Approximation Formula for Option Prices under Rough Heston Model and Short-Time Implied Volatility Behavior

Siow Woon Jeng,Adem Kilicman

doi:10.3390/sym12111878

Abstract

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.

Highlights

IntroductionStochastic volatility models and jump diffusion models such as classical Heston model [1]
Stochastic volatility models and jump diffusion models such as classical Heston model [1]and Merton jump diffusion model [2] have played an important role in the option pricing theory.The models have aimed at replicating the stochastic volatility effect and the jump-effect as displayed in the financial market
We find that, by comparing the implied volatility computed by Fourier inversion method with fractional Adams scheme [11,21] under rough Heston model, approximation formula for option prices and short-time term structure of the at-the-money skew equation have roughly the same order of explosive behavior for the term structure of at-the-money implied volatility skew

Summary

Introduction

Stochastic volatility models and jump diffusion models such as classical Heston model [1]. The authors of [11] obtained the characteristic function of the rough Heston model to compute the option price using the Fourier inversion method. As shown, we manage to verify that the option prices under rough Heston model computed using the approximation formula and the Fourier inversion method with fractional Adams scheme [11,21] are extremely consistent with one another for short maturity time. We find that, by comparing the implied volatility computed by Fourier inversion method with fractional Adams scheme [11,21] under rough Heston model, approximation formula for option prices (transformed back to implied volatility) and short-time term structure of the at-the-money skew equation have roughly the same order of explosive behavior for the term structure of at-the-money implied volatility skew. See pages 7–8 in [22], Proposition 3.1 in [23] or Proposition 2.1 in [24]

Approximation for Option Pricing Formula

Second-Order Approximation and Small-Time Behavior of Implied Volatility

Term Structure of At-the-Money Implied Volatility Skew

Numerical Experiments

Findings

Conclusions