金融数学5-6

Example 18 (“bears”) A di¤erent person wins £ 1000 in a lottery and has a …rm belief that a particular company is bad and that its shares will fall in value. Assume that the current spot price of its share is S0 = $100 and the price of the put with strike K = $100 and maturity T on its one share is D = $1: Further, suppose they buy a put for 11 shares. Then at time T : maximum loss is £ 11; if ST = $90, the pro…t is £ 99. As for the writer (of both call and put), he may hope that market will move in the right for him direction and he will keep the premium. There are special names for those acting on the assumption of a rise or fall of some asset. The dealers expecting prices to go up are called bulls. A bull opens a long position expecting to sell with pro…t afterwards when the market goes up. Dealers who expect the market to move downwards are called bears. A bear tends to sell securities he has (or even has not – by short selling). He hopes to close his short position by buying the traded items afterwards at lower prices. Futures and options are similar instruments for speculators in that they both provide a way in which a type of leverage can be obtained. However, there is an important di¤erence between the two. With futures the speculator’ s potential loss as well as the potential gain is very large. With options, no matter how bad things get, the speculator’ s loss is limited to the amount paid for the option.
rTwenku.baidu.com
S0 + C0
$2:34:
This suggests that the quoted put with D0 = $2 is underpriced relative to the call. Then we can take advantage and can be sure of a risk-free pro…t by using the following strategy. t=0 Buy one put Buy one stock unit Write call Borrow Total value of portfolio
4.5
Put - Call Parity
The key question is how to price options. What is a fair price for both holder and writer? Options are more complicated than forwards, so how do we price them? Do they have an arbitrage price? We will address these questions in the near future but …rstly consider another feature of European options. There is an important and surprising relationship that exists between the value of European puts and calls with the same expiration date and strike price that are written on the same underlying stock. De…nition 10 A portfolio is a collection of security holdings. Let Ct and Dt be the prices of a call and put, respectively, at time t with strike price K and maturity time T on a stock of price St ; Bt be the price of a bond at time t and r be continuously compounded interest rate. We assume that the stock is not paying dividends. Consider two portfolios:
r (T t) r (T t)
= Dt + St :
This is the put-call parity relationship for European options that pay no dividends. We note that in deriving the put-call parity we did not need to assume any model for the price behaviour St ; we used the no arbitrage principle only. If the put-call parity does not hold, we can …nd a strategy which will give us a risk-free pro…t. This is illustrated in the following example. Example 19 Let S0 = $40, K = $40, T = 1 3 (i.e. 4 months), r = 0:05, C0 = $3 and D0 = $2. As the call is selling at £ 3; we know from the put-call parity relationship that the put should be worth D0 = Ke
1 VT
K )+ + er(T
t)
Bt .
6
BT
K
2 2. Vt2 = Dt + St ;which becomes at t = T : VT = (K
2 VT
ST
ST )+ + ST :
6
K
K ST
If BT = K (i.e. Bt = Ke
r (T
t)
) then
1 2 VT = VT ;
independently of any randomness. Thus, these ‘ stock plus put’and ‘ bond plus call’combinations have the same payo¤s in all possible future states of the world. As agreed before, we are assuming no arbitrage opportunities. This implies that the current values of the portfolios should be the same as well. So Ct + Ke or Ct = Dt + St Ke
@ M. Tretyakov, University of Leicester, 2005-10
21
4. EUROPEAN OPTIONS
Introduction to Financial Mathematics
1 1. Vt1 = Ct + Bt ; which becomes at t = T : VT = (ST
Introduction to Financial Mathematics
This was about just one share. Had we expanded this strategy to 10 option contracts for 100 shares each on both sides, our pro…t would have been £ 345. We conclude that if we can value a European call option, we will value a European put option as well using the put-call parity.
4. EUROPEAN OPTIONS
Introduction to Financial Mathematics
We have already considered a number of examples illustrating the use of options for hedging. Let us now look at how they can be used for speculation. Example 17 (“bulls”) You win £ 1000 in a lottery and are wondering what to do with the money. You love a particular company and strongly believe that their shares will go up. We assume r = 0 to neglect the time value of money. Suppose that S0 = $10, you buy 100 shares. Then at some time T in future: you could lose up to £ 1000; if ST = $11, your pro…t is £ 100, i.e. 10%. Alternatively, suppose you can buy a call option with K = $10 and price C = $0:10. You buy this option on 110 shares, so you spend £ 11 out of your £ 1000: Then at time T : you could lose a maximum of £ 11 (the premium); if ST = $11, you get a pro…t of $110 from your win. $11 = $99, i.e. 900%! You also have the remaining £ 989
@ M. Tretyakov, University of Leicester, 2005-10
$2 $40 $3 $39 $0 22
ST 40
T = 1=3 40 ST > 40 ST 0 ST ST 0 40 ST $39:655 $39:655 $0:345 $0:345
5. BINARY MODEL OF PRICE EVOLUTION