The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model

Martin Forde; Antoine Jacquier; Roger Lee

doi:10.1137/110830241

CALIBRATION OF STOCHASTIC VOLATILITY MODELS VIA SECOND-ORDER APPROXIMATION: THE HESTON CASE

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024915500363 ◽

2015 ◽

Vol 18 (06) ◽

pp. 1550036 ◽

Cited By ~ 3

Author(s):

ELISA ALÒS ◽

RAFAEL DE SANTIAGO ◽

JOSEP VIVES

Keyword(s):

Term Structure ◽

Implied Volatility ◽

Order Approximation ◽

Calibration Procedure ◽

Computational Effort ◽

Heston Model ◽

Option Prices ◽

Volatility Models ◽

Short And Long Term ◽

Long Term Behavior

In this paper, we present a new, simple and efficient calibration procedure that uses both the short and long-term behavior of the Heston model in a coherent fashion. Using a suitable Hull and White-type formula, we develop a methodology to obtain an approximation to the implied volatility. Using this approximation, we calibrate the full set of parameters of the Heston model. One of the reasons that makes our calibration for short times to maturity so accurate is that we take into account the term structure for large times to maturity: We may thus say that calibration is not "memoryless," in the sense that the option's behavior far away from maturity does influence calibration when the option gets close to expiration. Our results provide a way to perform a quick calibration of a closed-form approximation to vanilla option prices, which may then be used to price exotic derivatives. The methodology is simple, accurate, fast and it requires a minimal computational effort.

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

Symmetry ◽

10.3390/sym12111878 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1878

Author(s):

Siow Woon Jeng ◽

Adem Kilicman

Keyword(s):

Term Structure ◽

Taylor Expansion ◽

Implied Volatility ◽

Order Approximation ◽

Heston Model ◽

Approximation Formula ◽

Option Prices ◽

Second Order Approximation ◽

Time Term ◽

Short Time

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.

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Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

Journal of Function Spaces ◽

10.1155/2019/9754679 ◽

2019 ◽

Vol 2019 ◽

pp. 1-12 ◽

Cited By ~ 3

Author(s):

Shican Liu ◽

Yanli Zhou ◽

Yonghong Wu ◽

Xiangyu Ge

Keyword(s):

Diffusion Process ◽

Option Pricing ◽

Stochastic Volatility ◽

Term Structure ◽

Implied Volatility ◽

Jump Diffusion ◽

Stochastic Volatility Model ◽

Heston Model ◽

Volatility Model ◽

Jump Diffusion Process

In financial markets, there exists long-observed feature of the implied volatility surface such as volatility smile and skew. Stochastic volatility models are commonly used to model this financial phenomenon more accurately compared with the conventional Black-Scholes pricing models. However, one factor stochastic volatility model is not good enough to capture the term structure phenomenon of volatility smirk. In our paper, we extend the Heston model to be a hybrid option pricing model driven by multiscale stochastic volatility and jump diffusion process. In our model the correlation effects have been taken into consideration. For the reason that the combination of multiscale volatility processes and jump diffusion process results in a high dimensional differential equation (PIDE), an efficient finite element method is proposed and the integral term arising from the jump term is absorbed to simplify the problem. The numerical results show an efficient explanation for volatility smirks when we incorporate jumps into both the stock process and the volatility process.

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SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL

International Journal of Theoretical and Applied Finance ◽

10.1142/s021902490900549x ◽

2009 ◽

Vol 12 (06) ◽

pp. 861-876 ◽

Cited By ~ 74

Author(s):

MARTIN FORDE ◽

ANTOINE JACQUIER

Keyword(s):

Implied Volatility ◽

Small Time ◽

Stochastic Volatility Model ◽

Probabilistic Methods ◽

Heston Model ◽

Closed Form Expression ◽

Standard Normal Distribution ◽

Legendre Transform ◽

Normal Distribution Function ◽

Volatility Model

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle. The rate function is computed as Fenchel-Legendre transform, and we show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise. We establish the corresponding result for implied volatility in Theorem 3.1, using well known bounds on the standard Normal distribution function. In Theorem 3.2 we compute the level, the slope and the curvature of the implied volatility in the small-maturity limit At-the-money, and the answer is consistent with that obtained by formal PDE methods by Lewis (2000) and probabilistic methods by Durrleman (2004).

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JOINING THE HESTON AND A THREE-FACTOR SHORT RATE MODEL: A CLOSED-FORM APPROACH

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024915500569 ◽

2015 ◽

Vol 18 (08) ◽

pp. 1550056 ◽

Cited By ~ 1

Author(s):

ROMAN HORSKY ◽

TILMAN SAYER

Keyword(s):

Term Structure ◽

Implied Volatility ◽

Heston Model ◽

Correlation Structure ◽

Market Model ◽

Characteristic Functions ◽

Rate Model ◽

Bond Price ◽

Equity Options ◽

Three Factor

In this paper, we present an innovative hybrid model for the valuation of equity options. Our approach includes stochastic volatility according to Heston (1993) [Review of Financial Studies 6 (2), 327–343] and features a stochastic interest rate that follows a three-factor short rate model based on Hull and White (1994) [Journal of Derivatives 2 (2), 37–48]. Our model is of affine structure, allows for correlations between the stock, the short rate and the volatility processes and can be fitted perfectly to the initial term structure. We determine the zero bond price formula and derive the analytic solution for European type options in terms of characteristic functions needed for fast calibration. We highlight the flexibility of our approach, by comparing the price and implied volatility surfaces of our model with the Heston model, where we in particular focus on the correlation structure. Our approach forms a comprehensive market model with an intuitive correlation structure that connects both the equity and interest market to the market volatility.

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