Exercise 8.1

Consider a model for the stock market where the short rate of interest r is a deterministic constant. We focus on a particular stock with price process S. Under the objective probability measure P we have the following dynamics for the price process.

dS(t)=aS(t)dt+σS(t)dW (t) +δS(t−)dN(t).

Here W is a standard Wiener process whereas N is a Poisson process with intensity A. We assume that α, σ, δ and λ are known to us. The dN term is to be interpreted in the following way:

Between the jump times of the Poisson process N, the S-process behaves just like ordinary geometric Brownian motion.

If N has a jump at time t this induces S to have a jump at time t. The size of the S-jump is given by

S(t)-S(t-) = δ-S(t-).

Discuss the following questions.

(a) Is the model free of arbitrage?

(b) Is the model complete?

(c) Is there a unique arbitrage free price for, say, a European call option?

(d) Suppose that you want to replicate a European call option maturing in January 1999. Is it posssible (theoretically) to replicate this asset by a portfolio consisting of bonds, the underlying stock and European call option maturing in December 2001?