14-Arbitrage Page 428 Wednesday, February 4, 2004 1:08 PM
428 The Mathematics of Financial Modeling and Investment Management
1 + rd–
q = ---------------------
ud–
u – 1 – r
1 – q = ---------------------
ud–
Let’s introduce in this market a derivative instrument. The condition
of absence of arbitrage univocally determines the price of this third secu-
rity. Consider first a European call option on the stock with expiration
date τ < T and with exercise price K > 0. Recall from Chapter 2 that a
European call option is a security that gives its holder the right but not
the obligation to purchase the stock at time τ at price K. Therefore, the
payoff process of the option is zero before time τ and, at time τ, is
Cττ = max(Sτ – K, 0 )
Let’s compute the value of the option Ct τ at any time 0 < t < τ. Given
that the binomial model is complete, the value Ct τ can be computed as
the discounted payoff at time t using the risk-neutral probabilities.
Using the formulas of the previous sections, we can therefore write
C
τ
C Q τ
t
τ
= Et -------------------------
( 1 + r)τ – t
This formula can be explicitly computed as follows. The distribution
of the payoff of the option at time τ under the risk-neutral probabilities is
the following:
PCττ = (ukdτ – t – k
+
[ S 0 – K) ] =
τ
k
- t
qk( 1 – q)τ – t – k
Therefore the conditional expectation under the risk-neutral probabili-
ties becomes
τ – t
τ 1 τ – t – k
Ct = -------------------------∑(u
k
d
τ – t – k
S 0 – K)
+
τ
k
- t
qk( 1 – q)
( 1 + r)
τ – tk = 0