Abstract. The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [S. Heston, Rev. Financ. Stud., 6 (1993) 1. Introduction. Stochastic volatility modeling emerged in the late nineties as a way to manage the smile. In this work, we focus on the Heston model, which is a lognormal model where the square of volatility follows a Cox-Ingersoll-Ross (CIR) 1 process. The call (and put) price has a closed formula in this model thanks to a Fourier inversion of the characteristic function (see Heston [22], Lewis [27], and Lipton [29]). When the parameters are piecewise constant, one can still derive a recursive closed formula using a PDE method (see Mikhailov and Nogel [31]) or a Markov argument in combination with affine models (see Elices [16]), but formula evaluation becomes increasingly time consuming. However, for general time dependent parameters there is no analytical formula and one usually has to perform Monte Carlo simulations. This explains the interest of recent works for designing more efficient Monte Carlo simulations: see Broadie and Kaya [13] for an exact simulation and bias-free scheme based on Fourier integral inversion; see Andersen [4] based on a Gaussian moment matching method and a user friendly algorithm; see Smith [40] relying on an almost exact scheme;
Abstract. The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier based methods is its rapidity (gain by a factor 100 or more), while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalent Heston models (extending some work of Piterbarg on stochastic volatility models [Pit05b]) and second, to the calibration procedure in terms of ill-posed problems.
Asymptotic expansion, Malliavin calculus, Volatility skew and smile, Small diffusion process, Small jump frequency/size, 60J75, 60HXX, G13,
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