Abstract
The Black-Scholes model does not account non-Markovian property and volatility smile or skew although asset price might depend on the past movement of the asset price and real market data can find a non-flat structure of the implied volatility surface. So, in this paper, we formulate an underlying asset model by adding a delayed structure to the constant elasticity of variance (CEV) model that is one of renowned alternative models resolving the geometric issue. However, it is still one factor volatility model which usually does not capture full dynamics of the volatility showing discrepancy between its predicted price and market price for certain range of options. Based on this observation we combine a stochastic volatility factor with the delayed CEV structure and develop a delayed hybrid model of stochastic and local volatilities. Using both a martingale approach and a singular perturbation method, we demonstrate the delayed CEV correction effects on the European vanilla option price under this hybrid volatility model as a direct extension of our previous work [12].
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References
Akgiray, D. Conditional heteroscedasticity in time series of stock returns. Journal of Business, 62(1): 55–80 (1989)
Arriojas, M., Hu, Y., Mohammed, S.-E., Pap, G. A delayed Black and Scholes formula. Stochastic Analysis and Applications, 25(2): 471–492 (2007)
Corazza, M., Malliaris, A.G. Multi-fractality in foreign currency market. Multinational Finance Journal, 6: 65–98 (2002)
Cox, J. Notes on option pricing I: constant elasticity of diffusions. Unpublished draft, Stanford University, 1975
Crech, D., Mazur, Z. Can one make any crash prediction in finance using the local Hurst exponent idea? Physica A, 336: 133–145 (2004)
Dibeh, G. Speculative dynamics in a time-delay model of asset prices. Physica A, 355(1): 199–208 (2005)
Fouque, J.-P., Papanicolaou, G.C., Sircar, K.R. Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge, 2000
Heston, S.L. Closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2): 327–343 (1993)
Ito, K., Nisio, M. On stationary solutions of a stochastic differential equation. Journal of Mathematics of Kyoto University, 4(1): 1–75 (1964)
Jackwerth, J.C., Rubinstein, M. Recovering probability distributions from contemporaneous security prices. Journal of Finance, 51: 1611–1631 (1996)
Kim, J.-H. Asymptotic theory of noncentered mixing stochastic differential equations. Stochastic Processes and their Applications, 114(1): 161–174 (2004)
Lee, M.-K., Kim, J.-H. A delay financial model with stochastic volatility; martingale method. Physica A: Statistical Mechanics and its Applications, 390(16): 2909–2919 (2011)
Lo, W. Long-term memory in stock market prices. Econometrica, 59(5): 1279–1313 (1991)
Mao, X. Stochastic Differential Equations and Applications. Horwood Publishing, West Sussex, 2007
Merton, R. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3: 125–144 (1976)
Oksendal, B. Stochastic Differential Equations. Springer, New York, 2003
Papanicolaou, G.C. Asymptotic analysis of stochastic equations. In: M. Rosenblatt (Ed.), Studies in Probability Theory, Mathematical Association of America, Washington DC, 111–179, 1978
Rubinstein, M. Nonparametric tests of alternative option pricing models. Journal of Finance, 40(2): 455–480 (1985)
Sheinkman, J., LeBaron, B. Nonlinear dynamics and stock returns. Journal of Business, 62(3): 311–337 (1989)
Stoica, G. A stochastic delay financial model. Proceedings of American Mathematical Society, 133(6): 1837–1841 (2004)
Swishchuk, A. Modeling and pricing of variance swaps for stochastic volatilities with delay. Wilmott Magazine, Issue 19, September, 63–73, 2005
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The work of J.-H. Kim was supported by the National Research Foundation of Korea NRF-2013R1A1A2A10006693.
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Lee, MK., Kim, JH. A delayed stochastic volatility correction to the constant elasticity of variance model. Acta Math. Appl. Sin. Engl. Ser. 32, 611–622 (2016). https://doi.org/10.1007/s10255-016-0588-3
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DOI: https://doi.org/10.1007/s10255-016-0588-3