scholarly journals Calibration and simulation of Heston model

2017 ◽  
Vol 15 (1) ◽  
pp. 679-704 ◽  
Author(s):  
Milan Mrázek ◽  
Jan Pospíšil
Keyword(s):  
Variance Reduction ◽  
Calibration Procedure ◽  
Optimization Techniques ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Approximation Formula ◽  
Market Data ◽  
Volatility Model ◽  
Calibration Methods ◽  
Real Market

Abstract We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.

scholarly journals SWITCHING TO NONAFFINE STOCHASTIC VOLATILITY: A CLOSED-FORM EXPANSION FOR THE INVERSE GAMMA MODEL

2016 ◽  
Vol 19 (05) ◽  
pp. 1650031 ◽  
Author(s):  
NICOLAS LANGRENÉ ◽  
GEOFFREY LEE ◽  
ZILI ZHU
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Time Dependent ◽  
Market Data ◽  
Affine Models ◽  
Inverse Gamma ◽  
Volatility Model ◽  
Black Scholes

We examine the inverse gamma (IGa) stochastic volatility model with time-dependent parameters. This nonaffine model compares favorably in terms of volatility distribution and volatility paths to classical affine models such as the Heston model, while being as parsimonious (only four stochastic parameters). In practice, this means more robust calibration and better hedging, explaining its popularity among practitioners. Closed-form volatility-of-volatility expansions are obtained for the price of vanilla options, which allow for very fast pricing and calibration to market data. Specifically, the price of a European put option with IGa volatility is approximated by a Black–Scholes price plus a weighted combination of Black–Scholes Greeks, with weights depending only on the four time-dependent parameters of the model. The accuracy of the expansion is illustrated on several calibration tests on foreign exchange market data. This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on nonaffine models favored by practitioners such as the IGa model.

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.

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.

scholarly journals INTEGRAL REPRESENTATION OF PROBABILITY DENSITY OF STOCHASTIC VOLATILITY MODELS AND TIMER OPTIONS

2017 ◽  
Vol 20 (08) ◽  
pp. 1750055 ◽  
Author(s):  
ZHENYU CUI ◽  
J. LARS KIRKBY ◽  
GUANGHUA LIAN ◽  
DUY NGUYEN
Keyword(s):  
Stochastic Volatility ◽  
Probabilistic Method ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Model Parameters ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Volatility Model ◽  
Probability Densities ◽  
Special Case

This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in [Z. Palmowski & T. Rolski (2002) A technique for exponential change of measure for Markov processes, Bernoulli 8(6), 767–785]. With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with nonzero correlations, namely the Heston 1993, [Formula: see text], and a special case of the [Formula: see text]-Hypergeometric stochastic volatility models recently proposed by [J. Da Fonseca & C. Martini (2016) The [Formula: see text]-Hypergeometric stochastic volatility model, Stochastic Processes and their Applications 126(5), 1472–1502]. Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the [Formula: see text] model and a special case of the [Formula: see text]-Hypergeometric model.

ON PRICING CONTINGENT CLAIMS UNDER THE DOUBLE HESTON MODEL

2012 ◽  
Vol 15 (05) ◽  
pp. 1250033 ◽  
Author(s):  
M. COSTABILE ◽  
I. MASSABÒ ◽  
E. RUSSO
Keyword(s):  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Contingent Claims ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Underlying Asset ◽  
State Variable ◽  
Proposed Model ◽  
Volatility Model ◽  
Asset Value

This article presents a lattice based approach for pricing contingent claims when the underlying asset evolves according to the double Heston (dH) stochastic volatility model introduced by Christoffersen et al. (2009). We discretize the continuous evolution of both squared volatilities by a "binomial pyramid", and consider the asset value as an auxiliary state variable for which a subset of possible realizations is attached to each node of the pyramid. The elements of the subset cover the range of asset prices at each time slice, and claim price is computed solving backward through the "binomial pyramid". Numerical experiments confirm the accuracy and efficiency of the proposed model.

scholarly journals Malliavin differentiability of the Heston volatility and applications to option pricing

2008 ◽  
Vol 40 (01) ◽  
pp. 144-162 ◽  
Author(s):  
Elisa Alòs ◽  
Christian-Oliver Ewald
Keyword(s):  
Option Pricing ◽  
Explicit Expression ◽  
Stochastic Volatility ◽  
Malliavin Calculus ◽  
Numerical Results ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Malliavin Derivative ◽  
Volatility Model

We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of Alòs (2006) in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.

MODELING AND SIMULATION OF SEQUENTIAL AUCTIONS: PRICING AND CALIBRATION ALGORITHMS

2012 ◽  
Vol 03 (03) ◽  
pp. 1250009 ◽  
Author(s):  
EMMANUEL M. TADJOUDDINE
Keyword(s):  
Market Price ◽  
Payoff Function ◽  
Stochastic Volatility Model ◽  
Sequential Auctions ◽  
Future Market ◽  
Market Data ◽  
Algorithmic Differentiation ◽  
Financial Products ◽  
Volatility Model ◽  
The Right

We consider sequential auctions wherein seller and bidder agents need to price goods on sale at the 'right' market price. We propose algorithms based on a binomial model for both the seller and buyer. Then, we consider the problem of calibrating pricing models to market data. To this end, we studied a stochastic volatility model used for option pricing, derived, and analyzed Monte Carlo estimators for computing the gradient of a certain payoff function using Finite Differencing and Algorithmic Differentiation. We then assessed the accuracy and efficiency of both methods as well as their impacts into the optimization algorithm. Numerical results are presented and discussed. This work can benefit those engaged in electronic trading or investors in financial products with the need for fast and more precise predictions of future market data.

scholarly journals A Semigroup Expansion for Pricing Barrier Options

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Takashi Kato ◽  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Expansion Method ◽  
Stochastic Volatility Model ◽  
Approximation Formula ◽  
Parabolic Partial Differential Equations ◽  
Barrier Options ◽  
Barrier Option ◽  
Volatility Model ◽  
Barrier Option Pricing ◽  
Asymptotic Expansion Method ◽  
Expansion Scheme

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.

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