Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

13 Pages Posted: 9 Jan 2015 Last revised: 10 Jan 2015

See all articles by Alexander Izmailov

Alexander Izmailov

Market Memory Trading L.L.C.

Brian Shay

Market Memory Trading, LLC

Date Written: December 10, 2014

Abstract

•The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publication “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” (see link http://ssrn.com/abstract=2489601).

•In this paper we report similar unique results for Asian Options, enabling complete analytical resolution of all problems associated with Asian Options. •Our discovery of the Asian Option probability density function enables exact closed-form analytical results for its expected value (price) for the first time without depending on Inverse Laplace or Fourier transforms that only abbreviate complex numerical integration procedures. •Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as Inverse Laplace or Fourier transforms, now used to represent the first moment. •Our formulation of the Asian Option probability density function is general enough to cover interesting and practical cases that are not addressed in the literature at all: for example, cases for which the averaging period is a terminal subset of the contract period, e.g. the last 3 months of the option lifetime. •Our formulation of the Asian Option probability density function is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that an Asian Option will be below or above any set of thresholds at any future time before or at termination. Such assessments are absolutely out of reach of current published methods for treating Asian Options.

•Resolution of the Asian Option Pricing problem enables knowledge of their linear correlation with Vanilla Options and, therefore, hedging in terms of Vanilla Options. Moreover, an opportunity to construct “synthetic” Asian Options from Vanilla Options for a given termination becomes a trivial exercise.

•All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Keywords: Asian Options, Options, Put Options, Call Options, Synthetic Options, Greeks, Black Scholes, Trading, Hedging, Market Making, Risk Management, VaR, Options’ Portfolio, Probability Density Function, Probability of Default, Insurance, Variable Annuity

JEL Classification: A10, A20, A22, A23, B40, C1, C10, C13, C15, C20, C30, C40, C50, C60, D40, D46, G1, G10, G20, G22

Suggested Citation

Izmailov, Alexander and Shay, Brian, Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach (December 10, 2014). Available at SSRN: https://ssrn.com/abstract=2546430 or http://dx.doi.org/10.2139/ssrn.2546430

Alexander Izmailov (Contact Author)

Market Memory Trading L.L.C. ( email )

19 East 71st Street, # 5D
New York, NY 10021
United States

Brian Shay

Market Memory Trading, LLC ( email )

805 Third Ave.
New York, NY 10017
United States

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