FEATURES OF NUMERICAL SIMULATION OF PRICE DYNAMICS BASED ON THE HESTON MODEL
Abstract
In the paper, we analyse the characteristics of a numerical study of assets and derivatives pricing in financial engineering. The result of the development of the onedimensional Black-Scholes model is stochastic volatility models, in particular a twodimensional Heston model. The appearance of two- and multidimensional models pushes a wider application of numerical methods in the analysis of stochastic pricing models since finding analytic solutions is considerably harder or even impossible. Although the Heston model has analytic solutions, their application requires precise numerical calculations of integral equations. For numerical analysis of the stochastic Heston model the Euler and Milstein schemes for solving stochastic differential equations are applied. Practical implementation is done using Mathematica software. Numerical arrays of values for the transitional probability density of asset price, volatility price, and option price values were found. Interpolation methods of Mathematica software were used, which are based on histogram data, giving the ability to receive values of transitional probability densities for a wider range of changes of asset price and volatility {S(t),V(t)}. However Heston model as well as other twodimensional models contains plenty of parameters. For that reason studying of properties of asset price dynamics and option price in the entire value range of parameters is quite a cumbersome problem. In the work, we propose a qualitative numerical analysis based on the Heston model. Since the Heston model is an extension of the Black-Scholes model for stochastic volatility, to understand pricing dynamics a comparative analysis with the Black-Scholes model is given. For this, in the Black-Scholes model, we set the value of constant volatility equal σ=√(V_0 ) , where V_0 – is the initial value of stochastic volatility in the Heston model. As a result, a qualitative analysis of option pricing in the Heston model based on studying of dynamics of average volatility 〈V(t)〉 is proposed. If the average volatility value √(〈V(t)〉 ) is greater than the initial value √(V_0 ) on the entire time interval t then the option price in the Heston model is higher than the Black-Scholes model implies. In case of a decrease in average volatility value compared to the initial value, the option price in the Heston model will be lower than the Black-Scholes model implies. Similar qualitative analysis can apply to problems of pricing in other models of greater dimensionality.
References
Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973. ISSN 0022-3808. DOI: 10.1086/260062.
Hull J.C. and White A.D., The pricing of options on assets with stochastic volatilities. J. Finance 42(2), 1987, 281–300. ISSN 1540-6261. DOI: 10.1111/j.1540-6261.1987.tb02568.x.
Scott L.O., Option pricing when the variance changes randomly: Theory, estimation, and an application. The Journal of Financial and Quantitative Analysis 22(4), 1987, 419–438. ISSN 0022-1090. DOI: 10.2307/2330793.
Stein J. and Stein E., Stock price distributions with stochastic volatility: An analytic approach. The Review of Financial Studies 4(4), 1991, 727–752. ISSN 0893-9454. DOI: 10.1093/rfs/4.4.727.
Heston S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2), 1993, 327–343. ISSN 0893-9454. DOI: 10.1093/rfs/6.2.327.
Higham D.J. and Mao X., Convergence of monte carlo simulations involving the mean-reverting square root process. Journal of Computational Finance 8(3), 2005, 35–61. ISSN 1460-1559. DOI: 10.21314/JCF.2005.136.
Lord R., Koekkoek R., and van Dijk D., A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance 10(2), 2010, 177–194. ISSN 1469-7688. DOI: 10.1080/14697680802392496.
Andersen L., Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance 11(3), 2008, 1–42. ISSN 1460-1559. DOI: 10.21314/JCF.2008.189.
Fabrice D. Rouah, Steven L. The Heston Model and its Extensions in Matlab and C#, Website. Wiley Finance Series. John Wiley & Sons, Inc., Hoboken, NJ, 2013. ISBN 9781118548257.
John C. Cox, Jonathan E. Ingersoll, Jr. and Stephen A. Ross, A theory of the term structure of interest rates. Econometrica 53(2), 1985, 385–407. ISSN 0012-9682. DOI: 10.2307/1911242.
Yuh-Dauh Lyuu. Financial Engineering and Computation: Principles, Mathematics, and Algorithms. Cambridge University Press, 648 p. (2004). ISBN: 9780521781718.
Gatheral J., The volatility surface: A practitioner’s guide. Wiley Finance. John Wiley & Sons, Hoboken, New Jersey, 2006. ISBN 9780470068250.
Martha L. Abell, James P. Braselton Mathematica by Example, 5th Edition, Academic Press, 2017. ISBN-13:978-0128124819.
L. A. Grzelak and C. W. Oosterlee, On the Heston model with Stochastic Interest Rate, SIAM Journal on Financial Mathematics, (2) (2011), 255-286. DOI. 10.1137/090756119
Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973. ISSN 0022-3808. DOI: 10.1086/260062.
Hull J.C. and White A.D., The pricing of options on assets with stochastic volatilities. J. Finance 42(2), 1987, 281–300. ISSN 1540-6261. DOI: 10.1111/j.1540-6261.1987.tb02568.x.
Scott L.O., Option pricing when the variance changes randomly: Theory, estimation, and an application. The Journal of Financial and Quantitative Analysis 22(4), 1987, 419–438. ISSN 0022-1090. DOI: 10.2307/2330793.
Stein J. and Stein E., Stock price distributions with stochastic volatility: An analytic approach. The Review of Financial Studies 4(4), 1991, 727–752. ISSN 0893-9454. DOI: 10.1093/rfs/4.4.727.
Heston S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2), 1993, 327–343. ISSN 0893-9454. DOI: 10.1093/rfs/6.2.327.
Higham D.J. and Mao X., Convergence of monte carlo simulations involving the mean-reverting square root process. Journal of Computational Finance 8(3), 2005, 35–61. ISSN 1460-1559. DOI: 10.21314/JCF.2005.136.
Lord R., Koekkoek R., and van Dijk D., A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance 10(2), 2010, 177–194. ISSN 1469-7688. DOI: 10.1080/14697680802392496.
Andersen L., Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance 11(3), 2008, 1–42. ISSN 1460-1559. DOI: 10.21314/JCF.2008.189.
Fabrice D. Rouah, Steven L. The Heston Model and its Extensions in Matlab and C#, Website. Wiley Finance Series. John Wiley & Sons, Inc., Hoboken, NJ, 2013. ISBN 9781118548257.
John C. Cox, Jonathan E. Ingersoll, Jr. and Stephen A. Ross, A theory of the term structure of interest rates. Econometrica 53(2), 1985, 385–407. ISSN 0012-9682. DOI: 10.2307/1911242.
Yuh-Dauh Lyuu. Financial Engineering and Computation: Principles, Mathematics, and Algorithms. Cambridge University Press, 648 p. (2004). ISBN: 9780521781718.
Gatheral J., The volatility surface: A practitioner’s guide. Wiley Finance. John Wiley & Sons, Hoboken, New Jersey, 2006. ISBN 9780470068250.
Martha L. Abell, James P. Braselton Mathematica by Example, 5th Edition, Academic Press, 2017. ISBN-13:978-0128124819.
L. A. Grzelak and C. W. Oosterlee, On the Heston model with Stochastic Interest Rate, SIAM Journal on Financial Mathematics, (2) (2011), 255-286. DOI. 10.1137/090756119
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