Fast Correlation Greeks by Adjoint Algorithmic Differentiation

Abstract

We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely efficient calculation of correlation Risk of option prices computed with Monte Carlo simulations. A key point in the construction is the use of binning to simultaneously achieve computational efficiency and accurate confidence intervals. We illustrate the method for a copula-based Monte Carlo computation of claims written on a basket of underlying assets, and we test it numerically for Portfolio Default Options. For any number of underlying assets or names in a portfolio, the sensitivities of the option price with respect to all the pairwise correlations is obtained at a computational cost which is at most 4 times the cost of calculating the option value itself. For typical applications, this results in computational savings of several order of magnitudes with respect to standard methods.One of the consequences of the current crisis of the Financial Markets is a renewed emphasis on rigorous Risk management practices. In order to quantify the financial exposure of financial firms, and to ensure an efficient capital allocation, and more effective hedging practices, regulators and senior management alike are insisting more and more on a thorough monitoring of Risk. Among all businesses, those dealing with complex, over the counter derivative securities are the ones receiving the most attention.A thorough calculation of the Risk exposure of portfolios of structured derivatives comes with a high operational cost because of the large amount of computer power required. Indeed, highly time consuming Monte Carlo (MC) simulations are very often the only tool available for pricing and hedging complex securities. Calculating the Greeks, or price sensitivities, by 'Bumping' i.e., by perturbing in turn the underlying model parameters, repeating the simulation and forming finite difference approximations results in a computational burden increasing linearly with the number of sensitivities computed. This easily becomes very significant when the models employed depend on a large number of parameters, as it is typically the case.A particularly challenging task is the calculation of correlation Risk, i.e., the calculation of the sensitivites of a security with respect to some measure of the correlations among the random factors it depends on. Indeed, calculating Risk with respect to all the independent pairwise correlations by Bumping requires repeating the MC simulation a large number of times, e.g. increasing quadratically with the number of random factors, and it is often unfeasible because of its high computational cost.Several alternative methods for the calculation of price * Electronic address: luca.capriotti@credit-suisse.com. † Electronic address: mike.giles@maths.ox.ac.uk sensitivities have been proposed in the literature (for a review see e.g., [1]). Among these, the Pathwise Derivative method [2] provides unbiased estimates at a computational cost that may be smaller than the one of Bumping. However, in many problems the standard Pathwise Derivat...

The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks

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A brief history of quantitative finance

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Adjoints and Automatic (Algorithmic) Differentiation in Computational Finance

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