Volume 16, no. 3Pages 5 - 19

Models with Uncertain Volatility

G.I. Beliavsky, N.V. Danilova
Models in which volatility is one of the possible trajectories are considered in the paper. As an example of a model with a certain volatility, the Black-Scholes model is considered. As an example of models with uncertain volatility three models are considered: the Heston model with random trajectories and two models with deterministic trajectories from a confidence set of possible trajectories. Three computational methods are proposed for finding the range of fair prices for a European option. The first method is based on solving viscosity equations using difference schemes. The second is the Monte-Carlo method, which is based on the simulation of the initial stock price process. The third is the tree method, which is based on approximating the original continuous model with a discrete model and obtaining recurrent formulas on a binary tree to calculate the upper and lower prices. The results of calculations using the listed methods are presented. It is shown that the ranges of fair prices obtained using the three numerical methods practically coincide.
Full text
Black-Scholes model; Heston model; uncertain volatility; viscosity equation; option; fair price.
1. Samuelson P. Rational Theory of Warrant Pricing. Industrial Management Review, 1965, vol. 6, no. 2, pp.13–31. DOI: 10.1007/978-3-319-22237-0-11
2. Black F., Scholes M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 1973, vol. 81, no. 3, pp. 637–659. DOI: 10.1142/9789814759588-0001
3. Merton R. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 1973, no. 4, pp. 141–183. DOI: 10.2307/3003143
4. Shryaev А. Osnovy stohasticheskih matematicheskih finansov [Basis of Stochastic Mathematical Finance]. Moscow, MCNMO, 2016. (in Russian)
5. Heston S. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 1993, no. 6, pp. 327–343. DOI: 10.1093/rfs/6.2.327
6. Rouah F., Steven L. The Heston Model and Its Extensions in Matlab and C. Hoboken, New Jersey, John Wiley and Sons, 2013.
7. Beliavsky G., Danilova N., Grober T. The Uncertainty Volatility Models and Tree Approximation. Applied Mathematical Sсiences, 2016, vol. 10, no. 19, pp. 921–930. DOI: 10.12988/ams.2016.6114
8. Avellaneda M., Levy A., Paras A. Pricing and Hedging Derivative Securities in Markets with Uncertain Volatilities. Applied Mathematical Finance, 1995, no. 2, pp. 73–88. DOI: 10.1080/13504869500000005
9. Hull J., White A. The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance, 1997, vol. 42, no. 2, pp. 281–300. DOI: 10.1111/j.1540-6261.1987.tb02568.x
10. Johnson H., Shanno D. Option Pricing when the Variance is Changing. Journal of Financial and Quantitative Analysis, 1987, vol. 22, no. 2, pp. 143–151. DOI: 10.2307/2330709
11. Meyer G. The Black-Scholes Barenblatt Equation for Options with Uncertain Volatility and its Application to Static Hedging. International Journal of Theoretical and Applied Finance,
2006, no. 9, pp. 673–703. DOI: 10.1142/S0219024906003755
12. Peng Shige. G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty. Probability, 2007, DOI: 10.48550/arXiv.0711.2834
13. Stein E., Stein J. Stock Price Distributions with Stochastic Volatility: an Analytic Approach. Reviews of Financial Studies, 1991, vol. 4, no. 4, pp. 727–752.
14. Tychonoff A. Theoremes d'unicite pour l’equation de la chaleur. Mathematics Sbornik, 1935, vol. 42, no. 2, pp. 199–216. (in French)
15. Scott L. Option Pricing when the Variance Changes Randomly. Theory, Estimation and an Application. Journal of Financial and Quantitative Analysis, 1987, vol. 22, no. 4, pp. 419–438. DOI: 10.2307/2330793
16. Beliavsky G., Danilova N. Control in Binary Models with Disorder. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2022, vol. 15, no. 3, pp. 67–82. (in Russian) DOI: 10.14529/mmp220305