scholarly journals SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL

2009 ◽  
Vol 12 (06) ◽  
pp. 861-876 ◽  
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).

scholarly journals Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
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.

Small-Time smile for the multifactor volatility heston model

2020 ◽  
Vol 57 (4) ◽  
pp. 1070-1087
Author(s):  
Dohyun Ahn ◽  
Kyoung-Kuk Kim ◽  
Younghoon Kim
Keyword(s):  
Stochastic Volatility ◽  
Small Time ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Mild Conditions ◽  
Time Asymptotics ◽  
Volatility Model ◽  
Small Time Asymptotics

AbstractWe extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

SMALL-TIME ASYMPTOTICS IN GEOMETRIC ASIAN OPTIONS FOR A STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL

2019 ◽  
Vol 22 (02) ◽  
pp. 1950005
Author(s):  
HOSSEIN JAFARI ◽  
GHAZALEH RAHIMI
Keyword(s):  
Stochastic Volatility ◽  
Lévy Process ◽  
Implied Volatility ◽  
Option Price ◽  
Small Time ◽  
Stochastic Volatility Model ◽  
Levy Process ◽  
Asian Option ◽  
Volatility Model ◽  
Jump Diffusion Model

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

scholarly journals Modified Mean-Variance Risk Measures for Long-Term Portfolios

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 111
Author(s):  
Hyungbin Park
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Investment Strategy ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Variance Risk ◽  
Investment Portfolios ◽  
Mean Variance

This paper proposes modified mean-variance risk measures for long-term investment portfolios. Two types of portfolios are considered: constant proportion portfolios and increasing amount portfolios. They are widely used in finance for investing assets and developing derivative securities. We compare the long-term behavior of a conventional mean-variance risk measure and a modified one of the two types of portfolios, and we discuss the benefits of the modified measure. Subsequently, an optimal long-term investment strategy is derived. We show that the modified risk measure reflects the investor’s risk aversion on the optimal long-term investment strategy; however, the conventional one does not. Several factor models are discussed as concrete examples: the Black–Scholes model, Kim–Omberg model, Heston model, and 3/2 stochastic volatility model.

scholarly journals Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
R. Company ◽  
L. Jódar ◽  
M. Fakharany ◽  
M.-C. Casabán
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Diffusion Reaction ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Volatility Model ◽  
Classical Technique ◽  
Domain Consistency ◽  
Positive Coefficients

This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, two-dimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed by means of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.

GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

2016 ◽  
Vol 19 (02) ◽  
pp. 1650014 ◽  
Author(s):  
INDRANIL SENGUPTA
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility ◽  
Gaussian Distribution ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Inverse Gaussian ◽  
Martingale Measures ◽  
Structure Preserving ◽  
Volatility Model ◽  
Equivalent Martingale

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

scholarly journals VARIANCE AND VOLATILITY SWAPS UNDER A TWO-FACTOR STOCHASTIC VOLATILITY MODEL WITH REGIME SWITCHING

2019 ◽  
Vol 22 (04) ◽  
pp. 1950009
Author(s):  
XIN-JIANG HE ◽  
SONG-PING ZHU
Keyword(s):  
Markov Chain ◽  
Characteristic Function ◽  
Stochastic Volatility ◽  
Regime Switching ◽  
Numerical Experiments ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Significant Difference ◽  
Cir Process

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.

A NOVEL ANALYTICAL APPROACH FOR PRICING DISCRETELY SAMPLED GAMMA SWAPS IN THE HESTON MODEL

2016 ◽  
Vol 57 (3) ◽  
pp. 244-268
Author(s):  
SANAE RUJIVAN
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Analytical Approach ◽  
Solution Procedure ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Variance Swaps ◽  
Closed Form Solutions ◽  
Volatility Model ◽  
Closed Form Formula

The main purpose of this paper is to present a novel analytical approach for pricing discretely sampled gamma swaps, defined in terms of weighted variance swaps of the underlying asset, based on Heston’s two-factor stochastic volatility model. The closed-form formula obtained in this paper is in a much simpler form than those proposed in the literature, which substantially reduces the computational burden and can be implemented efficiently. The solution procedure presented in this paper can be adopted to derive closed-form solutions for pricing various types of weighted variance swaps, such as self-quantoed variance and entropy swaps. Most interestingly, we discuss the validity of the current solutions in the parameter space, and provide market practitioners with some remarks for trading these types of weighted variance swaps.

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