A forward started jump-diffusion model and pricing of cliquet style exotics
Tóm tắt
In this paper we present an alternative model for pricing exotic options and structured products with forward-starting components. As presented in the recent study by Eberlein and Madan (Quantitative Finance 9(1):27–42, 2009), the pricing of such exotic products (which consist primarily of different variations of locally/globally, capped/floored, arithmetic/geometric etc. cliquets) depends critically on the modeling of the forward–return distributions. Therefore, in our approach, we directly take up the modeling of forward variances corresponding to the tenor structure of the product to be priced. We propose a two factor forward variance market model with jumps in returns and volatility. It allows the model user to directly control the behavior of future smiles and hence properly price forward smile risk of cliquet-style exotic products. The key idea, in order to achieve consistency between the dynamics of forward variance swaps and the underlying stock, is to adopt a forward starting model for the stock dynamics over each reset period of the tenor structure. We also present in detail the calibration steps for our proposed model.
Tài liệu tham khảo
Backus, D., Foresi, S., & Wu, L. (2004). Accounting for Biases in Back-Scholes, available at SSRN: http://ssrn.com/abstract=585623.
Bakshi G., Cao C., Chen Z. (1997) Empirical performance of alternative option pricing models. The Journal of Finance LII 5: 2003–2049
Barndorff-Nielsen O. E., Shephard N. (2001) Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial econometrics. Journal of the Royal Statistical Society 63: 167–241
Bates D. (1996) Jumps and stochastic volatility: The exchange rate processes implicit in Deutschemark options. Review of Financial Studies 9: 69–107
Bates D. (2000) Post-’87 crash fears in S&P500 futures options. Journal of Econometrics v.95: 181–238
Bergomi, L. (2004). Smile dynamics 1. Risk, September. London: Incisive Media.
Bergomi, L. (2005). Smile dynamics 2. Risk, October. London: Incisive Media.
Black F., Scholes M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654
Carr P., Ellis K., Gupta V. (1998) Static hedging of exotic options. Journal of Finance 53(3): 1165–1190
Carr P., Geman H., Madan D., Yor M. (2003) Stochastic volatility for Levy processes. Mathematical Finance 13(3): 345–382
Carr P., Madan D. (1998) Towards a theory of volatility trading, volatility. In: Jarrow R. (eds) Volatility: New estimation techniques for pricing derivatives. Risk Publications, London, pp 417–427
Cont, R., & Tankov, P. (2004). Financial modeling with jump processes. Chapman & Hall, CRC Financial Mathematics Series. ISBN 1-58488-413-4.
Duffie D., Pan J., Singleton K. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376
Eberlein E., Madan D. (2009) Sato processes and the valuation of structured products. Quantitative Finance 9(1): 27–42
Kemma A. G. Z., Vorst A. C. F. (1990) A pricing method for options based on average asset values. Journal of Banking and Finance 14: 113–129
Mehta, N. B., Molisch, A. F., Wu, J., & Zhang, J. (2006). Approximating the sum of correlated Lognormal or Lognormal-Rice random variables. IEEE International Conference on Communications (ICC) (Vol. 4, pp. 1605–1610). June, ISSN: 8164-9547.
Merton R. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3: 125–144
Milevsky M.A., Posner S.E. (1998) Asian options, the sum of lognormals and the reciprocal gamma distribution. The Journal of Financial and Quantitative Analysis 33(3): 409–422
Overhaus, M. et al. (2007). Equity hybrid derivatives. Wiley.
Poulsen R. (2006) Barrier options and their static hedges: Simple derivations and extensions. Quantitative Finance 6(4): 327–335
Schoutens, W., Simons, E., & Tistaert, J. (2004). A perfect calibration! Now what? Wilmott, March. Wiley.