Abstract
Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263, we studied the weak error of discretization schemes for the Heston model, which are based on exact simulation of the underlying volatility process. Both for an Euler- and a trapezoidal-type scheme for the log-asset price, we established weak order one for smooth payoffs without any assumptions on the Feller index of the volatility process. In our analysis, we also observed the usual trade off between the smoothness assumption on the payoff and the restriction on the Feller index. Moreover, we provided error expansions, which could be used to construct second order schemes via extrapolation. In this paper, we illustrate our theoretical findings by several numerical examples.
Highlights
The Heston Model Heston (1993) is a widely used stochastic volatility model to price financial options. It consists of two stochastic differential equations (SDEs) for an asset price process S and its volatility V: q dSt = μSt dt + Vt St ρdWt + 1 − ρ dBt, (1)
Schemes built on the Broadie-Kaya trick, i.e., Equation (3), have a different structure than schemes which arise by a direct discretization of the log-Heston model as, e.g., the schemes studied in Altmayer and Neuenkirch (2017); Lord et al (2009)
The presented numerical experiments explore whether the Theorems are valid under milder assumptions
Summary
The Heston Model Heston (1993) is a widely used stochastic volatility model to price financial options. We require smoothness assumptions for f that are not met by the payoffs in practice, which are at most Lipschitz continuous or even discontinuous This is a typical problem for weak approximation of SDEs as the Heston SDE, which do not satisfy the so-called standard assumptions on the coefficients. Schemes built on the Broadie-Kaya trick, i.e., Equation (3), have a different structure than schemes which arise by a direct discretization of the log-Heston model as, e.g., the schemes studied in Altmayer and Neuenkirch (2017); Lord et al (2009). The volatility V = (Vt )t∈[0,T ] is discretized by an Euler scheme, a fix for retaining the positivity is introduced by using the positive part, and the equation for the log-Heston price. For pricing American options, they can be beneficial
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