Calibration and simulation of Heston model

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.

[1]  Robert A. Jarrow,et al.  Bayesian analysis of contingent claim model error , 2000 .

[2]  Alan L. Lewis.  Option Valuation Under Stochastic Volatility: With Mathematica Code , 2000 .

[3]  Philipp Ziegler,et al.  Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure , 2018, Empirical Economics.

[4]  Jörg Kienitz,et al.  Financial Modelling: Theory, Implementation and Practice with MATLAB Source , 2013 .

[5]  S. Mikhailov,et al.  Heston ’ s Stochastic Volatility Model Implementation , Calibration and Some , 2003 .

[6]  Leif Andersen,et al.  Extended Libor Market Models with Stochastic Volatility , 2001 .

[7]  W. Feller.  TWO SINGULAR DIFFUSION PROBLEMS , 1951 .

[8]  Jan Pospíšil,et al.  Unifying pricing formula for several stochastic volatility models with jumps , 2017 .

[9]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

[10]  David H. Bailey,et al.  A Fast Method for the Numerical Evaluation of Continuous Fourier and Laplace Transforms , 1994, SIAM J. Sci. Comput..

[11]  T. Sobotka,et al.  On Optimization Techniques for Calibration of Stochastic Volatility Models , 2014 .

[12]  Antoine Jacquier,et al.  The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model , 2012, SIAM J. Financial Math..

[13]  F. Delbaen,et al.  Convergence of discretized stochastic (interest rate) processes with stochastic drift term , 1998 .

[14]  A. Pelsser,et al.  UvA-DARE ( Digital Academic Repository ) Efficient , almost exact simulation of the Heston stochastic volatility model , 2008 .

[15]  Oleksandr Zhylyevskyy,et al.  A fast Fourier transform technique for pricing American options under stochastic volatility , 2010 .

[16]  E. Stein,et al.  Stock Price Distributions with Stochastic Volatility: An Analytic Approach , 1991 .

[17]  Wim Schoutens,et al.  The little Heston trap , 2006 .

[18]  F. Rouah.  The Heston Model and Its Extensions in Matlab and C#: Rouah/The Heston Model and Its Extensions in Matlab and C# , 2013 .

[19]  Christian Kahl,et al.  Fast strong approximation Monte Carlo schemes for stochastic volatility models , 2006 .

[20]  Eric Benhamou,et al.  Time Dependent Heston Model , 2009, SIAM J. Financial Math..

[21]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[22]  Jan Pospíšil,et al.  Market calibration under a long memory stochastic volatility model , 2016 .

[23]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[24]  Mukarram Attari.  Option Pricing Using Fourier Transforms: A Numerically Efficient Simplification , 2004 .

[25]  Christian Kahl,et al.  Not-so-complex logarithms in the Heston model , 2006 .

[26]  Louis O. Scott.  Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

[27]  Leif Andersen.  Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[28]  Oleksandr Zhylyevskyy.  Efficient Pricing of European-Style Options Under Heston's Stochastic Volatility Model , 2012 .

[29]  M. Joshi,et al.  Fast and Accurate Long Stepping Simulation of the Heston Stochastic Volatility Model , 2010 .

[30]  Aurélien Alfonsi,et al.  High order discretization schemes for the CIR process: Application to affine term structure and Heston models , 2010, Math. Comput..

[31]  A. Elices,et al.  Models with time-dependent parameters using transform methods: application to Heston's model , 2007, 0708.2020.

[32]  S. Ben Hamida,et al.  Recovering Volatility from Option Prices by Evolutionary Optimization , 2004 .

[33]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[34]  Jim Gatheral,et al.  Pricing under rough volatility , 2015 .

[35]  Cornelis W. Oosterlee,et al.  A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions , 2008, SIAM J. Sci. Comput..

[36]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[37]  Jim Gatheral.  The Volatility Surface: A Practitioner's Guide , 2006 .

[38]  David H. Bailey,et al.  The Fractional Fourier Transform and Applications , 1991, SIAM Rev..

[39]  D. Dijk,et al.  A comparison of biased simulation schemes for stochastic volatility models , 2008 .

[40]  Luis Ortiz-Gracia,et al.  A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options , 2015, SIAM J. Sci. Comput..

[41]  R. Poulsen,et al.  Approximation behoves calibration , 2013 .

[42]  D. Higham,et al.  Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process , 2005 .

[43]  Antoine Jacquier,et al.  The large-maturity smile for the Heston model , 2011, Finance Stochastics.

[44]  Jan Pospísil,et al.  On calibration of stochastic and fractional stochastic volatility models , 2016, Eur. J. Oper. Res..

[45]  Rafael de Santiago,et al.  CALIBRATION OF STOCHASTIC VOLATILITY MODELS VIA SECOND-ORDER APPROXIMATION: THE HESTON CASE , 2015 .

[46]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

[47]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .