Multi-Currency Local Volatility Model

37 Pages Posted: 30 Jun 2008 Last revised: 24 Sep 2008

See all articles by Daniel Alexandre Bloch

Daniel Alexandre Bloch

Université Paris VI Pierre et Marie Curie

Yukio Nakashima

affiliation not provided to SSRN

Date Written: August 2008

Abstract

We establish the need for local volatility coupled with domestic and foreign stochastic interest rates to properly manage some exotic hybrid options. We then compute such a local volatility and identify a bias with respect to the local volatility with deterministic rates. Performing variance-covariance analysis on the logarithm of the underlying price together with the domestic and foreign spot rates we estimate that bias by calculating the variances of the logarithm of the underlying price with and without stochastic rates at fixed points in time and in space. Equating the resulting variances we express the local volatility with stochastic rates in terms of the one with deterministic rates plus a bias obtaining an exact, fast and robust way of calibrating any local volatility with stochastic rates to market prices. We calculate it by using a bootstrapping method requiring solving a quadratic equation at each maturity and strike and present results on the Japanese market.

Keywords: Local Volatility, Cross-Currency, Stochastic Rates, FX options, Malliavin Calculus, Calibration

Suggested Citation

Bloch, Daniel Alexandre and Nakashima, Yukio, Multi-Currency Local Volatility Model (August 2008). Available at SSRN: https://ssrn.com/abstract=1153337 or http://dx.doi.org/10.2139/ssrn.1153337

Daniel Alexandre Bloch (Contact Author)

Université Paris VI Pierre et Marie Curie ( email )

175 Rue du Chevaleret
Paris, 75013
France

Yukio Nakashima

affiliation not provided to SSRN ( email )

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