecture Tip: Delta is the first derivative of the OPM with respectto S. Gamma is the second derivative with respect to S.An option’s delta changes as S changes, and gamma measures therate of change. Delta is often used to determine how many optionsare needed to hedge a portfolio. As S changes, the number ofoptions needed will change because delta depends on S.The larger the gamma, the smaller the change in S required tocause a significant change in delta. The larger the change in delta,the greater the need to rebalance the portfolio and the higher thetrading costs. Therefore, portfolio managers will often look at boththe gamma and the delta when deciding which options to use forhedging.B.Varying the Time to ExpirationSlide 22: Varying Time to Expiration and ThetaSlide 23: Figure 24.2 Option Prices vs. Time to ExpirationFor American calls and puts, the value of the option increases asthe time to expiration increases.It is never optimal to exercise call options on non-dividend payingstocks early. Therefore, the value of a European call will alsoincrease as time increases.However, it may be optimal to exercise a put option early, and aEuropean put prevents early exercise. Therefore, there aresituations in which a shorter time to expiration would actually bemore valuable, and the relationship between European put valueand time is ambiguous.The relationship between option value and time to expiration iscalled theta.Intrinsic valuecall: max[S − E, 0]put: max[E − S, 0]Option value = Intrinsic value + Time premiumTime premium – option value associated with the time left toexpiration, decreases as expiration approachesSlide 24: Example: Time PremiumsExample: Consider the previous option valuation examples. Whatis the intrinsic value and the time premium for each option?Call: C = 6.03intrinsic value = max[35 − 30, 0] = 5time premium = 6.03 − 5 = 1.03Put: P = .44intrinsic value = max[30 − 35, 0] = 0time premium = .44 − 0 = .44C.Varying the Standard DeviationSlide 25: Varying Standard Deviation and VegaSlide 26: Figure 24.3 Option Prices vs. Return Standard DeviationThe relationship between volatility and option value is called vega.As volatility increases, the value of the option increases.The potential loss is limited to your premium. However, the greaterthe volatility, the larger the potential gain.D.Varying the Risk-Free RateSlide 27: Varying the Risk-free Rate and RhoSlide 28: Figure 24.4 Option Prices vs. Risk-free Interest RateThe value of a call increases as the risk-free rate increases. Theopposite is true for puts. However, the impact is very small,especially for “realistic” rates.The relationship between the risk-free rate and option value iscalled rho.E.Implied Standard DeviationsSlide 29: Implied Standard DeviationsSlide 30: Work the Web ExampleWe can observe option values, underlying asset values, exerciseprices, and risk-free rates in the market. The one variable that isnot observable is the standard deviation, or volatility.The OPM can be used to estimate the expected standard deviationof returns – called the implied standard deviation or impliedvolatility.There is not a closed-form solution—the easiest way to find theimplied volatility is to use an options calculator.24.4 Valuation of Equity and Debt in a Leveraged FirmSlide 31: Equity as a Call OptionEquity can be viewed as a call option on the assets of a business.When a debt payment is due, stockholders can choose not toexercise the option and the assets pass to the bondholders.Paying off the debt is the same as exercising the option.A.Valuing the Equity in a Leveraged FirmSlide 32: Valuing Equity and Changes in AssetsExample: For simplicity, assume a firm has a 5-year, zero coupon bond with a face value of $20 million. The firm’s assets have a market value of $30 million. The volatility of asset returns is .3, and the continuously compounded risk-free rate is 5%. What is the market value of equity? Of debt? S = 30; E = 20; t = 5; σ = .3; R = .05 N(1.31) = (.9032 + .9066) ⁄ 2 = .9049 N(.64) = .7389 Equity = 30(.9049) − 20e −.05(5)(.7389) = 15.63788 million Debt = 30 − 15.63788 = 14.36212 millionWhat is the firm’s cost of debt? 14.36212 = 20e −R(5) R = .06623 = 6.623%B. Options and the Valuation of Risky Bonds A protective put can be used to reduce the risk of bonds. Buy a put with an exercise price of $20 million Value of risky bond + put = value of risk-free bond 14.36212 + P = 20e −.05(5) P = 1.2139 million Increasing the value of the put decreases the value of the risky bond. Slide 33:Put-Call Parity and the Balance Sheet IdentityValue of debt = Value of risk-free debt − Put Debt = Ee −Rt − PPCP: S + P = C + Ee −RtS = C + (Ee −Rt − P)()()64.53.31.1d 31.153.523.05.2030ln d 221=-==⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛=Assets = Equity + DebtPCP and the balance sheet identity are the same.24.4 Options and Corporate Decisions: Some ApplicationsC.Mergers and DiversificationSlide 34: Mergers and DiversificationSlide 35: Extended Example – Part ISlide 36: Extended Example – Part IISlide 37: Extended Example – Part IIIUse option valuation to investigate whether diversification is agood reason for a merger from a stockholder’s viewpoint.If synergies do not exist, then a merger will reduce volatilitywithout increasing cash flow.Decreasing volatility decreases the value of the call option (equity)and the put option. Decreasing the value of the put increases thevalue of the debt.Slide 38: M&A ConclusionsSo, a merger for diversification reasons, transfers value from thestockholders to the bondholders.D.Options and Capital BudgetingSlide 39: Extended Example: Low NPV – Part ISlide 40: Extended Example: Low NPV – Part IISlide 41: Extended Example: Low NPV – Part IIIIf a firm has a substantial amount of debt, stockholders may preferriskier projects, even if they have a lower NPV.The riskier project increases the volatility of the asset returns. Theincreased volatility increases the value of the call (equity) and theput. The increased put value decreases the value of the debt. Thistransfers wealth from the bondholders to the stockholders.Slide 42: Extended Example: Negative NPV – Part ISlide 43: Extended Example: Negative NPV – Part IISlide 44: Extended Example: Negative NPV – Part IIISlide 45: Extended Example: Negative NPV – Part IVStockholders may even prefer a negative NPV project if itincreases volatility enough.The wealth transfer from bondholders to stockholders mayoutweigh the negative NPV.Slide 46: ConclusionsLecture Tip: Bondholders recognize the desire of stockholders totake on riskier projects. Consequently, provisions are put into thebond indentures to try to prevent this wealth transfer. Theseprovisions add to the firm’s cost either directly through a higherinterest rate or through additional monitoring costs. These costsare all considered agency costs.24.5 Summary and ConclusionsSlide 47: Quick QuizSlide 48: Ethics IssuesSlide 49: Comprehensive ProblemSlide 50: End of Chapter。

索提诺比率计算公式

索提诺比率计算公式

索提诺比率计算公式
普索提诺比率(Put-Call Parity)是由曼尼斯和特拉弗斯于1986年提出的一种金融算法,用来衡量股票期权和期货合约之间的关系。

普索提诺比率是投资者使用期权和期货对冲价值显著性差异的重要工具。

一、普索提诺比率的定义
普索提诺比率是指投资者利用股票期权和股票期货的组合,创造和保有无风险定价关系的一种比率。

普索提诺比率定义为:把股票期权的价格乘以在相同时间的期货价格,再加上未来的期货的价格减去股票期权的价格即可求出普索提诺比率。

二、普索提诺比率的应用
1、使用普索提诺比率来计算风险和资金成本:风险衡量和资金成本可以通过普索提诺比率来估计,普索提诺比率的应用可以使投资者更好的理解股票期权和股票期货的关系,使他们在投资期权和期货时可以更准确的判断。

2、使用普索提诺比率作为一种套利手段:普索提诺比率可以用作一种套利工具,投资者可以利用普索提诺比率来把握套利机会。

3、使用普索提诺比率来保持资产价值:普索提诺比率可用来维持某种资产价值,比如在投资期权和期货时,投资者可以使用普索提诺比率
来判断投资的风险。

三、求解普索提诺比率的公式
普索提诺比率的计算公式为:[Put Price+ Stock Price]/[Strike Price + Futures Price] = Put-Call Parity
其中:
Put Price:放弃权价格
Stock Price:股票价格
Strike Price:执行价格
Futures Price:期货价格。

投资学Chap021

投资学Chap021

CHAPTER 21: OPTION VALUATIONPROBLEM SETS1. The value of a put option also increases with the volatility of the stock. We see this from theput-call parity theorem as follows:P = C – S0 + PV(X) + PV(Dividends)Given a value for S and a risk-free interest rate, then, if C increases because of an increase in volatility, P must also increase in order to maintain the equality of the parity relationship.2. A $1 increase in a call option’s exercise price would lead to a decrease in the option’s value ofless than $1. The change in the call price would equal $1 only if: (i) there were a 100%probability that the call would be exercised, and (ii) the interest rate were zero.3. Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore,the value of the put option increases as beta increases.4. Holding beta constant, the stock with a lot of firm-specific risk has higher total volatility. Theoption on the stock with higher firm-specific risk is worth more.5. A call option with a high exercise price has a lower hedge ratio. This call option is less in themoney. Both d1 and N(d1) are lower when X is higher.6. a. Put A must be written on the stock with the lower price. Otherwise, given the lowervolatility of Stock A, Put A would sell for less than Put B.b. Put B must be written on the stock with the lower price. This would explain its higherprice.c. Call B must have the lower time to expiration. Despite the higher price of Stock B, CallB is cheaper than Call A. This can be explained by a lower time to expiration.d. Call B must be written on the stock with higher volatility. This would explain its higherprice.e. Call A must be written on the stock with higher volatility. This would explain its higherprice.7.Exercise Price Hedge Ratio 120 0/30 = 0.000 110 10/30 = 0.333 100 20/30 = 0.667 90 30/30 = 1.000As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0.8.S d 1 N(d 1) 45 -0.0268 0.4893 50 0.5000 0.6915 550.97660.83569.a. uS 0 = 130 ⇒ P u = 0 dS 0 = 80 ⇒ P d = 30 The hedge ratio is:5380130300dS uS P P H 00d u -=--=--=b.Riskless Portfolio S = 80 S = 130 Buy 3 shares 240 390 Buy 5 puts 150 0 Total390390Present value = $390/1.10 = $354.545c.The portfolio cost is: 3S + 5P = 300 + 5P The value of the portfolio is: $354.545 Therefore: P = $54.545/5 = $10.9110. The hedge ratio for the call is:5280130020dS uS C C H 00d u =--=--=Riskless Portfolio S = 80 S = 130 Buy 2 shares 160 260 Write 5 calls 0 -100 Total160160Present value = $160/1.10 = $145.455 The portfolio cost is: 2S – 5C = $200 – 5C The value of the portfolio is: $145.455 Therefore: C = $54.545/5 = $10.91 Does P = C + PV(X) – S?10.91 = 10.91 + 110/1.10 – 100 = 10.9111. d 1 = 0.3182 ⇒ N(d 1) = 0.6248d 2 = –0.0354 ⇒ N(d 2) = 0.4859 Xe - r T = 47.56 C = $8.1312. P = $5.69This value is derived from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity:P = C + PV(X) – S 0 = $8.13 + $47.56 - $50 = $5.6913. a. C falls to $5.5541b. C falls to $4.7911c. C falls to $6.0778d. C rises to $11.5066e. C rises to $8.718714. According to the Black-Scholes model, the call option should be priced at:[$55 ⨯ N(d1)] – [50 ⨯ N(d 2)] = ($55 ⨯ 0.6) – ($50 ⨯ 0.5) = $8Since the option actually sells for more than $8, implied volatility is greater than 0.30.15. A straddle is a call and a put. The Black-Scholes value would be:C + P = S0 N(d1) - Xe–rT N(d 2) + Xe–rT [1 - N(d 2)] - S0 [1 - N(d1)]= S0 [2N(d1) - 1] + Xe–rT [1 - 2N(d 2)]On the Excel spreadsheet (Spreadsheet 21.1), the valuation formula would be:B5*(2*E4 - 1) + B6*EXP(-B4*B3)*(1 - 2*E5)16. The rate of return of a call option on a long-term Treasury bond should be more sensitive tochanges in interest rates than is the rate of return of the underlying bond. The option elasticityexceeds 1.0. In other words, the option is effectively a levered investment and the rate of return on the option is more sensitive to interest rate swings.17. Implied volatility has increased. If not, the call price would have fallen as a result of thedecrease in stock price.18. Implied volatility has increased. If not, the put price would have fallen as a result of thedecreased time to expiration.19. The hedge ratio approaches one. As S increases, the probability of exercise approaches 1.0.N(d1) approaches 1.0.20. The hedge ratio approaches –1.0. As S decreases, the probability of exercise approaches 1.[N(d1) –1] approaches –1 as N(d1) approaches 0.21. A straddle is a call and a put. The hedge ratio of the straddle is the sum of the hedge ratios ofthe individual options: 0.4 + (–0.6) = –0.222. a. The spreadsheet appears as follows:The standard deviation is: 0.3213b. The spreadsheet below shows the standard deviation has increased to: 0.3568Implied volatility has increased because the value of an option increases with greatervolatility.c. Implied volatility increases to 0.4087 when expiration decreases to four months. Theshorter expiration decreases the value of the option; therefore, in order for the optionprice to remain unchanged at $8, implied volatility must increase.d. Implied volatility decreases to 0.2406 when exercise price decreases to $100. Thedecrease in exercise price increases the value of the call, so that, in order to the optionprice to remain at $8, implied volatility decreases.e. The decrease in stock price decreases the value of the call. In order for the option priceto remain at $8, implied volatility increases.23. a. The delta of the collar is calculated as follows:Position DeltaBuy stock 1.0Buy put, X = $45 N(d1) – 1 = –0.40Write call, X = $55 –N(d1) = –0.35Total 0.25If the stock price increases by $1, then the value of the collar increases by $0.25. Thestock will be worth $1 more, the loss on the purchased put will be $0.40, and the callwritten represents a liability that increases by $0.35.b.If S becomes very large, then the delta of the collar approaches zero. Both N(d1) termsapproach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the(present value of the) exercise price of the call, and is unaffected by small changes in thestock price.As S approaches zero, the delta also approaches zero: both N(d1) terms approach 0.For very small stock prices, the value of the portfolio is simply the (present value of the)exercise price of the put, and is unaffected by small changes in the stock price.24. Put X DeltaA 10 -0.1B 20 -0.5C 30 -0.925. a. Choice A: Calls have higher elasticity than shares. For equal dollar investments, a call’scapital gain potential is greater than that of the underlying stock.b. Choice B: Calls have hedge ratios less than 1.0, so the shares have higher profit potential.For an equal number of shares controlled, the dollar exposure of the shares is greaterthan that of the calls, and the profit potential is therefore greater.26. a. uS 0 = 110 ⇒ P u = 0 dS 0 = 90 ⇒ P d = 10 The hedge ratio is:2190110100dS uS P P H 00d u -=--=--=A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110/1.05 = $104.76 Therefore:S + 2P = $104.76$100 + 2P = $104.76 ⇒ P = $2.38b. Cost of protective put portfolio = $100 + $2.38 = $102.38c.Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is –0.5, the portfolio consists of (1 – 0.5) = 0.5 shares of stock, which costs $50, and the remaining funds ($52.38) invested in T-bills, earning 5% interest. Portfolio S = 90 S = 110 Buy 0.5 shares 45 55 Invest in T-bills 55 55 Total100110This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put.27. The put values in the second period are:P uu = 0P ud = P du = 110 − 104.50 = 5.50 P dd = 110 − 90.25 = 19.75To compute P u , first compute the hedge ratio:3150.10412150.50udSuuSP P H 0ud uu -=--=--=Form a riskless portfolio by buying one share of stock and buying three puts. The cost of the portfolio is: S + 3P u = $110 + 3P u The payoff for the riskless portfolio equals $121: Riskless Portfolio S = 104.50 S = 121 Buy 1 share 104.50 121.00 Buy 3 puts 16.50 0.00 Total121.00121.00Therefore, find the value of the put by solving:$110 + 3P u = $121/1.05 ⇒ P u = $1.746 To compute P d , compute the hedge ratio:0.125.9050.10475.1950.5ddSduSP P H 0dd du -=--=--=Form a riskless portfolio by buying one share and buying one put. The cost of the portfolio is: S + P d = $95 + P d The payoff for the riskless portfolio equals $110: Riskless Portfolio S = 90.25 S = 104.50 Buy 1 share 90.25 104.50 Buy 1 put 19.75 5.50 Total110.00110.00Therefore, find the value of the put by solving:$95 + P d = $110/1.05 ⇒ P d = $9.762 To compute P, compute the hedge ratio:5344.095110762.9746.1dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying 0.5344 of a share and buying one put. The cost of the portfolio is: 0.5344S + P = $53.44 + PThe payoff for the riskless portfolio equals $60.53:Riskless Portfolio S = 95 S = 110Buy 0.5344 share 50.768 58.784Buy 1 put 9.762 1.746Total 60.530 60.530Therefore, find the value of the put by solving:$53.44 + P = $60.53/1.05 ⇒ P = $4.208Finally, we verify this result using put-call parity. Recall from Example 21.1 that:C = $4.434Put-call parity requires that:P = C + PV(X) – S$4.208 = $4.434 + ($110/1.052) - $100Except for minor rounding error, put-call parity is satisfied.28. If r = 0, then one should never exercise a put early. There is no “time value cost” to waiting toexercise, but there is a “volatility benefit” from waiting. To show this more rigorously, consider the following portfolio: lend $X and short one share of stock. The cost to establish the portfolio is (X – S 0). The payoff at time T (with zero interest earnings on the loan) is (X – S T). Incontrast, a put option has a payoff at time T of (X – S T) if that value is positive, and zerootherwise. The put’s payoff is at least as large as the portfolio’s, and therefore, the put must cost at least as much as the portfolio to purchase. Hence, P ≥ (X – S 0), and the put can be sold for more than the proceeds from immediate exercise. We conclude that it doesn’t pay to exercise early.29. a. Xe-rTb. Xc. 0d. 0e. It is optimal to exercise immediately a put on a stock whose price has fallen to zero. Thevalue of the American put equals the exercise price. Any delay in exercise lowers valueby the time value of money.30. Step 1: Calculate the option values at expiration. The two possible stock prices and thecorresponding call values are:uS 0 = 120 ⇒ C u = 20 dS 0 = 80 ⇒ C d = 0 Step 2: Calculate the hedge ratio.2180120020dS uS C C H 00d u =--=--=Therefore, form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2CStep 3: Show that the payoff for the riskless portfolio equals $80: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Write 2 calls 0 -40 Total8080Therefore, find the value of the call by solving:$100 – 2C = $80/1.10 ⇒ C = $13.636Notice that we did not use the probabilities of a stock price increase or decrease. These are not needed to value the call option.31.The two possible stock prices and the corresponding call values are:uS 0 = 130 ⇒ C u = 30 dS 0 = 70 ⇒ C d = 0 The hedge ratio is:2170130030dS uS C C H 00d u =--=--=Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2CThe payoff for the riskless portfolio equals $70: Riskless Portfolio S = 70 S = 130 Buy 1 share 70 130 Write 2 calls 0 -60 Total7070Therefore, find the value of the call by solving:$100 – 2C = $70/1.10 ⇒ C = $18.182Here, the value of the call is greater than the value in the lower-volatility scenario.32. The two possible stock prices and the corresponding put values are:uS 0 = 120 ⇒ P u = 0dS 0 = 80 ⇒ P d = 20 The hedge ratio is:2180120200dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P = 100 + 2PThe payoff for the riskless portfolio equals $120: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Buy 2 puts 40 0 Total120120Therefore, find the value of the put by solving:$100 + 2P = $120/1.10 ⇒ P = $4.545 According to put-call parity: P + S = C + PV(X) Our estimates of option value satisfy this relationship:$4.545 + $100 = $13.636 + $100/1.10 = $104.54533. If we assume that the only possible exercise date is just prior to the ex-dividend date, then therelevant parameters for the Black-Scholes formula are:S 0 = 60r = 0.5% per month X = 55 σ = 7%T = 2 monthsIn this case: C = $6.04If instead, one commits to foregoing early exercise, then we reduce the stock price by the present value of the dividends. Therefore, we use the following parameters:S 0 = 60 – 2e − (0.005 ⨯ 2) = 58.02 r = 0.5% per month X = 55 σ = 7%T = 3 months In this case, C = $5.05The pseudo-American option value is the higher of these two values: $6.0434. True. The call option has an elasticity greater than 1.0. Therefore, the call’s percentage rate ofreturn is greater than that of the underlying stock. Hence the GM call responds more thanproportionately when the GM stock price changes in response to broad market movements.Therefore, the beta of the GM call is greater than the beta of GM stock.35. True. The elasticity of a call option is higher the more out of the money is the option. (Eventhough the delta of the call is lower, the value of the call is also lower. The proportionalresponse of the call price to the stock price increases. You can confirm this with numerical examples.) Therefore, the rate of return of the call with the higher exercise price respondsmore sensitively to changes in the market index, and therefore it has the higher beta.36. As the stock price increases, conversion becomes increasingly more assured. The hedge ratioapproaches 1.0. The price of the convertible bond will move one-for-one with changes in the price of the underlying stock.37. Salomon believes that the market assessment of volatility is too high. Therefore, Salomonshould sell options because the analysis suggests the options are overpriced with respect to true volatility. The delta of the call is 0.6, while that of the put is 0.6 – 1 = –0.4. Therefore, Salomon should sell puts and calls in the ratio of 0.6 to 0.4. For example, if Salomon sells 2 calls and 3 puts, the position will be delta neutral:Delta = (2 ⨯ 0.6) + [3 ⨯ (–0.4)] = 038. If the stock market index increases 1%, the 1 million shares of stock on which the options arewritten would be expected to increase by:0.75% ⨯ $5 ⨯ 1 million = $37,500The options would increase by:delta ⨯ $37,500 = 0.8 ⨯ $37,500 = $30,000In order to hedge your market exposure, you must sell $3,000,000 of the market indexportfolio so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.39. S = 100; current value of portfolioX = 100; floor promised to clients (0% return)σ = 0.25; volatilityr = 0.05; risk-free rateT = 4 years; horizon of programa. Using the Black-Scholes formula, we find that:d1 = 0.65, N(d1) = 0.7422, d 2 = 0.15, N(d 2) = 0.5596Put value = $10.27Therefore, total funds to be managed equals $110.27 million: $100 million portfolio valueplus the $10.27 million fee for the insurance program.The put delta is: N(d1) – 1 = 0.7422 – 1 = –0.2578Therefore, sell off 25.78% of the equity portfolio, placing the remaining funds in T-bills. Theamount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is:$110.27 million – $74.22 million = $36.05 millionb.At the new portfolio value, the put delta becomes: –0.2779This means that you must reduce the delta of the portfolio by:0.2779 – 0.2578 = 0.0201You should sell an additional 2.01% of the equity position and use the proceeds to buyT-bills. Since the stock price is now at only 97% of its original value, you need to sell: $97 million ⨯ 0.0201 = $1.950 million of stock40. Using the true volatility (32%) and time to expiration T = 0.25 years, the hedge ratio for Exxonis N(d1) = 0.5567. Because you believe the calls are under-priced (selling at an impliedvolatility that is too low), you will buy calls and short 0.5567 shares for each call you buy.41. The calls are cheap (implied σ = 0.30) and the puts are expensive (impliedσ = 0.34). Therefore, buy calls and sell puts. Using the “true” volatility ofσ = 0.32, the call delta is 0.5567 and the put delta is: 0.5567 – 1.0 = –0.4433Therefore, for each call purchased, buy: 0.5567/0.4433 = 1.256 puts42. a.To calculate the hedge ratio, suppose that the market index increases by 1%. Then the stock portfolio would be expected to increase by:1% ⨯ 1.5 = 1.5% or 0.015 ⨯ $1,250,000 = $18,750Given the option delta of 0.8, the option portfolio would increase by:$18,750 ⨯ 0.8 = $15,000Salomon’s liability from writing these options would increase by the same amount. The market index portfolio would increase in value by 1%. Therefore, SalomonBrothers should purchase $1,500,000 of the market index portfolio in order to hedge its position so that a 1% change in the index would result in a $15,000 change in the value of the portfolio.b.The delta of a put option is:0.8 – 1 = –0.2Therefore, for every 1% the market increases, the index will rise by 10 points and the value of the put option contract will change by:delta ⨯ 10 ⨯ contract multiplier = –0.2 ⨯ 10 ⨯ 100 = –$200 Therefore, Salomon should write: $12,000/$200 = 60 put contractsCFA PROBLEMS1. Statement a: The hedge ratio (determining the number of futures contracts to sell) ought to beadjusted by the beta of the equity portfolio, which is 1.20. The correct hedge ratio would be:400,22.1000,2β2,000β500100$million 100$=⨯=⨯=⨯⨯Statement b: The portfolio will be hedged, and should therefore earn the risk-free rate, not zero, as the consultant claims. Given a futures price of 100 and an equity price of 100, the rate of return over the 3-month period is: (100 - 99)/99 = 1.01% = approximately 4.1% annualized 2.a.The value of the call option is expected to decrease if the volatility of the underlying stock price decreases. The less volatile the underlying stock price, the less the chance of extreme price movements and the lower the probability that the option expires in the money. This makes the participation feature on the upside less valuable.The value of the call option is expected to increase if the time to expiration of the option increases. The longer the time to expiration, the greater the chance that the option will expire in the money resulting in an increase in the time premium component of the option’s value.b. i. When European options are out of the money, investors are essentially saying that theyare willing to pay a premium for the right, but not the obligation, to buy or sell theunderlying asset. The out-of-the-money option has no intrinsic value, but, since optionsrequire little capital (just the premium paid) to obtain a relatively large potential payoff,investors are willing to pay that premium even if the option may expire worthless. TheBlack-Scholes model does not reflect investors’ demand for any premium above thetime value of the option. Hence, if investors are willing to pay a premium for an out-of-the-money option above its time value, the Black-Scholes model does not value thatexcess premium.ii. With American options, investors have the right, but not the obligation, to exercise theoption prior to expiration, even if they exercise for non-economic reasons. This increasedflexibility associated with American options has some value but is not considered in theBlack-Scholes model because the model only values options to their expiration date(European options).3. a. American options should cost more (have a higher premium). American options give theinvestor greater flexibility than European options since the investor can choose whetherto exercise early. When the stock pays a dividend, the option to exercise a call early canbe valuable. But regardless of the dividend, a European option (put or call) never sellsfor more than an otherwise-identical American option.b. C = S0 + P - PV(X) = $43 + $4 - $45/1.055 = $4.346Note: we assume that Abaco does not pay any dividends.c. i) An increase in short-term interest rate ⇒ PV(exercise price) is lower, and call valueincreases.ii) An increase in stock price volatility ⇒ the call value increases.iii) A decrease in time to option expiration ⇒ the call value decreases.4. a. The two possible values of the index in the first period are:uS0 = 1.20 × 50 = 60dS0 = 0.80 × 50 = 40The possible values of the index in the second period are:uuS0 = (1.20)2 × 50 = 72udS0 = 1.20 × 0.80 × 50 = 48duS0 = 0.80 × 1.20 × 50 = 48ddS0 = (0.80)2 × 50 = 32C uu = 72 − 60 = 12 C ud = C du = C dd = 0 Since C ud = C du = 0, then C d = 0.To compute C u , first compute the hedge ratio:214872012udSuuSC C H 0ud uu =--=--=Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C u = $60 – 2C u The payoff for the riskless portfolio equals $48: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Write 2 calls 0 -24 Total4848Therefore, find the value of the call by solving:$60 – 2C u = $48/1.06 ⇒ C u = $7.358 To compute C, compute the hedge ratio:3679.040600358.7dS uS C C H 00d u =--=--=Form a riskless portfolio by buying 0.3679 of a share and writing one call. The cost of the portfolio is: 0.3679S – C = $18.395 – C The payoff for the riskless portfolio equals $14.716: Riskless Portfolio S = 40 S = 60 Buy 0.3679 share 14.716 22.074 Write 1 call 0.000 −7.358 Total14.71614.716Therefore, find the value of the call by solving:$18.395 – C = $14.716/1.06 ⇒ C = $4.512P uu = 0P ud = P du = 60 − 48 = 12 P dd = 60 − 32 = 28To compute P u , first compute the hedge ratio:214872120udSuuSP P H 0ud uu -=--=--=Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P u = $60 + 2P u The payoff for the riskless portfolio equals $72: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Buy 2 puts 24 0 Total7272Therefore, find the value of the put by solving:$60 + 2P u = $72/1.06 ⇒ P u = $3.962 To compute P d , compute the hedge ratio:0.132482812ddSduSP P H 0dd du -=--=--=Form a riskless portfolio by buying one share and buying one put. The cost of the portfolio is: S + P d = $40 + P d The payoff for the riskless portfolio equals $60: Riskless Portfolio S = 32 S = 48 Buy 1 share 32 48 Buy 1 put 28 12 Total6060Therefore, find the value of the put by solving:$40 + P d = $60/1.06 ⇒ P d = $16.604 To compute P, compute the hedge ratio:6321.04060604.16962.3dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying 0.6321 of a share and buying one put.The cost of the portfolio is: 0.6321S + P = $31.605 + PThe payoff for the riskless portfolio equals $41.888:Riskless Portfolio S = 40 S = 60Buy 0.6321 share 25.284 37.926Buy 1 put 16.604 3.962Total 41.888 41.888Therefore, find the value of the put by solving:$31.605 + P = $41.888/1.06 ⇒ P = $7.912d. According to put-call-parity:C = S0 + P - PV(X) = $50 + $7.912 - $60/(1.062 ) = $4.512This is the value of the call calculated in part (b) above.5. a. (i) Index increases to 1402. The combined portfolio will suffer a loss. The written callsexpire in the money; the protective put purchased expires worthless. Let’s analyze theoutcome on a per-share basis. The payout for each call option is $52, for a total cashoutflow of $104. The stock is worth $1,402. The portfolio will thus be worth: $1,402 -$104 = $1,298The net cost of the portfolio when the option positions are established is:$1,336 + $16.10 (put) - [2 ⨯ $8.60] (calls written) = $1,334.90(ii) Index remains at 1336. Both options expire out of the money. The portfolio will thus beworth $1,336 (per share), compared to an initial cost 30 days earlier of $1,334.90. Theportfolio experiences a very small gain of $1.10.(iii) Index declines to 1270. The calls expire worthless. The portfolio will be worth $1,330,the exercise price of the protective put. This represents a very small loss of $4.90 comparedto the initial cost 30 days earlier of $1,334.90b. (i) Index increases to 1402. The delta of the call approaches 1.0 as the stock goes deep intothe money, while expiration of the call approaches and exercise becomes essentially certain.The put delta approaches zero.(ii) Index remains at 1336. Both options expire out of the money. Delta of eachapproaches zero as expiration approaches and it becomes certain that the options willnot be exercised.(iii) Index declines to 1270. The call is out of the money as expiration approaches. Deltaapproaches zero. Conversely, the delta of the put approaches -1.0 as exercise becomescertain.c. The call sells at an implied volatility (11.00%) that is less than recent historical volatility(12.00%); the put sells at an implied volatility (14.00%) that is greater than historicalvolatility. The call seems relatively cheap; the put seems expensive.。

【经典】期货期权及其他衍生品第九版答案(英文)HullOFOD9eSolutionsCh20

【经典】期货期权及其他衍生品第九版答案(英文)HullOFOD9eSolutionsCh20

CHAPTER 20Volatility SmilesPractice QuestionsProblem 20.1.What volatility smile is likely to be observed when(a)Both tails of the stock price distribution are less heavy than those of the lognormaldistribution?(b)The right tail is heavier, and the left tail is less heavy, than that of a lognormaldistribution?(a) A smile similar to that in Figure 20.7 is observed.(b)An upward sloping volatility smile is observedProblem 20.2.What volatility smile is observed for equities?A downward sloping volatility smile is usually observed for equities.Problem 20.3.What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern likely to be more pronounced for a two-year option than for a three-month option?Jumps tend to make both tails of the stock price distribution heavier than those of the lognormal distribution. This creates a volatility smile similar to that in Figure 20.1. The volatility smile is likely to be more pronounced for the three-month option.Problem 20.4.A European call and put option have the same strike price and time to maturity. The call has an implied volatility of 30% and the put has an implied volatility of 25%. What trades would you do?The put has a price that is too low relative to the call’s price. The correct trading strategy is to buy the put, buy the stock, and sell the call.Problem 20.5.Explain carefully why a distribution with a heavier left tail and less heavy right tail than the lognormal distribution gives rise to a downward sloping volatility smile.The heavier left tail should lead to high prices, and therefore high implied volatilities, forout-of-the-money (low-strike-price) puts. Similarly the less heavy right tail should lead to low prices, and therefore low volatilities for out-of-the-money (high-strike-price) calls. A volatility smile where volatility is a decreasing function of strike price results.Problem 20.6.The market price of a European call is $3.00 and its price given by Black-Scholes-Merton model with a volatility of 30% is $3.50. The price given by this Black-Scholes-Merton modelfor a European put option with the same strike price and time to maturity is $1.00. What should the market price of the put option be? Explain the reasons for your answer.With the notation in the text bs bs rT qT c Ke p Se --+=+ mkt mkt rT qT c Ke p Se --+=+ It follows that bs mkt bs mkt c c p p -=-In this case, mkt 300c =.; bs 350c =.; and bs 100p =.. It follows that mkt p should be 0.50.Problem 20.7.Explain what is meant by crashophobia.The crashophobia argument is an attempt to explain the pronounced volatility skew in equity markets since 1987. (This was the year equity markets shocked everyone by crashing more than 20% in one day). The argument is that traders are concerned about another crash and as a result increase the price of out-of-the-money puts. This creates the volatility skew.Problem 20.8.A stock price is currently $20. Tomorrow, news is expected to be announced that will either increase the price by $5 or decrease the price by $5. What are the problems in using Black –Scholes-Merton to value one-month options on the stock?The probability distribution of the stock price in one month is not lognormal. Possibly it consists of two lognormal distributions superimposed upon each other and is bimodal.Black –Scholes is clearly inappropriate, because it assumes that the stock price at any future time is lognormal.Problem 20.9.What volatility smile is likely to be observed for six-month options when the volatility is uncertain and positively correlated to the stock price?When the asset price is positively correlated with volatility, the volatility tends to increase as the asset price increases, producing less heavy left tails and heavier right tails. Implied volatility then increases with the strike price.Problem 20.10.What problems do you think would be encountered in testing a stock option pricing model empirically?There are a number of problems in testing an option pricing model empirically. These include the problem of obtaining synchronous data on stock prices and option prices, the problem of estimating the dividends that will be paid on the stock during the option’s life, the pr oblem of distinguishing between situations where the market is inefficient and situations where the option pricing model is incorrect, and the problems of estimating stock price volatility.Problem 20.11.Suppose that a central bank’s policy is to allow an exchange rate to fluctuate between 0.97and 1.03. What pattern of implied volatilities for options on the exchange rate would you expect to see?In this case the probability distribution of the exchange rate has a thin left tail and a thin right tail relative to the lognormal distribution. We are in the opposite situation to that describedfor foreign currencies in Section 20.2. Both out-of-the-money and in-the-money calls and puts can be expected to have lower implied volatilities than at-the-money calls and puts. The pattern of implied volatilities is likely to be similar to Figure 20.7.Problem 20.12.Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?A deep-out-of-the-money option has a low value. Decreases in its volatility reduce its value. However, this reduction is small because the value can never go below zero. Increases in its volatility, on the other hand, can lead to significant percentage increases in the value of the option. The option does, therefore, have some of the same attributes as an option on volatility. Problem 20.13.A European call option on a certain stock has a strike price of $30, a time to maturity of one year, and an implied volatility of 30%. A European put option on the same stock has a strike price of $30, a time to maturity of one year, and an implied volatility of 33%. What is the arbitrage opportunity open to a trader? Does the arbitrage work only when the lognormal assumption underlying Black–Scholes–Merton holds? Explain carefully the reasons for your answer.Put–call parity implies that European put and call options have the same implied volatility. If a call option has an implied volatility of 30% and a put option has an implied volatility of 33%, the call is priced too low relative to the put. The correct trading strategy is to buy the call, sell the put and short the stock. This does not depend on the lognormal assumption underlying Black–Scholes–Merton. Put–call parity is true for any set of assumptions. Problem 20.14.Suppose that the result of a major lawsuit affecting a company is due to be announced tomorrow. The company’s stock price is currently $60. If the ruling is favorable to the company, the stock price is expected to jump to $75. If it is unfavorable, the stock is expected to jump to $50. What is the risk-neutral probability of a favorable ruling? Assume that the volatility of the company’s stock will be 25% for six months after the ruling if the ruling is favorable and 40% if it is unfavorable. Use DerivaGem to calculate the relationship between implied volatility and strike price for six-month European options on the company today. The company does not pay dividends. Assume that the six-month risk-free rate is 6%. Consider call options with strike prices of $30, $40, $50, $60, $70, and $80.Suppose that p is the probability of a favorable ruling. The expected price of the company’s stock tomorrow isp p p+-=+7550(1)5025This must be the price of the stock today. (We ignore the expected return to an investor over one day.) Hence+=p502560or 04p =..If the ruling is favorable, the volatility, σ, will be 25%. Other option parameters are075S =, 006r =., and 05T =.. For a value of K equal to 50, DerivaGem gives the value of a European call option price as 26.502.If the ruling is unfavorable, the volatility, σ will be 40% Other option parameters are050S =, 006r =., and 05T =.. For a value of K equal to 50, DerivaGem gives the value of a European call option price as 6.310.The value today of a European call option with a strike price today is the weighted average of 26.502 and 6.310 or:042650206631014387.⨯.+.⨯.=. DerivaGem can be used to calculate the implied volatility when the option has this price. The parameter values are 060S =, 50K =, 05T =., 006r =. and 14387c =.. The implied volatility is 47.76%.These calculations can be repeated for other strike prices. The results are shown in the table below. The pattern of implied volatilities is shown in Figure S20.1.Figure S20.1: Implied Volatilities in Problem 20.14Problem 20.15.An exchange rate is currently 0.8000. The volatility of the exchange rate is quoted as 12% and interest rates in the two countries are the same. Using the lognormal assumption,estimate the probability that the exchange rate in three months will be (a) less than 0.7000, (b) between 0.7000 and 0.7500, (c) between 0.7500 and 0.8000, (d) between 0.8000 and 0.8500, (e) between 0.8500 and 0.9000, and (f) greater than 0.9000. Based on the volatility smile usually observed in the market for exchange rates, which of these estimates would you expectto be too low and which would you expect to be too high?As pointed out in Chapters 5 and 17 an exchange rate behaves like a stock that provides a dividend yield equal to the foreign risk-free rate. Whereas the growth rate in anon-dividend-paying stock in a risk-neutral world is r , the growth rate in the exchange rate in a risk-neutral world is f r r -. Exchange rates have low systematic risks and so we can reasonably assume that this is also the growth rate in the real world. In this case the foreign risk-free rate equals the domestic risk-free rate (f r r =). The expected growth rate in the exchange rate is therefore zero. If T S is the exchange rate at time T its probability distribution is given by equation (14.3) with 0=μ:()T T S S T 220,2/ln ~ln σσϕ-where 0S is the exchange rate at time zero and σ is the volatility of the exchange rate. In this case 008000S =. and 012=.σ, and 025T =. so that()25.012.0,2/25.012.08.0ln ~ln 22⨯⨯-ϕT Sor ()206.0,2249.0~ln -ϕT Sa) ln 0.70 = –0.3567. The probability that 070T S <. is the same as the probability thatln 03567T S <-.. It is0356702249(21955)006N N -.+.⎛⎫=-. ⎪.⎝⎭This is 1.41%.b) ln 0.75 = –0.2877. The probability that 075T S <. is the same as the probability thatln 02877T S <-.. It is028*******(10456)006N N -.+.⎛⎫=-. ⎪.⎝⎭This is 14.79%. The probability that the exchange rate is between 0.70 and 0.75 is therefore 14791411338%.-.=..c) ln 0.80 = –0.2231. The probability that 080T S <. is the same as the probability thatln 02231T S <-.. It is022*******(00300)006N N -.+.⎛⎫=. ⎪.⎝⎭This is 51.20%. The probability that the exchange rate is between 0.75 and 0.80 is therefore 512014793641%.-.=..d) ln 0.85 = –0.1625. The probability that 085T S <. is the same as the probability thatln 01625T S <-.. It is0162502249(10404)006N N -.+.⎛⎫=. ⎪.⎝⎭This is 85.09%. The probability that the exchange rate is between 0.80 and 0.85 istherefore 850951203389%.-.=..e) ln 0.90 = –0.1054. The probability that 090T S <. is the same as the probability thatln 01054T S <-.. It is010*******(19931)006N N -.+.⎛⎫=. ⎪.⎝⎭This is 97.69%. The probability that the exchange rate is between 0.85 and 0.90 is therefore 976985091260%.-.=..f) The probability that the exchange rate is greater than 0.90 is 1009769231%-.=..The volatility smile encountered for foreign exchange options is shown in Figure 20.1 of the text and implies the probability distribution in Figure 20.2. Figure 20.2 suggests that wewould expect the probabilities in (a), (c), (d), and (f) to be too low and the probabilities in (b) and (e) to be too high.Problem 20.16.A stock price is $40. A six-month European call option on the stock with a strike price of $30 has an implied volatility of 35%. A six-month European call option on the stock with a strike price of $50 has an implied volatility of 28%. The six-month risk-free rate is 5% and no dividends are expected. Explain why the two implied volatilities are different. UseDerivaGem to calculate the prices of the two options. Use put –call parity to calculate the prices of six-month European put options with strike prices of $30 and $50. Use DerivaGem to calculate the implied volatilities of these two put options.The difference between the two implied volatilities is consistent with Figure 20.3 in the text. For equities the volatility smile is downward sloping. A high strike price option has a lower implied volatility than a low strike price option. The reason is that traders consider that the probability of a large downward movement in the stock price is higher than that predicted by the lognormal probability distribution. The implied distribution assumed by traders is shown in Figure 20.4.To use DerivaGem to calculate the price of the first option, proceed as follows. Select Equity as the Underlying Type in the first worksheet. Select Black-Scholes European as the Option Type. Input the stock price as 40, volatility as 35%, risk-free rate as 5%, time to exercise as 0.5 year, and exercise price as 30. Leave the dividend table blank because we are assuming no dividends. Select the button corresponding to call. Do not select the implied volatility button. Hit the Enter key and click on calculate. DerivaGem will show the price of the option as 11.155. Change the volatility to 28% and the strike price to 50. Hit the Enter key and click on calculate. DerivaGem will show the price of the option as 0.725. Put –call parity is 0rT c Ke p S -+=+ so that 0rT p c Ke S -=+-For the first option, 11155c =., 040S =, 0054r =., 30K =, and 05T =. so that005051115530400414p e -.⨯.=.+-=.For the second option, 0725c =., 040S =, 006r =., 50K =, and 05T =. so that00505072550409490p e -.⨯.=.+-=.To use DerivaGem to calculate the implied volatility of the first put option, input the stock price as 40, the risk-free rate as 5%, time to exercise as 0.5 year, and the exercise price as 30. Input the price as 0.414 in the second half of the Option Data table. Select the buttons for a put option and implied volatility. Hit the Enter key and click on calculate. DerivaGem will show the implied volatility as 34.99%.Similarly, to use DerivaGem to calculate the implied volatility of the first put option, input the stock price as 40, the risk-free rate as 5%, time to exercise as 0.5 year, and the exercise price as 50. Input the price as 9.490 in the second half of the Option Data table. Select the buttons for a put option and implied volatility. Hit the Enter key and click on calculate. DerivaGem will show the implied volatility as 27.99%.These results are what we would expect. DerivaGem gives the implied volatility of a put with strike price 30 to be almost exactly the same as the implied volatility of a call with a strike price of 30. Similarly, it gives the implied volatility of a put with strike price 50 to be almost exactly the same as the implied volatility of a call with a strike price of 50.Problem 20.17.“The Black–Scholes–Merton model is used by traders as an interpolation tool.” Discuss this view.When plain vanilla call and put options are being priced, traders do use theBlack-Scholes-Merton model as an interpolation tool. They calculate implied volatilities for the options whose prices they can observe in the market. By interpolating between strike prices and between times to maturity, they estimate implied volatilities for other options. These implied volatilities are then substituted into Black-Scholes-Merton to calculate prices for these options. In practice much of the work in producing a table such as Table 20.2 in the over-the-counter market is done by brokers. Brokers often act as intermediaries between participants in the over-the-counter market and usually have more information on the trades taking place than any individual financial institution. The brokers provide a table such as Table 20.2 to their clients as a service.Problem 20.18.Using Table 20.2 calculate the implied volatility a trader would use for an 8-month option with K/S0 = 1.04.The implied volatility is 13.45%. We get the same answer by (a) interpolating between strike prices of 1.00 and 1.05 and then between maturities six months and one year and (b) interpolating between maturities of six months and one year and then between strike prices of 1.00 and 1.05.Further QuestionsProblem 20.19.A company’s stock is selling for $4. The company has no outstanding debt. Analysts consider the liquidation value of the company to be at least $300,000 and there are 100,000 shares outstanding. What volatility smile would you expect to see?In liquidation the company’s stock price must be at least 300,000/100,000 = $3. The company’s stock price should therefore always be at least $3. This means that the stock price distribution that has a thinner left tail and fatter right tail than the lognormal distribution. Anupward sloping volatility smile can be expected.Problem 20.20.A company is currently awaiting the outcome of a major lawsuit. This is expected to be known within one month. The stock price is currently $20. If the outcome is positive, the stock price is expected to be $24 at the end of one month. If the outcome is negative, it is expected to be $18 at this time. The one-month risk-free interest rate is 8% per annum.a.What is the risk-neutral probability of a positive outcome?b.What are the values of one-month call options with strike prices of $19, $20, $21, $22,and $23?e DerivaGem to calculate a volatility smile for one-month call options.d.Verify that the same volatility smile is obtained for one-month put options.a.If p is the risk-neutral probability of a positive outcome (stock price rises to $24),we must have00800833+-=p p e.⨯.2418(1)20so that 0356p=.b.The price of a call option with strike price K is 008008333(24)-when 24K pe-.⨯.K<.Call options with strike prices of 19, 20, 21, 22, and 23 therefore have prices 1.766,1.413, 1.060, 0.707, and 0.353, respectively.c.From DerivaGem the implied volatilities of the options with strike prices of 19, 20, 21,22, and 23 are 49.8%, 58.7%, 61.7%, 60.2%, and 53.4%, respectively. The volatilitysmile is therefore a “frown” with the volatilities for deep-out-of-the-money anddeep-in-the-money options being lower than those for close-to-the-money options.d.The price of a put option with strike price K is 008008333--. PutK p e-.⨯.(18)(1)options with strike prices of 19, 20, 21, 22, and 23 therefore have prices of 0.640,1.280, 1.920,2.560, and3.200. DerivaGem gives the implied volatilities as 49.81%,58.68%, 61.69%, 60.21%, and 53.38%. Allowing for rounding errors these are thesame as the implied volatilities for put options.Problem 20.21. (Excel file)A futures price is currently $40. The risk-free interest rate is 5%. Some news is expected tomorrow that will cause the volatility over the next three months to be either 10% or 30%. There is a 60% chance of the first outcome and a 40% chance of the second outcome. Use DerivaGem to calculate a volatility smile for three-month options.The calculations are shown in the following table. For example, when the strike price is 34, the price of a call option with a volatility of 10% is 5.926, and the price of a call option when the volatility is 30% is 6.312. When there is a 60% chance of the first volatility and 40% of.⨯.+.⨯.=.. The implied volatility given by this the second, the price is 0659260463126080price is 23.21. The table shows that the uncertainty about volatility leads to a classic volatility smile similar to that in Figure 20.1 of the text. In general when volatility is stochastic with the stock price and volatility uncorrelated we get a pattern of implied volatilities similar to that observed for currency options.Problem 20.22. (Excel file)Data for a number of foreign currencies are provided on the author’s Web site:http://www.rotman.utoronto.ca/~hull/dataChoose a currency and use the data to produce a table similar to Table 20.1.The data provided for this problem is new to the 8th edition. The following table shows the percentage of daily returns greater than 1, 2, 3, 4, 5, and 6 standard deviations for each currency. The pattern is similar to that in Table 20.1.Problem 20.23. (Excel file)Data for a number of stock indices are provided on the author’s Web site:http://www.rotman.utoronto.ca/~hull/dataChoose an index and test whether a three standard deviation down movement happens more often than a three standard deviation up movement.The data provided for this problem is new to the 8Th edition. The percentage of times up and down movements happen are shown in the table below.As might be expected from the shape of the volatility smile large down movements occur more often than large up movements. However, the results are not significant at the 95% level. (The standard error of the Average >3sd down percentage is 0.185% and the standard error of the Average >3sd up percentage is 0.161%. The standard deviation of the difference between the two is 0.245%)Problem 20.24.Consider a European call and a European put with the same strike price and time to maturity.Show that they change in value by the same amount when the volatility increases from a level, 1σ, to a new level, 2σ within a short period of time. (Hint Use put –call parity.)Define 1c and 1p as the values of the call and the put when the volatility is 1σ. Define 2c and 2p as the values of the call and the put when the volatility is 2σ. From put –call parity101qT rT p S e c Ke --+=+202qT rT p S e c Ke --+=+If follows that1212p p c c -=-Problem 20.25.An exchange rate is currently 1.0 and the implied volatilities of six-month European options with strike prices 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, and 1.3 are 13%, 12%, 11%, 10%, 11%, 12%, and 13%. The domestic and foreign risk free rates are both 2.5%. Calculate the implied probability distribution using an approach similar to that used in the appendix for Example 20A.1. Compare it with the implied distribution where all the implied volatilities are 11.5%.Define: 1()for 0708T T g S g S =.≤<. 2()for 0809T T g S g S =.≤<. 3()for 0910T T g S g S =.≤<.4()for 1011T T g S g S =.≤<.5()for 1112T T g S g S =.≤<.6()for 1213T T g S g S =.≤<.The value of 1g can be calculated by interpolating to get the implied volatility for a six-month option with a strike price of 0.75 as 12.5%. This means that options with strike prices of 0.7, 0.75, and 0.8 have implied volatilities of 13%, 12.5% and 12%, respectively. From DerivaGem their prices are $0.2963, $0.2469, and $0.1976, respectively. Using equation (20A.1) with 075K =. and 005δ=. we get00250512(0296301976202469)00315005e g .⨯..+.-⨯.==.. Similar calculations show that 207241g =., 340788g =., 436766g =., g 5 = 0.7285, and 600898g =.. The total probability between 0.7 and 1.3 is the sum of these numbersmultiplied by 0.1 or 0.9329. If the volatility had been flat at 11.5% the values of 1g , 2g , 3g , 4g , 5g , and 6g would have been 0.0239, 0.9328, 4.2248, 3.7590, 0.9613, and 0.0938. The total probability between 0.7 and 1.3 is in this case 0.9996. This shows that the volatility smile gives rise to heavy tails for the distribution.Problem 20.26.Using Table 20.2 calculate the implied volatility a trader would use for an 11-month option with K/S0= 0.98Interpolation gives the volatility for a six-month option with a strike price of 98 as 12.82%. Interpolation also gives the volatility for a 12-month option with a strike price of 98 as 13.7%.A final interpolation gives the volatility of an 11-month option with a strike price of 98 as13.55%. The same answer is obtained if the sequence in which the interpolations are done is reversed.。

期权期货与其他衍生产品第九版课后习题与答案Chapter (26)

期权期货与其他衍生产品第九版课后习题与答案Chapter (26)
Since cdo c cdi
cdo c S0e qT ( H S0 )2 N ( y) Ke rT ( H S0 )2 2 N ( y T ) From the formulas in the text y1 y when H K . The two expression for cdo are therefore equivalent when H K
cdo S0 N ( x1 )e qT Ke rT N ( x1 T ) S0e qT ( H S0 )2 N ( y1 )
Ke rT ( H S0 )2 2 N ( y1 T ) Substituting H K and noting that r q 2 2 2 we obtain x1 d1 so that
cdo c S0e qT ( H S0 )2 N ( y1 ) Ke rT ( H S0 )2 2 N ( y1 T ) The formula for cdi when H K is
cdi S0e qT ( H S0 )2 N ( y) Ke rT ( H S0 )2 2 N ( y T )
e q (T2 T1 ) call options with strike price Ke( r q )(T2 T1 ) and maturity T1 .
Problem 26.6. Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, H , is less than or equal to the strike price, K . The second applies to the situation where H K . Show that the two formulas are the same when H K . Consider the formula for cdo when H K

《金融理论与公司政策(第四版)》课后答案

《金融理论与公司政策(第四版)》课后答案

= .18 d2 = .18 – ( .0961) (.6538) = –.02 Substituting these values into the Black-Scholes Formula, C = 44.375N(.18) – 45e N(–.02)
–.03
Using the Table of Normal Areas, we can determine N(.18) and N(–.02). Substituting these values into the formula yields C = 44.375(.5714) – 45(.9704)(.4920) = 25.3559 – 21.4847 = $3.87 5. Compare the payoffs at maturity of two portfolios. The first is a European put option with exercise price X1, and the second is a European put option written on the same stock, with the same time to maturity, but with exercise price X2 < X1. The payoffs are given in Table S7.1. Because portfolio A has a value either greater than or equal to the value of portfolio B in every possible state of nature, the put with a higher exercise price is more valuable. P(S, T, X1) > P(S, T, X2) Table S7.1 Portfolio a) P(S, T, X1) b) P(S, T, X2) Comparative Value of A and B S < X2 X1 – S X2 – S VA > VB if X2 < X1. X1 ≤ S 0 0 VA = VB

金融分析师(CFA)一级考试课本总结---Fix-income-derevat

金融分析师(CFA)一级考试课本总结---Fix-income-derevat

金融分析师(CFA)一级考试课本总结---Fix-income-derevat金融分析师(CFA)一级考试课本总结- Fix income derevatIndenture是借贷双方的合约。

zero-coupon bonds,到期付par value,中间不付息,所以高折价发行,一般用半年期折现accrual bonds,类似zero coupon,以par value发行,有coupon rate,按利息按复利,到期结算step-up notes,coupon rate逐渐上涨。

Fix income derevative and alternative investmentsIndenture是借贷双方的合约。

zero-coupon bonds,到期付par value,中间不付息,所以高折价发行,一般用半年期折现accrual bonds,类似zero coupon,以par value发行,有coupon rate,按利息按复利,到期结算step-up notes,coupon rate逐渐上涨deferred-coupon bonds,第一次付息推迟。

浮息债券new coupon rate=reference rate+-quoted margin,upper limit 叫cap,lower limit叫floor,组合叫collar accrual bonds在付息日之间交易,有clean price和full price,计算交易日为止未付的利息。

Bond中的Optioncall feature,发债人可以以高于par value的价格买回,在call protection时期不能买回。

prepayment option,允许发债人提前支付本金给amortizing securityput feature,允许bondholder 提前收回principalconversion option,允许bondholder转换一定数量的普通股;如允许交换别的公司股票,叫做 exchange option回购协议,repo,卖证券的公司承诺在特定时间特定价格买回证券。

CFA考试重点:Put call parity的问题

CFA考试重点:Put call parity的问题

CFA考试重点:Put call parity的问题
公式:
l Note that the options much be European-style and the puts and calls mush have the same exercise price for these relations to hold.
l Fiduciary call: buy riskless bond that pays X at maturity and a call with exercise price X.
l Protective put: buy security and long put.
l Put-call parity 的作用:
ü 为option定价
ü 合成option。

ü 无风险套利。

在无风险套利中,遵循long小边,short大边原则
如:

套利方法是,long security, long put, short call, short bonds.
各位考生,2015年CFA备考已经开始,为了方便各位考生能更加系统地掌握考试大纲的重点知识,帮助大家充分备考,体验实战,网校开通了全免费的高顿题库(包括精题真题和全真模考系统),题库里附有详细的答案解析,学员可以通过多种题型加强练习,通过针对性地训练与模考,对学习过程进行全面总结。

put-call parity exercises

put-call parity exercises

MULTIPLE CHOICE PROBLEMS(d) 1 A one year call option has a strike price of 55, expires in 6 months, and has a priceof $5.04. If the risk free rate is 5%, and the current stock price is $50, whatshould the corresponding put be worth?a) $3.04b) $4.64c) $6.08d) $8.46e) None of the above(d) 2 A one year call option has a strike price of 50, expires in 6 months, and has a priceof $4.74. If the risk free rate is 6%, and the current stock price is $45, whatshould the corresponding put be worth?b) $2.74a) $5.48c) $6.08d) $8.30e) None of the above(c) 3 A one year call option has a strike price of 60, expires in 6 months, and has a priceof $4.54. If the risk free rate is 7%, and the current stock price is $55, whatshould the corresponding put be worth?a) $3.54b) $4.56c) $7.54d) $7.08e) None of the above(d) 4 A one year call option has a strike price of 70, expires in 6 months, and has a priceof $7.34. If the risk free rate is 6%, and the current stock price is $62, whatshould the corresponding put be worth?a) $5.34b) $8.00c) $10.68d) $13.33e) None of the aboveUSE THE FOLLOWING INFORMATION FOR THE NEXT THREE PROBLEMSCurrently, a September corn futures contract is priced at 315 cents per bushel, with one contract equal to 5,000 bushels. Your broker requires an initial margin of 5 percent on futures contracts.(e) 5 How much must your deposit in a margin account if you wish to purchase 5contracts?a) $315.00b) $700.50c) $750.50d) $1,500.00e) $3,937.50(c) 6 Suppose the contract expires at a price of 320 cents per bushel. Disregardingtransaction costs, what is your percentage return?a) 1.07%b) 2.38%c) 31.75%d) 50.00%e) None of the above(d) 7 Suppose the contract expires at 325 cents per bushel. Disregarding transactionscosts, what is your percentage return?a) 2.14%b) 4.76%c) 31.75%d) 63.49%e) None of the aboveUSE THE FOLLOWING INFORMATION FOR THE NEXT THREE PROBLEMSCurrently, a September corn futures contract is priced at 205 cents per bushel, with one contract equal to 5,000 bushels. Your broker requires an initial margin of 6 percent on futures contracts.(e) 8 How much must you deposit in a margin account if you wish to purchase 3 contracts?a) $205b) $615c) $750d) $1,500e) $1,845(c) 9 Suppose the contract expires at a price of 220 cents per bushel. Disregardingtransaction costs, what is your percentage return?a) 107%b) 238%c) 121.95%d) 163.52%e) None of the above(d) 10 Suppose the contract expires at 210 cents per bushel. Disregarding transactionscosts, what is your percentage return?a) 2.14%b) 5.62%c) 46.89%d) 40.65%e) None of the aboveUSE THE FOLLOWING INFORMATION FOR THE NEXT TWO PROBLEMSOn the last day of October, Bruce Springsteen is considering the purchase of 100 shares of Olivia Corporation common stock selling at $37 1/2 per share and also considering an Olivia option. Calls PutsPrice December March December March35 3 3/4 5 1 1/4 240 2 1/2 3 1/2 4 1/2 4 3/4(d) 11 If Bruce decides to buy a March call option with an exercise price of 35, what is hisdollar gain (loss) if he closes his position when the stock is selling at 43 1/2?a) $225.00 lossb) $350.00 lossc) $225.00 gaind) $350.00 gaine) $850.00 gain(b) 12 If Bruce buys a March put option with an exercise price of 40, what is his dollargain (loss) if he closes his position when the stock is selling at 43 1/2?a) $825.00 lossb) $475.00 lossc) $350.00 lossd) $25.00 losse) He has a gainUSE THE FOLLOWING INFORMATION FOR THE NEXT TWO PROBLEMSRick Thompson is considering the following alternatives for investing in Davis Industries which is now selling for $44 per share:1) Buy 500 shares, and2) Buy six month call options with an exercise price of 45 for $3.25 premium.(a) 13 Assuming no commissions or taxes what is the annualized percentage gain if thestock reaches $50 in four months and a call was purchased?a) 161.54% gainb) 53.85% gainc) 161.54% lossd) 11.11% gaine) 53.85% loss(b) 14 Assuming no commissions or taxes, what is the annualized percentage gain if thestock is at $30 in four months and the stock was purchased?a) 9.54% lossb) 95.45% lossc) 0.9545% gaind) 95.45% gaine) 9.54% gain(a) 15 Tom Gettback buys 100 shares of Johnson Walker stock for $87.00 per share and a3-month Johnson Walker put option with an exercise price of $105.00 for $20.00.What is his dollar gain if at expiration the stock is selling for $80.00 per share?a) $200 lossb) $700 lossc) $200 gaind) $700 gaine) None of the above(b) 16 Tom Gettback buys 100 shares of Johnson Walker stock for $87.00 per share and a3-month Johnson Walker put option with an exercise price of $105.00 for $20.00.What is Tom’s dollar gain/loss if at expiration the stock is selling for $105.00 pershare?a) $1000 gainb) $200 lossc) $1000 lossd) $200 gaine) None of the aboveUSE THE FOLLOWING INFORMATION FOR THE NEXT FOUR PROBLEMSSarah Kling bought a 6-month Peppy Cola put option with an exercise price of $55 for apremium of $8.25 when Peppy was selling for $48.00 per share.(d) 17 If at expiration Peppy is selling for $42.00, what is Sarah’s dollar gain or loss?a) $420 gainb) $420 lossc) $475 lossd) $475 gaine) None of the above(b) 18 What is Sarah’s annualized gain/loss?a) 11.51% gainb) 115.15% gainc) 11.51% lossd) 115.15% losse) None of the above(a) 19 If at expiration Peppy is selling for $47.00, what is Sarah’s dollar gain or loss?a) $25 lossb) $250 lossc) $25 gaind) $250 gaine) None of the above(b) 20 What is Sarah’s annualized gain/loss?a) 60.60% gainb) 6.06% lossc) 60.60% lossd) 6.06% gaine)None of the above(a) 21 A stock currently trades for $25. January call options with a strike price of $30sell for $6. The appropriate risk free bond has a price of $30. Calculate the priceof the January put option.a) $11b) $24c) $19d) $30e) $25(e) 22 A stock currently trades for $115. January call options with a strike price of $100sell for $16, and January put options a strike price of $100 sell for $5. Estimatethe price of a risk free bond.a) $120b) $15c) $105d) $116e) $104(d) 23 Assume that you have purchased a call option with a strike price $60 for $5. Atthe same time you purchase a put option on the same stock with a strike price of$60 for $4. If the stock is currently selling for $75 per share, calculate the dollarreturn on this option strategy.a) $10b) -$4c) $5d) $6e) $15(c) 24 Assume that you purchased shares of a stock at a price of $35 per share. At thistime you purchased a put option with a $35 strike price of $3. The stock currentlytrades at $40. Calculate the dollar return on this option strategy.a) $3b) -$2c) $2d) -$3e) $0(c) 25 Assume that you purchased shares of a stock at a price of $35 per share. At thistime you wrote a call option with a $35 strike and received a call price of $2. Thestock currently trades at $70. Calculate the dollar return on this option strategy.a) $25b) -$2c) $2d) -$25e) $0(b) 26 A stock currently trades at $110. June call options on the stock with a strike priceof $105 are priced at $4. Calculate the arbitrage profit that you can earna) $0b) $1c) $5d) $4e) None of the above(c) 27 Datacorp stock currently trades at $50. August call options on the stock with astrike price of $55 are priced at $5.75. October call options with a strike price of$55 are priced at $6.25. Calculate the value of the time premium between theAugust and October options.a) -$0.50b) $0c) $0.50d) $5e) -$5(a) 28 A stock currently trades at $110. June put options on the stock with a strike priceof $115 are priced at $5.25. Calculate the dollar return on one put contract.a) -$25b) $500c) $0d) -$75e) $525(d) 29 A stock currently trades at $110. June call options on the stock with a strike priceof $105 are priced at $5.75. Calculate the dollar return on one call contract.a) -$50b) $500c) $575d) -$75e) $0(e) 30 Consider a stock that is currently trading at $50. Calculate the intrinsic value for aput option that has an exercise price of $55.a) $0b) $50c) $55d) $105e) $5(a) 31 Consider a stock that is currently trading at $50. Calculate the intrinsic value for aput option that has an exercise price of $35.a) $0b) $50c) $35d) $15e) $85(e) 32 Consider a stock that is currently trading at $25. Calculate the intrinsic value for acall option that has an exercise price of $35.a) $25b) $35c) $10d) -$10e) $0(c) 33 Consider a stock that is currently trading at $25. Calculate the intrinsic value for acall option that has an exercise price of $15.a) $25b) $35c) $10d) $60e) $0CHAPTER 21ANSWERS TO PROBLEMS1 p(t) = $5.04 - $50+ $55(1 + .05)-½= $5.04 - $50 + $53.42 = $8.462 p(t) = $4.74 - $45+ $50(1 + .06)-½= $4.74 - $45 + $48.56 = $8.303 p(t) = $4.54 - $55+ $60(1 + .07) -½= $4.54 - $55 + $58.00 = $7.544 p(t) = $7.34 - $62+ $70(1 + .06) -½= $7.34 - $62 + $67.99 = $13.335 (5)(3.15)(5000)(0.05) = $3,937.56 (320 - 315) ÷ (0.05)(315) = 31.75%7 (325 - 315) ÷ (0.05)(315) = 63.49%8 (3)(2.05)(5000)(0.06) = $1,845.009 (220 - 205) ÷ (0.06)(205) = 121.95%10 (210 - 205) ÷ (0.06)(205) = 40.65%11 43 1/2 - 35 = 8.58.5 - 5 = 3.5 $3.5/share x 100 shares/contract = $350.0012 The option is worthless so he loses the $475 he paid for the contract.13 [(50 - 45 - 3.25) ÷ 3.25] x 3 = 161.54% gain14 [(30 - 44) ÷ 44] x 3 = 95.45% loss15 Profit on put = 105 - 80 - 20 = 55 x 100 = $500.00Loss on stock = $700.00Net loss = $700.00 - 500.00 = $200.00 (loss) 16 Put value = 0, therefore, loss = $2,000.00Stock (105 - 87)(100) = $1,800.00Net loss = $2,000 - 1,800 = $200.00 (loss)17 [(55 - 42 - 8.25) x 100] = $475 gain18 [(55 - 42 - 8.25) ÷ 8.25] x 2 = 115.15% gain19 [(55 - 47 - 8.25) x 100] = $25 loss20 [(55 - 47 - 8.25) ÷ 8.25] x 2 = 6.06% loss21 P = 6 + 30 –25 = $1122 Bond price = 115 + 5 – 16 = $10423 Profit on call = (75 – 60) – 5 = 10Profit on put = -4Total = $624 Profit on stock = 40 – 35 = 5Profit on put = -3Total = $225 Profit on stock = 70 – 35 = 25Profit on call = 35 – 70 + 2 = -23Total = $226 Arbitrage profit = 110 – 105 – 4 = $127 Time premium = 6.25 – 5.75 = $0.5028 Dollar return = (115 – 110 – 5.25)(100) = -$2529 Dollar return = (110 – 105 – 5.75)(100) = -$7530 Put = Max[55 – 50, 0] = $531 Put = Max[35 – 50, 0] = $032 Call = Max[25 – 35, 0] = $033 Call = Max[25 – 15, 0] = $1021 - 11。

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