This paper considers the pricing of turbo warrants under a hybrid stochastic and local volatility model. The model consists of the constant elasticity of variance model incorporated by a fast fluctuating Ornstein-Uhlenbeck process for stochastic volatility. The sensitive structure of the turbo warrant price is revealed by asymptotic analysis and numerical computation based on the observation that the elasticity of variance controls leverage effects and plays an important role in characterizing various phases of volatile markets. 1. Introduction Turbo warrants, which appeared first in Germany in late 2001, have experienced a considerable growth in Northern Europe and Hong Kong. They are special types of knockout barrier options in which the rebate is calculated as another exotic option. For one thing, this contract has a low vega so that the option price is less sensitive to the change of the implied volatility of the security market, and for another, it is highly geared owing to the possibility of knockout. Closed form expression for the price has been presented by Eriksson [1] under geometric Brownian motion (GBM) for the underlying security. It is well-known that the assumption of the GBM for the underlying security price in the Black-Scholes model [2] does not capture many empirical lines of evidence appeared in financial markets. Maybe, the two most significant shortcomings of the model's assumption lie in flat implied volatility, whereas the volatility fluctuates depending upon market conditions and the underestimation of extreme moves, yielding tail risk. So, there have been a lot of alternative underlying models developed to extend the GBM and overcome these problems. Local volatility models are one type of them, where the volatility depends on the price level of the underlying security itself. The most well-known local volatility model is the constant elasticity of variance (CEV) model in which the volatility is given by a power function of the underlying security price. It has been proposed by Cox [3] and Cox and Ross [4]. Another version of volatility models has been developed by assuming security's volatility to be a random process governed by another state variables such as the tendency of volatility to revert to some long-run mean value, the variance of the volatility process itself, and so forth. This type of models is called (pure) stochastic volatility models. The models developed by Heston [5] and Fouque et al. [6] are popular ones among others in this category. Also, there is another type of generalized models for the underlying
References
[1]
J. Eriksson, On the pricing equations of some path-dependent options [Ph.D. dissertation], Uppsala University, Uppsala, Sweden, 2006.
[2]
F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–654, 1973.
[3]
J. Cox, “Notes on option pricing I: constant elasticity of variance diffusions,” Working Paper, Stanford University, 1975, reprinted in The Journal of Portfolio Management, vol. 22, pp. 15–17, 1996.
[4]
J. C. Cox and S. A. Ross, “The valuation of options for alternative stochastic processes,” Journal of Financial Economics, vol. 3, no. 1-2, pp. 145–166, 1976.
[5]
S. L. Heston, “Closed-form solution for options with stochastic volatility with applications to bond and currency options,” The Review of Financial Studies, vol. 6, pp. 327–343, 1993.
[6]
J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, “Asymptotics of a two-scale stochastic volatility model,” in Equations Aux Derivees Partielles et Applications, Articles Dedies a Jacques-Louis Lions, pp. 517–525, Gauthier-Villars, Paris, France, 1998.
[7]
P. Carr, H. Geman, D. B. Madan, and M. Yor, “Stochastic volatility for Lévy processes,” Mathematical Finance, vol. 13, no. 3, pp. 345–382, 2003.
[8]
P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, Managing Smile Risk, Wilmott Magazine, 2002.
[9]
S.-Y. Choi, J.-P. Fouque, and J.-H. Kim, “Option pricing under hybrid stochastic and local volatility,” Quantitative Finance, vol. 13, no. 8, pp. 1157–1165, 2013.
[10]
B.-H. Bock, S.-Y. Choi, and J.-H. Kim, “The pricing of European options under the constant elasticity of variance with stochastic volatility,” Fluctuation and Noise Letters, vol. 12, no. 1, Article ID 135004, 13 pages, 2013.
[11]
M.-K. Lee, J. -H. Kim, and K. -H. Jang, “Pricing arithmetic Asian options under hybrid stochastic and local volatility,” Journal of Applied Mathematics, Article ID 784386, 2014.
[12]
H. Y. Wong and C. M. Chan, “Turbo warrants under stochastic volatility,” Quantitative Finance, vol. 8, no. 7, pp. 739–751, 2008.
[13]
A. Domingues, The valuation of turbo warrants under the CEV Model [M.S. thesis], ISCTE Business School, University Institute of Lisbon, Lisbon, Portugal, 2012.
[14]
H. Y. Wong and K. Y. Lau, “Analytical valuation of turbo warrants under double exponential jump diffusion,” Journal of Derivatives, vol. 15, no. 4, pp. 61–73, 2008.
[15]
G. Tataru and F. Travis, Stochastic Local Volatility, Quantitative Development Group, Bloomberg, 2010, version 1.
[16]
J.-H. Kim, J. Lee, S.-P. Zhu, and S.-H. Yu, “A multiscale correction to the Black-Scholes formula,” Applied Stochastic Models in Business and Industry. In press.
[17]
B. ？ksendal, Stochastic Differential Equations, Springer, New York, NY, USA, 6th edition, 2003.
[18]
H. Y. Wong and C. M. Chan, “Lookback options and dynamic fund protection under multiscale stochastic volatility,” Insurance, vol. 40, no. 3, pp. 357–385, 2007.
[19]
J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. S？lna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, UK, 2011.
[20]
D. Davydov and V. Linetsky, “Pricing and hedging path-dependent options under the CEV process,” Management Science, vol. 47, no. 7, pp. 949–965, 2001.
[21]
S.-H. Park and J.-H. Kim, “Asymptotic option pricing under the CEV diffusion,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 490–501, 2011.
[22]
J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, “Singular perturbations in option pricing,” SIAM Journal on Applied Mathematics, vol. 63, no. 5, pp. 1648–1665, 2003.