A finite element approach to the pricing of discrete lookbacks with stochastic volatility

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Abstract
Finite element methods are described for valuing lookback options under stochastic volatility. Particular attention is paid to the method for handling the boundary equations. For some boundaries, the equations reduce to first-order hyperbolic equations which must be discretized to ensure that outgoing waves are correctly modelled. Some example computations show that for certain choices of parameters, the option price computed for a lookback under stochastic volatility can differ from the price under the usual constant volatility assumption by as much as 35% (i.e. $7.30 compared with $5.45 for an at-the-money put), even though the models are calibrated so as to produce exactly the same price for an at-the-money vanilla European option with the same time remaining until expiry.