ñòð. 29 |

Long Put

10

5

Stock price

0

0 5 10 15 20 25 30 35 40

âˆ’5

âˆ’10

Short Put

âˆ’15

âˆ’20

Figure 9.1 The option profits for the strike price of $25 and the option

premium of $5: (a) calls, (b) puts.

98 Option Pricing

If, however, the IBM share price stays put through 3-Aug-03, an

option buyer incurs losses of $2.45 (i.e., 100%). In the mean time, a

share buyer has no losses and may continue to hold shares, hoping

that their price will grow in future.

At market closing on 7-Jul-03, the put option for the IBM share

with the strike price of $80 at maturity on 3-Aug-03 was $1.50. Hence,

buyers of this put option bet on price falling below $(80Ã€1.50) Â¼

$78.50. If, say the IBM stock price falls to $75, the buyer of the put

option has a gain of $(78:50 Ã€ 75) Â¼ $3.50.

Now, consider hedging in which the investor buys simultaneously

one share for $83.95 and a put option with the strike price of $80 for

$1.50. The investor has gains only if the stock price rises above

$(83:95 Ã¾ 1:50) Â¼$85:45. However, if the stock price falls to say $75,

the investorâ€™s loss is $(80 Ã€ 85:45) Â¼ Ã€$5:45 rather than the loss of

$(75 Ã€ 83:95) Â¼ Ã€$8:95 incurred without hedging with the put

option. Hence, in the given example, the hedging expense of $1.50

allows the investor to save $(Ã€5:45 Ã¾ 8:95) Â¼$3:40.

9.3 BINOMIAL TREES

Let us consider a simple yet instructive method for option pricing

that employs a discrete model called the binomial tree. This model is

based on the assumption that the current stock price S can change at

the next moment only to either the higher value Su or the lower value

Sd (where u > 1 and d < 1). Let us start with the first step of the

binomial tree (see Figure 9.2). Let the current option price be equal to

F and denote it with Fu or Fd at the next moment when the stock price

moves up or down, respectively. Consider now a portfolio that con-

sists of D long shares and one short option. This portfolio is risk-free

if its value does not depend on whether the stock price moves up or

down, that is,

SuD Ã€ Fu Â¼ SdD Ã€ Fd (9:3:1)

Then the number of shares in this portfolio equals

D Â¼ (Fu Ã€ Fd )=(Su Ã€ Sd) (9:3:2)

The risk-free portfolio with the current value (SD Ã€ F) has the future

value (SuD Ã€ Fu ) Â¼ (SdD Ã€ Fd ). If the time interval is t and the risk-

99

Option Pricing

Su2

Fuu

Su

Fu

Sud

S

F

Fud

Sd

Fd

Sd2

Fdd

Figure 9.2 Two-step binomial pricing tree.

free interest rate is r, the relation between the portfolioâ€™s present value

and future value is

(SD Ã€ F) exp(rt) Â¼ SuD Ã€ Fu (9:3:3)

Combining (9.3.2) and (9.3.3) yields

F Â¼ exp(Ã€rt)[pFu Ã¾ (1 Ã€ p)Fd ] (9:3:4)

where

p Â¼ [ exp (rt) Ã€ d]=(u Ã€ d) (9:3:5)

The factors p and (1 Ã€ p) in (9.3.4) have the sense of the probabilities

for the stock price to move up and down, respectively. Then, the

expectation of the stock price at time t is

E[S(t)] Â¼ E[pSu Ã¾ (1 Ã€ p)Sd] Â¼ S exp (rt) (9:3:6)

This means that the stock price grows on average with the risk-free

rate. The framework within which the assets grow with the risk-free

rate is called risk-neutral valuation. It can be discussed also in terms of

the arbitrage theorem [4]. Indeed, violation of the equality (9.3.3)

100 Option Pricing

implies that the arbitrage opportunity exists for the portfolio. For

example, if the left-hand side of (9.3.3) is greater than its right-hand

side, one can immediately make a profit by selling the portfolio and

buying the risk-free asset.

Let us proceed to the second step of the binomial tree. Using

equation (9.3.4), we receive the following relations between the option

prices on the first and second steps

Fu Â¼ exp (Ã€rt)[pFuu Ã¾ (1 Ã€ p)Fud ] (9:3:7)

Fd Â¼ exp (Ã€rt)[pFud Ã¾ (1 Ã€ p)Fdd ] (9:3:8)

The combination of (9.3.4) with (9.3.7) and (9.3.8) yields the current

option price in terms of the option prices at the next step

F Â¼ exp (Ã€2rt)[p2 Fuu Ã¾ 2p(1 Ã€ p)Fud Ã¾ (1 Ã€ p)2 Fdd ] (9:3:9)

This approach can be generalized for a tree with an arbitrary number

of steps. Namely, first the stock prices at every node are calculated

by going forward from the first node to the final nodes. When the

stock prices at the final nodes are known, we can determine the

option prices at the final nodes by using the relevant payoff relation

(e.g., (9.2.1) for the long call option). Then we calculate the option

prices at all other nodes by going backward from the final nodes to

the first node and using the recurrent relations similar to (9.3.7) and

(9.3.8).

The factors that determine the price change, u and d, can be

estimated from the known stock price volatility [1]. In particular, it

is generally assumed that prices follow the geometric Brownian

motion

dS Â¼ mSdt Ã¾ sSdW (9:3:10)

where m and s are the drift and diffusion parameters, respectively, and

dW is the standard Wiener process (see Section 4.2). Hence, the price

changes within the time interval [0, t] are described with the lognor-

mal distribution

pï¬ƒï¬ƒ

ln S(t) Â¼ N( ln S0 Ã¾ (m Ã€ s2 =2)t, s t) (9:3:11)

In (9.3.11), S0 Â¼ S(0), N(m, s) is the normal distribution with mean

m and standard deviation s. It follows from equation (9.3.11) that the

expectation of the stock price and its variance at time t equal

101

Option Pricing

ñòð. 29 |