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Smart expansion and fast calibration for jump diffusions. (English) Zbl 1195.91153
The authors use Malliavin calculus techniques to derive an analytic formula for the price of European options, for any model including local volatility and Poisson jump processes. To perform a rigorous analysis, they use a suitable parameterization that is just a tool to derive convenient representations. By using an asymptotic expansion in the context of small diffusions and small jumps (relative to the frequency or to the size), estimates for the derivatives are established. This allows making an explicit contribution at given order and to control the error. It is proved that the accuracy depends on the smoothness of the payoff function. It is also demonstrated that under realistic parameters, the accuracy is good enough, and model calibration becomes very rapid. It is observed that one may use the approximation price and obtain a volatility smile for short maturities and a volatility skew for long maturities.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60J75 Jump processes (MSC2010) 60H07 Stochastic calculus of variations and the Malliavin calculus 91G80 Financial applications of other theories
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##### References:
 [1] Albanese, C., Campolieti, G., Carr, P., Lipton, A.: Black–Scholes goes hypergeometric. Risk Mag. 14(12), 99–103 (2001) [2] Andersen, L., Andreasen, J.: Jump diffusion process: volatility smile fitting and numerical methods for pricing. Rev. Deriv. Res. 4, 231–262 (2000) · Zbl 1274.91398 [3] Andersen, L., Andreasen, J.: Volatile volatilities. Risk Mag. 15(12), 163–168 (2002) [4] Antonelli, F., Scarlatti, S.: Pricing options under stochastic volatility: a power series approach. Finance Stoch. 13, 269–303 (2009) · Zbl 1199.91200 [5] Benhamou, E., Gobet, E., Miri, M.: Closed forms for European options in a local volatility model. Int. J. Theor. Appl. Finance (2008, to appear). Available at http://hal.archives-ouvertes.fr/hal-00325939/fr/ · Zbl 1205.91153 [6] Benhamou, E., Gobet, E., Miri, M.: Time-dependent Heston model. SSRN preprint (2009). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1367955 · Zbl 1198.91203 [7] Benhamou, E., Miri, M.: Predictor corrector methods applied to PIDE and Monte Carlo simulations. Working Paper, Pricing Partners (2006). Available on request. [8] Scholes, M., Black, F.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) · Zbl 1092.91524 [9] Bouchard, B., Elie, R.: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Process. Appl. 118, 53–75 (2008) · Zbl 1136.60048 [10] Cass, T.: Smooth densities for solutions to stochastic differential equations with jumps. Stoch. Process. Appl. 119, 1416–1435 (2008) · Zbl 1161.60321 [11] Cont, R., Tankov, P.: Non-parametric calibration of jump diffusion option pricing models. J. Comput. Finance 7(3), 1–49 (2003) [12] Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43, 1596–1626 (2005) · Zbl 1101.47059 [13] Dupire, B.: Pricing with a smile. Risk Mag. 7(1), 18–20 (1994) [14] Fouque, J.P., Papanicolaou, G., Sircar, R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000) · Zbl 0954.91025 [15] Fournié, E., Lasry, J.M., Lebuchoux, J., Lions, P.L., Touzi, N.: Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3, 391–412 (1999) · Zbl 0947.60066 [16] Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25, 71–106 (1985) · Zbl 0575.60065 [17] Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley Finance, New York (2006) [18] Gobet, E.: Revisiting the Greeks for European and American options. In: Akahori, J., Ogawa, S., Watanabe, S. (eds.) Proceedings of the International Symposium on Stochastic Processes and Mathematical Finance, Ritsumeikan University, Kusatsu, Japan, pp. 53–71. World Scientific, Singapore (2004). · Zbl 1191.91054 [19] Gobet, E., Munos, R.: Sensitivity analysis using Itô–Malliavin calculus and martingales. Application to stochastic control problem. SIAM J. Control Optim. 43, 1676–1713 (2005) · Zbl 1116.60033 [20] Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Mag., 84–108 (2002) [21] Hagan, P.S., Woodward, D.E.: Equivalent Black volatilities. Appl. Math. Finance 6, 147–157 (1999) · Zbl 1009.91033 [22] Henry Labordère, P.: A general asymptotic implied volatility for stochastic volatility models (2005). http://arxiv.org/abs/cond-mat/0504317 [23] Lewis, A.: Option Valuation under Stochastic Volatility. Finance Press, New York (2000) · Zbl 0937.91060 [24] Matache, A.M., von Petersdorff, T., Schwab, C.: Fast deterministic pricing of options on Lévy driven assets. Math. Modell. Numer. Anal. 38, 37–71 (2004) · Zbl 1072.60052 [25] Matytsin, A.: Perturbative analysis of volatility smiles. Working paper (2000). http://www.math.columbia.edu/$$\sim$$smirnov/matytsin2000.pdf [26] Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1–2), 125–144 (1976) · Zbl 1131.91344 [27] Miri, M.: Stochastic expansions and closed pricing formulas of European options (tentative title). PhD thesis, Université de Grenoble (2009, in preparation). http://www-ljk.imag.fr/MATHFI/ [28] Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006) · Zbl 1099.60003 [29] Piterbarg, V.V.: A multi-currency model with FX volatility skew. SSRN working paper (2005). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=685084 [30] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge University Press, Cambridge (1992) · Zbl 0778.65003 [31] Rubinstein, M.: Implied binomial trees. J. Finance 49, 771–818 (1994) [32] Rudin, W.: Real and Complex Analysis. McGraw-Hill, Toronto (1966) · Zbl 0142.01701 [33] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) · Zbl 0973.60001 [34] Siopacha, M., Teichmann, J.: Weak and strong Taylor methods for numerical solutions of stochastic differential equations (2007). http://arxiv.org/abs/0704.0745 · Zbl 1214.91136
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