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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. $$ d S(t)=(L S(t) d t+o S(t) d W(t)+\delta S(t-) d N(t) $$ - 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 $$ S(t)-S(t-)=\delta \cdot S(t-) . $$ 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 ? Here \(W\) is a standard Wiener process whereas \(N\) is a Poisson process with intensity \(\lambda\). We assume that \(\alpha, \sigma, \delta\) and \(\lambda\) are known to us. The \(d N\) term is to be interpreted in the following way.

Consider the standard Black-Scholes model, and \(n\) different simple contingent claims with contract functions \(\Phi_{1}, \ldots, \Phi_{k}\). Let $$ V=\sum_{i=1}^{n} h_{i}(t) S_{j}(t) $$ denote the value process of a self-financing, Markovian (see Definition 5.2) portfolio. Because of the Markovian Assumption, \(V\) will be of the form \(V(t, S(t))\). Show that \(V\) satisfics the Black-Scholes equation.