A Risk-Neutral Stochastic Volatility Model

1998 ◽  
Vol 01 (02) ◽  
pp. 289-310 ◽  
Author(s):  
Yingzi Zhu ◽  
Marco Avellaneda
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Monte Carlo Study ◽  
Stochastic Volatility Model ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
The Difference ◽  
Log Normal

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

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.

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 A Comparative Analysis of the Black-Scholes- Merton Model and the Heston Stochastic Volatility Model

2019 ◽  
Vol 39 ◽  
pp. 127-140
Author(s):  
Tahmid Tamrin Suki ◽  
ABM Shahadat Hossain
Keyword(s):  
Comparative Analysis ◽  
Stochastic Volatility ◽  
Stock Prices ◽  
Computer Algebra System ◽  
Stochastic Volatility Model ◽  
Affecting Factors ◽  
Pricing Models ◽  
Merton Model ◽  
Volatility Model ◽  
Black Scholes

This paper compares the performance of two different option pricing models, namely, the Black-Scholes-Merton (B-S-M) model and the Heston Stochastic Volatility (H-S-V) model. It is known that the most popular B-S-M Model makes the assumption that volatility of an asset is constant while the H-S-V model considers it to be random. We examine the behavior of both B-S-M and H-S-V formulae with the change of different affecting factors by graphical representations and hence assimilate them. We also compare the behavior of some of the Greeks computed by both of these models with changing stock prices and hence constitute 3D plots of these Greeks. All the numerical computations and graphical illustrations are generated by a powerful Computer Algebra System (CAS), MATLAB. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 127-140

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.

CURRENCY DERIVATIVES UNDER A MINIMAL MARKET MODEL WITH RANDOM SCALING

2005 ◽  
Vol 08 (08) ◽  
pp. 1157-1177 ◽  
Author(s):  
DAVID HEATH ◽  
ECKHARD PLATEN
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Market Model ◽  
European Options ◽  
Bessel Processes ◽  
Exchange Rate Dynamics ◽  
Volatility Model ◽  
Minimal Market Model ◽  
Implied Volatility Surfaces

This paper uses an alternative, parsimonious stochastic volatility model to describe the dynamics of a currency market for the pricing and hedging of derivatives. Time transformed squared Bessel processes are the basic driving factors of the minimal market model. The time transformation is characterized by a random scaling, which provides for realistic exchange rate dynamics. The pricing of standard European options is studied. In particular, it is shown that the model produces implied volatility surfaces that are typically observed in real markets.

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