Efficient Valuation of Options on VIX under Gatheral’s Double Lognormal Stochastic Volatility Model: An Asymptotic Expansion Approach

2010 ◽  
Author(s):  
Chenxu Li
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Volatility Model

A control variate method for weak approximation of SDEs via discretization of numerical error of asymptotic expansion

2019 ◽  
Vol 25 (3) ◽  
pp. 239-252
Author(s):  
Yusuke Okano ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Diffusion Processes ◽  
Stochastic Volatility Model ◽  
Weak Approximation ◽  
Numerical Examples ◽  
Control Variate ◽  
Volatility Model ◽  
Multidimensional Diffusion

Abstract The paper shows a new weak approximation method for stochastic differential equations as a generalization and an extension of Heath–Platen’s scheme for multidimensional diffusion processes. We reformulate the Heath–Platen estimator from the viewpoint of asymptotic expansion. The proposed scheme is implemented by a Monte Carlo method and its variance is much reduced by the asymptotic expansion which works as a kind of control variate. Numerical examples for the local stochastic volatility model are shown to confirm the efficiency of the method.

IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY

2001 ◽  
Vol 04 (01) ◽  
pp. 45-89 ◽  
Author(s):  
ROGER W. LEE
Keyword(s):  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Asymptotic Methods ◽  
Stochastic Volatility Model ◽  
Slow Variation ◽  
Option Prices ◽  
Local Volatility ◽  
Volatility Model ◽  
Zero Correlation

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

scholarly journals A VOLATILITY-OF-VOLATILITY EXPANSION OF THE OPTION PRICES IN THE SABR STOCHASTIC VOLATILITY MODEL

2020 ◽  
Vol 23 (03) ◽  
pp. 2050018
Author(s):  
OLESYA GRISHCHENKO ◽  
XIAO HAN ◽  
VICTOR NISTOR
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Heat Kernel ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Transition Probability ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Volatility Model ◽  
The One

We propose a new type of asymptotic expansion for the transition probability density function (or heat kernel) of certain parabolic partial differential equations (PDEs) that appear in option pricing. As other, related methods developed by Costanzino, Hagan, Gatheral, Lesniewski, Pascucci, and their collaborators, among others, our method is based on the computation of the truncated asymptotic expansion of the heat kernel with respect to a “small” parameter. What sets our method apart is that our small parameter is possibly different from the time to expiry and that the resulting commutator calculations go beyond the nilpotent Lie algebra case. In favorable situations, the terms of this asymptotic expansion can quickly be computed explicitly leading to a “closed-form” approximation of the solution, and hence of the option price. Our approximations tend to have much fewer terms than the ones obtained from short time asymptotics, and are thus easier to generalize. Another advantage is that the first term of our expansion corresponds to the classical Black-Scholes model. Our method also provides equally fast approximations of the derivatives of the solution, which is usually a challenge. A full theoretical justification of our method seems very difficult at this time, but we do provide some justification based on the results of (Siyan, Mazzucato, and Nistor, NWEJ 2018). We therefore mostly content ourselves to demonstrate numerically the efficiency of our method by applying it to the solution of the mean-reverting SABR stochastic volatility model PDE, commonly referred to as the [Formula: see text]SABR PDE, by taking the volatility of the volatility parameter [Formula: see text] (vol-of-vol) as a small parameter. For this PDE, we provide extensive numerical tests to gauge the performance of our method. In particular, we compare our approximation to the one obtained using Hagan’s formula and to the one obtained using a new, adaptive finite difference method. We provide an explicit asymptotic expansion for the implied volatility (generalizing Hagan’s formula), which is what is typically needed in concrete applications. We also calibrate our model to observed market option price data. The resulting values for the parameters [Formula: see text], [Formula: see text], and [Formula: see text] are realistic, which provides more evidence for the conjecture that the volatility is mean-reverting.

A weak approximation with Malliavin weights for local stochastic volatility model

2017 ◽  
Vol 04 (01) ◽  
pp. 1750002
Author(s):  
Toshihiro Yamada
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Stochastic Differential Equation ◽  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Weak Approximation ◽  
Volatility Model ◽  
Sabr Model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

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