General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases. Permanent repository linkKey words : arithmetic Asian options; CEV diffusion; stochastic volatility models; Lévy processes; discrete average; continuous average MSC2000 subject classification : Primary: 91B70, 91B25, 60J25; Secondary: 60H30, 91B24 OR/MS subject classification : Primary: asset pricing, diffusion, Markov processes, stochastic model applications 1. Introduction. We develop accurate analytical pricing formulae for discretely and continuously monitored arithmetic Asian options under general stochastic asset models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. The payoff of the arithmetic Asian option depends on the arithmetic average price of the underlying asset monitored over a pre-specified period. For more than two decades, much effort has been put into the research on efficient methodologies for computing the price of this option or, in general, expected values of functionals of the average value, under different model assumptions for the underlying. Developing such methods is of considerable practical importance as arithmetic averages see wide application in many fields of finance. Amongst others, we mention uses in computing net present value in project valuation (see [72]), optimal capacity planning under average demand uncertainty for a single firm (see [32]) and stock-swap merger proposals (see [60]). Weighted arithmetic averages also appear in technical analysis and in algorithmic trading; for example, we recall the moving average trading rule and its use from an asset allocation perspective (see [75]). Moving average automatic trading strategies set buying and selling orders depending on the position of the average price for a given period with respect to the current market price (see [50]). Finally, weighted arithmetic average indexes are used as trading benchmarks in pension plans (see [11]).Arithmetic Asian options are very popular among derivatives traders and risk managers. Their...

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