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2015 | OriginalPaper | Buchkapitel

32. Consequences for Option Pricing of a Long Memory in Volatility

verfasst von : Stephen J. Taylor

Erschienen in: Handbook of Financial Econometrics and Statistics

Verlag: Springer New York

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Abstract

Conditionally heteroscedastic time series models are used to describe the volatility of stock index returns. Volatility has a long memory property in the most general models and then the autocorrelations of volatility decay at a hyperbolic rate; contrasts are made with popular, short memory specifications whose autocorrelations decay more rapidly at a geometric rate.

Options are valued for ARCH volatility models by calculating the discounted expectations of option payoffs for an appropriate risk-neutral measure. Monte Carlo methods provide the expectations. The speed and accuracy of the calculations is enhanced by two variance reduction methods, which use antithetic and control variables. The economic consequences of a long memory assumption about volatility are documented, by comparing implied volatilities for option prices obtained from short and long memory volatility processes.

Results are given for options on the S & P 100-share index, with lives up to 2 years. The long memory assumption is found to have a significant impact upon the term structure of implied volatilities and a relatively minor impact upon smile shapes. These conclusions are important because evidence for long memory in volatility has been found in the prices of many assets.

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A conservative robust standard error for our estimate of d is 0.12, using information provided by Bollerslev and Mikkelsen (1996).

The conditional expectations of r _t for measures P and Q differ by \( \lambda \sqrt{h_t\kern0.1em } \) and a typical average value of \( \sqrt{h_t} \) is 0.00858. Assuming 253 trading days in 1 year gives the stated value of λ.

The filter coefficients sum to b ₁ + … + b _1,000 = 0.983. After b ₁ = 1 and b ₂ = − 0.12, all of the coefficients are near zero, with b ₁₀₀ = 0.00017 and b _1,000 = 7 × 10⁻⁶.

The results support the conjecture that IV(T) ≅ a ₁ + a ₂ T ^2d−1 for large T with a₂ determined by the history of observed returns.

An estimate of the constant a₁ (defined in the previous footnote) is 16.0 %. An estimate of 15.0 % follows by supposing the long memory limit is 105 % of the short memory limit, based on the limit of ln(h _t) being higher by 0.1 for the long memory process as noted in Appendix 3. The difference in the limits is a consequence of the risk premium obtained by owning the asset; its magnitude is mainly determined by the pronounced asymmetry in the volatility shock function g(z _t).

The dependence of moments of h _t on the measure is shown by Duan (1995, p. 19) for the GARCH(1,1) model.

When z ∼ N(0,1), \( E\left[\left|z-\lambda \right|\right]=\sqrt{2/\pi } \exp \left(-{\scriptscriptstyle \frac{1}{2}}{\lambda}^2\right)+\lambda \left(2\Phi \left(\lambda \right)-1\right) \) with Φ the cumulative distribution function of z.

As (1 − L)^d1 = 0 for d > 0, it follows from Eqs. 32.31 and 32.32 that \( {\displaystyle \sum_{j=1}^{\infty }{a}_j=}{\displaystyle \sum_{j=1}^{\infty }{b}_j=}\ 1. \)

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Titel: Consequences for Option Pricing of a Long Memory in Volatility
verfasst von: Stephen J. Taylor
Verlag: Springer New York
Buch: Handbook of Financial Econometrics and Statistics
Print ISBN: 978-1-4614-7749-5

Electronic ISBN: 978-1-4614-7750-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-1-4614-7750-1_32