Quantitative Finance for Physicists

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 $(801.50) ¼

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

option has a gain of $(78:50  75) ¼ $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 þ 1:50) ¼$85:45. However, if the stock price falls to say $75,

the investor’s loss is $(80  85:45) ¼$5:45 rather than the loss of

$(75  83:95) ¼$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 $(5:45 þ 8:95) ¼ $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 F

u

or F

d

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  F

u

¼ SdD  F

d

(9:3:1)

Then the number of shares in this portfolio equals

D ¼ (F

u

 F

d

)=(Su  Sd) (9:3:2)

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

value (SuD  F

u

) ¼ (SdD  F

d

). If the time interval is t and the risk-

98 Option Pricing

free interest rate is r, the relation between the portfolio’s present value

and future value is

(SD  F) exp(rt) ¼ SuD  F

u

(9:3:3)

Combining (9.3.2) and (9.3.3) yields

F ¼ exp(rt)[pF

u

þ (1  p)F

d

](9:3:4)

where

p ¼ [ exp (rt)  d]=(u  d) (9:3:5)

The factors p and (1  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)] ¼ E[pSu þ (1  p)Sd] ¼ 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)

Su

2

Fuu

Su

Fu

Sud

S

F

Fud

Sd

Fd

Sd

2

Fdd

Figure 9.2 Two-step binomial pricing tree.

Option Pricing 99