The Heston Model with Time-Dependent Correlation Driven by Isospectral Flows

Long Teng

doi:10.3390/math9090934

Abstract

In this work, we extend the Heston stochastic volatility model by including a time-dependent correlation that is driven by isospectral flows instead of a constant correlation, being motivated by the fact that the correlation between, e.g., financial products and financial institutions is hardly a fixed constant. We apply different numerical methods, including the method for backward stochastic differential equations (BSDEs) for a fast computation of the extended Heston model. An example of calibration to market data illustrates that our extended Heston model can provide a better volatility smile than the Heston model with other considered extensions.

Highlights

IntroductionLicensee MDPI, Basel, Switzerland.The Heston model [1] is one of the most widely used affine stochastic volatility models for equity prices, which is an extension of the Black and Schloes model [2] by taking into account stochastic volatility that is driven by a Cox-Ingersoll-Ross (CIR) process [3].it is well-known that, in many cases, the Heston model cannot provide enough skews or smiles in the implied volatility as market requires, in particular with short maturity.To tackle this problem, several extensions have been proposed in the literature: the Heston model by allowing time-dependent parameters [4,5,6]; the double Heston model with an additional volatility process [7]; the Heston model extended with a stochastic interest rate [8]; the Heston model that is extended by imposing a stochastic correlation [9,10]; and, the Heston model with a time-dependent correlation function [11]
Several extensions have been proposed in the literature: the Heston model by allowing time-dependent parameters [4,5,6]; the double Heston model with an additional volatility process [7]; the Heston model extended with a stochastic interest rate [8]; the Heston model that is extended by imposing a stochastic correlation [9,10]; and, the Heston model with a time-dependent correlation function [11]
Note that the risk-neutral probability measure is not needed in the backward stochastic differential equations (BSDEs)-based method; we still have μS in the model, and set μS = 0.05

Summary

Introduction

Licensee MDPI, Basel, Switzerland.The Heston model [1] is one of the most widely used affine stochastic volatility models for equity prices, which is an extension of the Black and Schloes model [2] by taking into account stochastic volatility that is driven by a Cox-Ingersoll-Ross (CIR) process [3].it is well-known that, in many cases, the Heston model cannot provide enough skews or smiles in the implied volatility as market requires, in particular with short maturity.To tackle this problem, several extensions have been proposed in the literature: the Heston model by allowing time-dependent parameters [4,5,6]; the double Heston model with an additional volatility process [7]; the Heston model extended with a stochastic interest rate [8]; the Heston model that is extended by imposing a stochastic correlation [9,10]; and, the Heston model with a time-dependent correlation function [11]. The Heston model [1] is one of the most widely used affine stochastic volatility models for equity prices, which is an extension of the Black and Schloes model [2] by taking into account stochastic volatility that is driven by a Cox-Ingersoll-Ross (CIR) process [3]. It is well-known that, in many cases, the Heston model cannot provide enough skews or smiles in the implied volatility as market requires, in particular with short maturity. Our aim is to improve the pure Heston model to provide better smiles in the implied volatility as market requires, and to add an economic concept of nonlinear relationship between the asset and its volatility

Objectives

Results

Conclusion