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 $(801.50) 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

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