Exotic Option Pricing using Heston Simulation (Presentation Slides)

Abstract

Exotic, bespoke and long-dated options require careful model selection, calibration and pricing. Local Volatility models struggle to fit the smile and skew observed in financial markets and smile tends to flatten for long-dated maturities. Stochastic Volatility (SV) models such as the Heston model don't exhibit such issues and are more suitable for long-dated option pricing.

Heston SV model calibration is well discussed academic literature with excellent papers such as "Calibration and simulation of Heston model" by Mrazek and Pospisil (2017). In this document we outline the process for pricing exotic options using the Heston SV model, namely we calibrate the model to liquid European options quoted in the market using Heston closed-form solutions to derive the Heston model parameters with high speed and accuracy. We then use the calibration parameters to perform Heston Monte Carlo simulation for exotic and long-dated option pricing, which typically have no analytical solution.

We present the Heston model and outline its properties; we then proceed to adjust the model into a log-normal process for better convergence using Ito's Lemma and outline how to correlate Brownian motions using Cholesky Decomposition. We then show how to perform Exact Simulation of the CIR variance process, which is part of the Heston model, and illustrate how to manage negative variance using the absorption and reflection techniques.

The Euler discretization scheme is discussed; it requires many small time steps and can be computationally intensive. We outline how exact simulation can improve upon this and allow model practitioners to take large time steps when performing Monte Carlo simulation with no loss of accuracy. This gives better speed and performance, as we can simulate asset paths on the observation dates of interest rather than on every time step to trade maturity.

We proceed to present the Almost Exact Simulation (AES) approach, which greatly simplifies the Heston discretization required for Monte Carlo simulation using an integral freezing approximation. This facilitates speedy pricing with negligible loss of accuracy. We conclude by presenting benchmark pricing and convergence results. We present this work in python, which can be found at https://bit.ly/HestonSV. All that remains is the trivial task of defining and scripting the exotic payoff.

Keywords: Exotic Options, Smile, Skew, Stochastic Volatility, Heston Model, Simulation, Monte Carlo, Euler Discretization, Exact Simulation, Almost Exact Simulation, Negative Variance, Absorption, Reflection, Cholesky Decomposition, Correlation, Integral Freezing, Pricing, Valuation, Python

JEL Classification: G12, G15, G19, G20, G21, G23, G24, G29