Aurora Borealis
1 May 2013 CFA Institute Journal Review

Easy Gram–Charlier Valuations of Options (Digest Summary)

  1. Biharilal Deora, CFA, CIPM

The Gram–Charlier method can be used to achieve computationally efficient closed-form approximations for Asian options and their path-dependent sensitivities. The authors examine an alternative solution for modeled processes in derivative securities in which closed-form valuation formulas either do not exist or are difficult to implement.

What’s Inside?

The Gram–Charlier (GC) method allows the construction of closed-form valuations and hedging approximations for a variety of financial processes. The authors’ analysis includes further exploring the GC estimator of the risk-neutral density and stressing its timewise advantage when it comes to valuing a chain of Asian options, for which each claim in the chain requires extensive processing time to compute because of the sample average method.

How Is This Article Useful to Practitioners?

The authors present an alternative to the popular Monte Carlo simulation whose distributional properties are not known. They incorporate the GC method into several old and new models—such as the Black, Scholes, and Merton; constant elasticity of variance; and variance-gamma models—to show accurate results for closed-form valuations and hedging approximations. They note how the GC method can be extended by preprocessing the volatility parameter estimates to obtain the sensitivity vega. The approach is similar to finding variance-stabilizing transformations when faced with heteroscedasticity in statistics.

In exploring the GC estimator of a risk-neutral density, the authors find the GC method can save time. They also determine when it can achieve affirmative outcomes or nonsensical outcomes, such as when hedging Asian contingent claims that are independent of the path properties on which many sensitivities depend.

The GC method does not impose a known rigid approximating probability density function on the analyzed problem and can be improved by appropriate variance reduction techniques at the point when the model moments are estimated.

How Did the Authors Conduct This Research?

For a chosen process, the authors construct estimates of the initial six moments of the distribution of the arithmetic average via a simulation, which are then used as inputs to a GC expansion to approximate the average’s probability density function. Next, they extend the results using effective variance reduction techniques, such as constant elasticity of variance (CEV) diffusions. They verify the GC approximation using a proof that gives the exact expressions for the first two moments of the arithmetic average of an underlying geometric Brownian motion process.

To clarify the construction of the GC estimate and its use in valuing Asian options, the authors provide various examples to expand on each part of the process. They then provide an example of representative valuations of Asian claims and compare the standard simulation technology with the GC probability density function closed-form approach. Finally, they go over how option sensitivities are calculated.

The authors conclude by illustrating how the GC operates in a stochastic process using the CEV class of diffusions. They focus on arithmetic averages for which the closed-form moment formulas are unavailable. They test the results on exact limiting higher-order moments in a GC estimator. But the GC estimator fails to provide an appropriate probability density function when used with positive point probability.

Abstractor’s Viewpoint

The authors provide a technique that, when applied correctly, can produce accurate and faster results without imposing rigid probability density function restrictions on the problem. They offer an interesting way of incorporating the GC method into existing stochastic processes with the benefit of reduced processing time, particularly when pricing typical Asian options.

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