Consider a four-period model binomial asset pricing model, with S0 = 4, u = 2, d

Question

Consider a four-period model binomial asset pricing model, with S0 = 4, u =
2, d =1/2, and take the interest rate r =1/4. For simplicity, r is measured with a
compounding frequency reflecting the length of each period, so that p = q =1/2. For
n = 0, 1, 2, 3, 4, define Yn =SIGMA k=0 Sk to be the sum of the stock prices between times zero and n. Consider an Asian call option that expires at time four and has strike K = 4 (i.e. whose payoff at time four is max(1/5Y4 ? 4, 0). This is like an European call, except the payoff of the option is based on the average stock price rather than
the final stock price. Let vn(s, y) denote the price of this option at time n if Sn = s and Yn = y. In particular,v4(s, y) = max(1/5 Y4 ? 4, 0).

1. Develop an algorithm for computing vn recursively. In particular, write a formula for vn in term of vn+1.
2. Apply the algorithm developed in (1) to compute v0(4, 4), the price of the Asian option at time zero.
3. Provide a formula for ?n(s, y), the number of shares of stock that should be held by the replicating portfolio at time n if Sn = s and Yn = y.

Explanation / Answer

3Ans:

Let Yn =

Pn

k=0 Sk. An Asian option expiring at time n and with

the strike price K is de¯ned via the payo®

Vn = (

1

n

Yn ¡ K)+:

(a) Let vk(s; y) denote the price of this option at time k and ±n(s; y) the number of

shares of stock that should be held by the replicating portfolio at time k, if Sk = s; Yk =

y. In particular vn(s; y) = (¡K + y=n)+. Develop an algorithm for computing vk, ±k

recursively, that is, write vk, ±k in terms of vk+1.

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