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Edgeworth expansions of stochastic trading time

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  • Decamps, Marc
  • De Schepper, Ann

Abstract

Under most local and stochastic volatility models the underlying forward is assumed to be a positive function of a time-changed Brownian motion. It relates nicely the implied volatility smile to the so-called activity rate in the market. Following Young and DeWitt-Morette (1986) [8], we propose to apply the Duru–Kleinert process-cum-time transformation in path integral to formulate the transition density of the forward. The method leads to asymptotic expansions of the transition density around a Gaussian kernel corresponding to the average activity in the market conditional on the forward value. The approximation is numerically illustrated for pricing vanilla options under the CEV model and the popular normal SABR model. The asymptotics can also be used for Monte Carlo simulations or backward integration schemes.

Suggested Citation

  • Decamps, Marc & De Schepper, Ann, 2010. "Edgeworth expansions of stochastic trading time," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3179-3192.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:16:p:3179-3192
    DOI: 10.1016/j.physa.2010.04.014
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    References listed on IDEAS

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    1. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    2. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    3. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
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