Characteristic Function Evaluation in Heston's Model on Stochastic Volatility

2007 ◽  
Author(s):  
Andrew Rieck
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Function Evaluation ◽  
Heston's Model

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.

LOCAL STOCHASTIC VOLATILITY WITH JUMPS: ANALYTICAL APPROXIMATIONS

2013 ◽  
Vol 16 (08) ◽  
pp. 1350050 ◽  
Author(s):  
STEFANO PAGLIARANI ◽  
ANDREA PASCUCCI
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Fourier Space ◽  
Stochastic Volatility Models ◽  
Fourier Methods ◽  
Analytical Approximations ◽  
Volatility Models ◽  
Approximation Formulas ◽  
Real Market ◽  
Lévy Jumps

We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function, which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

Stochastic volatility for utility maximizers — A martingale approach

2018 ◽  
Vol 05 (01) ◽  
pp. 1850007 ◽  
Author(s):  
Simon Ellersgaard ◽  
Martin Tegnér
Keyword(s):  
Stochastic Volatility ◽  
Optimal Strategies ◽  
Optimal Consumption ◽  
Investment Strategies ◽  
Relative Risk Aversion ◽  
Welfare Gains ◽  
Martingale Approach ◽  
Wealth Maximization ◽  
Heston's Model ◽  
Real Market

Using Martingale methods, we study the problem of optimal consumption-investment strategies in a complete financial market characterized by stochastic volatility. With Heston’s model as the working example, we derive optimal strategies for a constant relative risk aversion (CRRA) investor with particular attention to the cases where (i) she solely seeks to optimize her utility for consumption, and (ii) she solely seeks to optimize her bequest from investing in the market. Furthermore, we test the practical utility of our work by conducting an empirical study based on real market-data from the S&P500 index. Here, we concentrate on wealth maximization and investigate the degree to which the inclusion of derivatives facilitates higher welfare gains. Our experiments show that this is indeed the case, although we do not observe realized wealth-equivalents as high as expected. Indeed, if we factor in the increased transaction costs associated with including options, the results are somewhat less convincing.

scholarly journals From characteristic functions to implied volatility expansions

2015 ◽  
Vol 47 (03) ◽  
pp. 837-857 ◽  
Author(s):  
Antoine Jacquier ◽  
Matthew Lorig
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Characteristic Functions ◽  
Volatility Smile ◽  
Model Free ◽  
Implied Volatility Surface ◽  
Volatility Model ◽  
Volatility Surface

For any strictly positive martingaleS= eXfor whichXhas a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

scholarly journals CAM Stochastic Volatility Model for Option Pricing

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Wanwan Huang ◽  
Brian Ewald ◽  
Giray Ökten
Keyword(s):  
Monte Carlo ◽  
Characteristic Function ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
The Other ◽  
Stochastic Volatility Models ◽  
Control Variate ◽  
Volatility Models ◽  
Volatility Model ◽  
Cam Model

The coupled additive and multiplicative (CAM) noises model is a stochastic volatility model for derivative pricing. Unlike the other stochastic volatility models in the literature, the CAM model uses two Brownian motions, one multiplicative and one additive, to model the volatility process. We provide empirical evidence that suggests a nontrivial relationship between the kurtosis and skewness of asset prices and that the CAM model is able to capture this relationship, whereas the traditional stochastic volatility models cannot. We introduce a control variate method and Monte Carlo estimators for some of the sensitivities (Greeks) of the model. We also derive an approximation for the characteristic function of the model.

scholarly journals INFORMATION-THEORETIC ANALYSIS OF STOCHASTIC VOLATILITY MODELS

2019 ◽  
Vol 22 (01) ◽  
pp. 1850025
Author(s):  
OLIVER PFANTE ◽  
NILS BERTSCHINGER
Keyword(s):  
Machine Learning ◽  
Mutual Information ◽  
Stochastic Volatility ◽  
Asset Prices ◽  
Theoretic Analysis ◽  
Stochastic Volatility Models ◽  
Information Theoretic ◽  
Random Dynamics ◽  
Volatility Models ◽  
Heston's Model

Stochastic volatility models describe asset prices [Formula: see text] as driven by an unobserved process capturing the random dynamics of volatility [Formula: see text]. We quantify how much information about [Formula: see text] can be inferred from asset prices [Formula: see text] in terms of Shannon’s mutual information in a twofold way: theoretically, by means of a thorough study of Heston’s model; from a machine learning perspective, by means of investigating a family of exponential Ornstein–Uhlenbeck (OU) processes fitted on S&P 500 data.

scholarly journals From characteristic functions to implied volatility expansions

2015 ◽  
Vol 47 (3) ◽  
pp. 837-857 ◽  
Author(s):  
Antoine Jacquier ◽  
Matthew Lorig
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Characteristic Functions ◽  
Volatility Smile ◽  
Model Free ◽  
Implied Volatility Surface ◽  
Volatility Model ◽  
Volatility Surface

For any strictly positive martingale S = eX for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

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