Analytic Approximations for Multi-Asset Option Pricing

ICMA Centre Discussion Papers in Finance DP2009-05

44 Pages Posted: 25 Jun 2009 Last revised: 16 Aug 2010

See all articles by Carol Alexander

Carol Alexander

University of Sussex Business School; Peking University HSBC Business School

Aanand Venkatramanan

University of Reading - ICMA Centre

Multiple version iconThere are 2 versions of this paper

Date Written: December 11, 2009

Abstract

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option's price as a sum of prices of various compound exchange options, each with different pairs of sub-ordinate multi- or single-asset options. For some multi-asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single-asset option under a modified measure, and in such cases an almost exact analytic price exists for the multi-asset option. The underlying asset prices are assumed to follow lognormal processes, although the strong condition can be extended to certain other price processes for the underlying. More generally, approximate analytic prices for multi-asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the sub-ordinate basket options. The analytic formulae for multi-asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to sub-ordinate multi-asset options, and the deltas of these sub-ordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations, and we demonstrate how to calibrate these parameters to market data so that multi-asset option prices are consistent with the implied volatility and correlation skews of the assets.

Keywords: basket options, rainbow options, best-of and worst-of options, compound exchange options, analytic approximation

JEL Classification: C02, C3, G63

Suggested Citation

Alexander, Carol and Venkatramanan, Aanand, Analytic Approximations for Multi-Asset Option Pricing (December 11, 2009). ICMA Centre Discussion Papers in Finance DP2009-05, Available at SSRN: https://ssrn.com/abstract=1424985 or http://dx.doi.org/10.2139/ssrn.1424985

Carol Alexander

University of Sussex Business School ( email )

Falmer, Brighton BN1 9SL
United Kingdom

HOME PAGE: http://www.coalexander.com

Peking University HSBC Business School ( email )

Aanand Venkatramanan (Contact Author)

University of Reading - ICMA Centre ( email )

Whiteknights Park
P.O. Box 242
Reading RG6 6BA
United Kingdom

HOME PAGE: http://www.icmacentre.ac.uk/index.php?id=146

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