Explicit Heston Solutions and Stochastic Approximation for Path-Dependent Option Pricing

Abstract

New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: (1) stochastic approximation replaces regression in the LSM algorithm; (2) explicit weak solutions to stochastic differential equations are developed and applied to Heston model simulation; and (3) importance sampling expands these explicit solutions. The approach complements Heston (1993) and Broadie & Kaya (2006) by handling the case of path-dependence in the option's execution strategy. Numeric comparison against standard Monte Carlo methods demonstrate up to two orders of magnitude speed improvement. The general ideas will extend beyond the important Heston setting.2010 Mathematics Subject Classification. Primary 91G60, 65C05; Secondary 91G20, 60H10.

Branching Particle Pricers With Heston Examples

“…Both of these methods are based upon characterizing and evolving the distribution of (S t , V t ). Later, Kouritzin (2018) showed that the Heston SDEs were actually explicitly solvable in a way that is very convenient for simulation, which facilitates the pricing of exotic options whose prices cannot be written in closed-form. Specifically, if we let…”

“…We first recall the bootstrap algorithm, which will later be shown to be less suitable for American option pricing. We then present two branching algorithms that can be used to increase the performance of the Heston simulation procedure of Kouritzin (2018).…”

“…Option prices that do not allow closed-form solutions are often priced by Monte Carlo simulation. Using Kouritzin (2018)'s explicit solutions, it is possible to simulate multiple Heston paths {(S j t , V j t ), t ≥ 0} N j=1 efficiently. For reasons that will become apparent in the next paragraphs, we often refer to each of these simulated paths as particles.…”

“…Each particle (S j t , V j t ) has the desired distribution (the one under which we want to price) under the pricing measure P , and the associated likelihood (or Radon-Nikodym derivative) L j t = P P j Ft can be seen as its weight. For this reason, the simulation algorithm for option pricing of Kouritzin (2018) can be thought of as the analog of a weighted particle filter in sequential Monte Carlo methods, producing a weighted empirical measure of the form σ N [0,t]…”

“…Herein, we focus on two of these new branching algorithms and introduce them in the Heston simulation framework of Kouritzin (2018). Separately, we show empirically that the bootstrap algorithm is less suitable for pricing options that are strongly path-dependent, mainly because its excessive resampling affects the distribution of path estimates.…”