Pricing Algorithms of Multivariate Path Dependent Options

Abstract

Financial derivatives which are multivariate in nature are abundant in the financial markets. The underlying state variables may be the stock prices, interest rates, exchange rates, stochastic volatility, average of stock prices, extremum values of stock priors, etc. Option contracts whose life and payoff depend on the stochastic movement of the underlying asset prices are termed path dependent options. In this paper, we examine the pricing methods of several prototype path dependent options. These include options with sequential barriers, options with an external barrier and two-asset lookback options. The governing equations for the option prices are seen to resemble the diffusion type equations but with cross derivative terms, a feature which differs from the usual diffusion equations in engineering. Various techniques to reduce the complexity of the multivariate nature of these prototype option pricing models are discussed. It is illustrated that the dimensionality of a path dependent option model may be reduced by some ingenious choices of similarity variables. We also examine the design of pricing algorithms of these multivariate options, in particular, with regard to the treatment of discrete monitoring feature and the prescription of numerical boundary conditions. The possible generalizations of the numerical techniques presented in this paper to other models with more complicated path dependent payoff structures are also discussed.

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