| Investment Knowledge |
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Mortgage Swap The payment amount in a mortgage swap can be modeled as the value of a European discrete Asian call option on a basket of indices. Here the basket price consists of an arithmetic average of various stock and bond indices. The call option payoff at maturity is equal to maximum of zero and an arithmetic average of basket values at certain points in time prior to option expiry less a fixed strike. We assume that each index price follows, under the corresponding risk neutral probability measure, geometric Brownian motion with drift. Since the call option pays in Canadian dollars, however, the index price process is represented with respect to the Canadian risk neutral probability measure by means of a quanto adjustment. Observe that the basket price process does not follow geometric Brownian motion with drift; furthermore, the arithmetic average of basket values is not lognormally distributed. We approximate the basket price process, based on an analytical moment matching technique consistent with [Levy, 1992], using a single geometric Brownian motion with drift. Relevant defining drift and volatility values for this single geometric Brownian motion are computed. The arithmetic average of the resulting approximate basket price process is further approximated, based on a different analytical moment matching technique, using a shifted lognormal random variable. The call option price is then computed as a discounted expected value of the maximum of zero and the shifted lognormal random variable value less the fixed strike. Here, relevant defining parameters for the shifted, lognormal random variable are computed.
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