Algorithmic Hessians and the Fast Computation of Cross-Gamma Risk

21 Pages Posted: 19 Jun 2010 Last revised: 2 Dec 2010

Mark S. Joshi

University of Melbourne - Centre for Actuarial Studies

Chao Yang

University of Melbourne - Centre for Actuarial Studies

Date Written: June 18, 2010

Abstract

We introduce a new methodology for computing Hessians from algorithms for function evaluation, using backwards methods. We show that the complexity of the Hessian calculation is a linear function of the number of state variables times the complexity of the original algorithm. We apply our results to computing the Gamma matrix of multi-dimensional financial derivatives including Asian Baskets and cancellable swaps. In particular, our algorithm for computing Gammas of Bermudan cancellable swaps is order O(n^2) per step in the number of rates. We present numerical results demonstrating that the computing all n(n 1)/2 Gammas in the LMM takes roughly n/3 times as long as computing the price.

Keywords: automatic differentiation, Monte Carlo simulation, Greeks, Gamma, LIBOR market model, cancellable

JEL Classification: G13

Suggested Citation

Joshi, Mark S. and Yang, Chao, Algorithmic Hessians and the Fast Computation of Cross-Gamma Risk (June 18, 2010). Available at SSRN: https://ssrn.com/abstract=1626547 or http://dx.doi.org/10.2139/ssrn.1626547

Mark Joshi (Contact Author)

University of Melbourne - Centre for Actuarial Studies ( email )

Melbourne, 3010
Australia

Chao Yang

University of Melbourne - Centre for Actuarial Studies ( email )

Melbourne, 3010
Australia

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