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A Department of Statistics seminar by Dr Gerald Cheang Hock Lye, Associate Director, Centre for Financial Engineering and Risk Management, Nanyang Business School, Nanyang Technological University, Singapore.
Margrabe provides a pricing formula for an exchange option where the distributions of both stock prices are log-normal with correlated Wiener components. Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. We use Merton's analysis to extend Margrabe's results to the case of exchange options where both stock price processes also contain compound Poisson jump components. A Radon-Nikodym derivative process that induces the change of measure from the market measure to an equivalent martingale measure is introduced. The choice of parameters in the Radon-Nikodym derivative allows us to price the option under different financial-economic scenarios. In the case of the American version of the exchange option, we decompose its price into a sum of the European price and an early exercise premium. A probabilistic intepretation for the early exercise premium is provided.
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