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.

STOCHASTIC VOLATILITY AND JUMP-DIFFUSION — IMPLICATIONS ON OPTION PRICING

1999 ◽  
Vol 02 (04) ◽  
pp. 409-440 ◽  
Author(s):  
GEORGE J. JIANG
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Dynamic Properties ◽  
Jump Diffusion ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
Pricing Errors ◽  
Volatility Model ◽  
Random Jump

This paper conducts a thorough and detailed investigation on the implications of stochastic volatility and random jump on option prices. Both stochastic volatility and jump-diffusion processes admit asymmetric and fat-tailed distribution of asset returns and thus have similar impact on option prices compared to the Black–Scholes model. While the dynamic properties of stochastic volatility model are shown to have more impact on long-term options, the random jump is shown to have relatively larger impact on short-term near-the-money options. The misspecification risk of stochastic volatility as jump is minimal in terms of option pricing errors only when both the level of kurtosis of the underlying asset return distribution and the level of volatility persistence are low. While both asymmetric volatility and asymmetric jump can induce distortion of option pricing errors, the skewness of jump offers better explanations to empirical findings on implied volatility curves.

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

2001 ◽  
Vol 04 (04) ◽  
pp. 651-675 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
GEORGE PAPANICOLAOU ◽  
K. RONNIE SIRCAR
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Stochastic Volatility ◽  
Free Boundary Problem ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
One Dimensional ◽  
Volatility Model ◽  
Black Scholes

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

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.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

A STOCHASTIC VOLATILITY MODEL FOR RISK-REVERSALS IN FOREIGN EXCHANGE

2009 ◽  
Vol 12 (06) ◽  
pp. 877-899 ◽  
Author(s):  
CLAUDIO ALBANESE ◽  
ALEKSANDAR MIJATOVIĆ
Keyword(s):  
Stochastic Volatility ◽  
Foreign Exchange ◽  
Continuous Time ◽  
Efficient Algorithms ◽  
Stochastic Volatility Model ◽  
Foreign Exchange Markets ◽  
Local Volatility ◽  
Volatility Model ◽  
Exchange Markets

It is a widely recognized fact that risk-reversals play a central role in the pricing of derivatives in foreign exchange markets. It is also known that the values of risk-reversals vary stochastically with time. In this paper we introduce a stochastic volatility model with jumps and local volatility, defined on a continuous time lattice, which provides a way of modeling this kind of risk using numerically stable and relatively efficient algorithms.

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 MULTISCALE STOCHASTIC VOLATILITY MODEL FOR DERIVATIVES ON FUTURES

2014 ◽  
Vol 17 (07) ◽  
pp. 1450043 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
YURI F. SAPORITO ◽  
JORGE P. ZUBELLI
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Order Approximation ◽  
Perturbation Technique ◽  
Stochastic Volatility Model ◽  
Spot Price ◽  
Additional Hypothesis ◽  
First Order ◽  
Volatility Model ◽  
First Order Approximation

In this paper, we present a new method for computing the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility studied in Fouque et al. (2011). It provides an alternative method to the singular perturbation technique presented in Hikspoors & Jaimungal (2008). The main features of our method are twofold: firstly, it does not rely on any additional hypothesis on the regularity of the payoff function, and secondly, it allows an effective and straightforward calibration procedure of the group market parameters to implied volatilities. These features were not achieved in previous works. Moreover, the central argument of our method could be applied to interest rate derivatives and compound derivatives. The only pre-requisite of our approach is the first-order approximation of the underlying derivative. Furthermore, the model proposed here is well-suited for commodities since it incorporates mean reversion of the spot price and multiscale stochastic volatility. Indeed, the model was validated by calibrating the group market parameters to options on crude-oil futures, and it displays a very good fit of the implied volatility.

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