期货与期权 中央财经大学 11 模块七：股票期权的性质——期权价格的上下限 (11.1.1) 模块七课件

主讲教师

?课程名称：期货与期权

?宋斌博士副教授

–中央财经大学管理科学与工程学院投资系–办公室：沙河校区4号楼308

Chapter 7

Property of Stock Options-Upper and Lower bound of Option Prices

Methods-Arbitrage Restriction

?1. Constructing Portfolios-Portfolio Comparison ?2.Direct Arbitrage Analysis

The Upper Bound for a Call Option Price

-European Call and American Call

Call option price can never be worth more than the stock price.

?Otherwise, arbitrage(riskless profit) will be possible.?How? C t>S t?

-European Call and American Call

–Suppose you see a call option selling for c t= $65, and the underlying stock is selling for St= $60.

–The Arbitrage: sell the call, and buy the stock：c-St=5 ＞0 (at time t)?Worst case:S T≥K? The option is exercised and you pocket K-S T+S T

=K>0(at time T).

?Best case:S T＜K? The stock sells for less than K at option expiration, you keep 0+S T =S T>0

–Zero cash outlay today, no possibility of loss, and potential for gain.

-European Call and American Call ?Calls

–The upper bound is C

Lower Bound for European Call Option Prices (No Dividends)

–If there are no dividend payouts, the lower bound is C >max [ 0, S -K e -rT ]

C S

0PV(K)C=S

C=S-PV(K)

K

(No Dividends)

?Consider the two portfolios

Portfolio A: One European Call option plus cash = Ke–rT

Portfolio B: One Share

?For PF A at expiration, T

–If the cash is invested, it will grow to K at T

–If S T> K, then the option is exercised; PF A is worth S T ?We pay for the share with the cash and have a share worth S T –If S T< K, then the option is worthless; PF A is worth K

–Hence PF A is worth max (S T, K)

(No Dividends)

?PF A is worth max (S T , K ) at expiration

?PF B at expiration is worth S T

?At expiration, PF A has value at least as great as PF B

?In the absence of arbitrage, this must also be true today

?Hence c + Ke -rT ≥S 0 or c ≥S 0–Ke

-rT ?In addition, since the worst that can happen is that

the call will expire worthless,()

0max ,0rT c S Ke ?≥?

Theoretical Implications of these Bounds ?Because the price of an in-the-money call must

exceed S-PV(K), in the money calls on non-

dividend paying stocks will always have some

time value before expiration.

?An American call on a non-dividend paying stock

will never be exercised before expiration. Why

exercise for S-K, when you can sell it for S-

PV(K) or more?

?The value of an American call on a non-

dividend paying stock is the same as the value

of a European call on the same stock, all else

equal.

When will an American call be

Exercised Early? The Logic ?Exercising an option early destroys its time value, so you will only want to exercise early if the option has no time value.?Exercising early also destroys the downside protection provided by a call (i.e., if S T< K), and requires spending $K earlier(thereby losing interest that can be earned on $K).

?So, the dividend that will be paid “tomorrow” must be sufficiently great to compensate you for these “costs” of early exercise.

?Assume the stock pays no dividends.

?Upper bounds are

–American: C

–European: c

?Lower bounds are

–American: C >max (0, S-K e-rT)

–European: c >max (0, S-K e-rT)

?Although we can obtain the American call option’s weak lower bound by arbitrage restriction, we still have C≥c.Therefore we choose the strict lower bound

American: C >max (0, S-K e-rT)

?From the analysis, we know the American call option (no dividend ) will never exercise early, we conclude that C=c.

The Upper Bound for a Put

Option Price

Put option price can never be worth more than the strike price. P < K

?Otherwise, arbitrage will be possible. How P ＞K? Suppose there is a put option with a strike price of K=$50 and this put is selling for P= $60.

?Suppose there is a put option with a strike price of K=$50 and this put is selling for P= $60.

?The Arbitrage: Sell the put, and invest the $60 in the bank. (Note you have zero cash outlay).+ Pt(sell the put option )-Pt(riskless lending)=0

–Worse case----K≥S T? Stock price goes to zero S T→0.

?You short put option payoff is S T–K, because you were the put writer).

?But, you have from the sale of the put (plus interest)

?Total payoff is: S T-K+Pt e rT>0

–Best case---K≤S T? Stock price is at least $50 at expiration.

?The put expires with zero value (and you are off the hook).

?You keep the entire $60, plus interest.

?Total payoff is: 0+Pt e rT>0

for European Put only

?European put option price can never be worth more than the strike price. P < K e-rT

?Otherwise, arbitrage will be possible.

for European Put only

?How P ＞K e-rT?Suppose there is a put option with a strike price of K and this put is selling for P ＞K e-rT.

?The Arbitrage: Sell the put, and invest the money in the bank. (Note you have zero cash outlay). ).+ Pt(sell the put option )-Pt(riskless lending)=0

–Worse case----K≥S T? Stock price goes to zero S T→0.

?You short put option payoff is S T–K, because you were the put writer.

?But, you have from the sale of the put (plus interest)

?Total payoff is: S T-K+Pt e rT>0 (Pt e rT>K)

–Best case---K≤S T? Stock price is at least $50 at expiration.

?The put expires with zero value (and you are off the hook).

?You keep the entire $60, plus interest.

?Total payoff is: 0+Pt e rT>0

?Lower bounds are

–American: P >max (0, K-S)

–European: P >max (0, K e-rT–S)

(No Dividends)

?Consider the two portfolios

Portfolio C: One European Put option plus One Share

Portfolio D: Cash = Ke–rT

?For PF C at expiration T

–If S T< K, then the option is exercised; PF C is worth K ?We deliver the share and receive K

–If S T> K, then the option is worthless; PF C is worth S T –Hence PF C is worth max (S T, K)

(No Dividends)

?PF C is worth max (S T , K ) at expiration

?PF D at expiration is worth K

?At expiration, PF C has value at least as great as PF D

?In the absence of arbitrage, this must also be

true today

?Hence p + S 0≥Ke -rT or p ≥Ke -rT –S 0

?Again, since the worst that can happen is that the put expires worthless,()

0max ,0rT p Ke S ?≥?