The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 429 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 429

More generally, we give the following definition: A simple European

derivative instrument with expiration time τ is a financial instrument

whose payoff is zero for 0 ≤ t < τ and is an ℑτ-measurable random vari-

able Vτ at time τ. Recall from Chapter 6 that in this finite-state context, a

variable is ℑt-measurable if it assumes a constant value on each of the sets

of the partition It.

Given the risk-neutral probability measure Q, the value at time t of

the simple European derivative instrument can be computed as follows:

Q Vτ

Vt = Et -------------------------

( 1 + r)τ – t

If the underlying stock is represented by a binomial model, the value of

the European derivative instrument can be explicitly computed as:

τ – t

(^1) τ – t – k

Vt = -------------------------∑ Vτ

τ – t

qk( 1 – q)

( 1 + r)τ – tk = 0

k 

VALUATION OF AMERICAN OPTIONS

In order to define American options we have first to define the concept

of a stopping time. In fact, American options can be exercised at any

moment prior to expiration date in function of some exercising policy.

These policies define a stopping time. A stopping time is a random time

s, i.e., a random variable s such that

{ω ∈ Ω; s () ω = k} ∈ ℑk

Consider now an adapted process Xt and a stopping time s. Define a

payoff process ds as d s

t

s = 0 if t ≠ s and d

τ = Xs. Under the risk-neutral

probabilities we can write a valuation formula:

V Q Xs

t

s = E

t -----------------------– -

( 1 + r)st

These formulas allow the valuation of American securities in complete

markets.