# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 60J75 Jump processes 60H07 Stochastic calculus of variations and the Malliavin calculus 91G80 Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
##### 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 · doi:10.1023/A:1011354913068 [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 · doi:10.1007/s00780-008-0086-4 [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/ [6] Benhamou, E., Gobet, E., Miri, M.: Time-dependent Heston model. SSRN preprint (2009). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1367955 [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) · doi:10.1086/260062 [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 · doi:10.1016/j.spa.2007.03.010 [10] Cass, T.: Smooth densities for solutions to stochastic differential equations with jumps. Stoch. Process. Appl. 119, 1416–1435 (2008) · Zbl 1161.60321 · doi:10.1016/j.spa.2008.07.005 [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 · doi:10.1137/S0036142903436186 [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) [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 · doi:10.1007/s007800050068 [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) [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). [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 · doi:10.1137/S0363012902419059 [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 · doi:10.1080/135048699334500 [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) [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 · doi:10.1051/m2an:2004003 [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 · doi:10.1016/0304-405X(76)90022-2 [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) [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) [31] Rubinstein, M.: Implied binomial trees. J. Finance 49, 771–818 (1994) · doi:10.1111/j.1540-6261.1994.tb00079.x [32] Rudin, W.: Real and Complex Analysis. McGraw-Hill, Toronto (1966) [33] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) [34] Siopacha, M., Teichmann, J.: Weak and strong Taylor methods for numerical solutions of stochastic differential equations (2007). http://arxiv.org/abs/0704.0745