The Mathematics of Financial Modelingand Investment Management

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