Time Dependent Heston Model

Abstract

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;

Asymptotic and non asymptotic approximations for option valuation

“…As an illustration of flexibility of the stochastic analysis approach, it has been possible to handle Call/Put/digital options in local volatility models with Gaussian jumps [45], Call/Put options in local volatility models with stochastic Gaussian interest rates [7], Call/Put options in time-dependent Heston model [46], general average options (including Asian and Basket options) in local volatility models [42], and more recently local stochastic volatility models [47].…”

Calibration and simulation of Heston model

“…First possible extension can involve time-dependent parameters. A model with piece-wise constant parameters is studied by Mikhailov and Nögel [10], a linear time dependent Heston model is studied by Elices [45] and a more general case is introduced by Benhamou et al [46], who calculate also an approximation to the option price. However, Bayer, Fritz and Gatheral [47] claim that the overall shape of the volatility surface does not change in time and that also the parameters should remain constant.…”

Option Pricing for Stochastic Volatility Models: Vol-of-Vol Expansion

“…Proof: First, from the way we obtained the expression of ψ(t, ξ; ω) from that ofφ(t, ξ, 0; ω), it is easy to see that ψ ≡φ X [2], whereφ X (t, ξ; ω) :=φ(t, ξ, 0; ω). It follows that ψ ≡ ϕ X [2] (becauseφ X ≡ ϕ X [2]).…”

“…It follows that ψ ≡ ϕ X [2] (becauseφ X ≡ ϕ X [2]). Now, to show that F −1 (ϕ X ) ≡ F −1 (ψ) [2], it suces to show that we can dierentiate under with respect to ω. i.e.…”

“…For example, in [11] , Fouque et al propose an asymptotic expansion with respect to the parameter of mean reversion. Lewis [21], Benhamou et al [2] propose approximation methods based on the asymptotic expansion with respect to the volatility of volatility. In all the works cited above, the scope is often limited and the results can only be applied in a specic context, outside which we lose either the analytical formulas or the quality (accuracy) of these formulas.…”