Computational Methods for Quantitative Finance by Norbert Hilber Oleg Reichmann Christoph Schwab & Christoph Winter

Author:Norbert Hilber, Oleg Reichmann, Christoph Schwab & Christoph Winter
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

A key task in financial engineering is the fast and accurate calculation of sensitivities of market models with respect to model parameters. This becomes necessary, for example, in model calibration, risk analysis and in the pricing and hedging of certain derivative contracts. Classical examples are variations of option prices with respect to the spot price or with respect to time-to-maturity, the so-called “Greeks” of the model. For classical, diffusion type models and plain vanilla type contracts, the Greeks can be obtained analytically. With the trends to more general market models of jump–diffusion type and to more complicated contracts, closed form solutions are generally not available for pricing and calibration. Thus, prices and model sensitivities have to be approximated numerically.

Here, we consider the general class of Markov processes X, including stochastic volatility and Lévy models as described before. We distinguish between two classes of sensitivities. The sensitivity of the solution V to variation of a model parameter, like the Greek Vega (∂ σ V) and the sensitivity of the solution V to a variation of state spaces such as the Greek Delta (∂ x V). We show that an approximation for the first class can be obtained as a solution of the pricing PIDE with a right hand side depending on V. For the second class, a finite difference like differentiation procedure is presented which allows obtaining the sensitivities from the forward price without additional calls to the forward solver.

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