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Analytical approximation of the transition density in a local volatility model. (English) Zbl 1246.91137

The paper is mainly concerned with one-dimensional local volatility market models of the form \[ \text{d}S_t=\mu(t, S_t)\text{d} t+\sigma(t, S_t)S_t\text{d}W_t. \] Exploiting the analytical approximation of function \(\sigma\) the authors obtain an expansion of the transition density \(\Gamma(t, S_0; T, S)\) of \(S\) of the form \[ \Gamma(t, S_0; T, S)\approx \Gamma^0(t, S_0; T, S)+\sum_{n=1}^NJ_{S_0}^n\Gamma^0(t, S_0; T, S), \quad N\geq1, \] where the main term \(\Gamma^0(t, S_0; T, S)\) is the density in a suitable Black-Scholes model, while \(J_{S_0}^n\) is a differential operator containing derivatives w.r.t. \(S_0\). This is the expansion in terms of BS-Greeks. An iterative algorithm for the explicit expressions of the operators \(J_{S_0}^n\) is constructed. This algorithm is straightforward to implement by using a symbolic computation software. The analytical approximation formulae are tested with a Monte-Carlo simulation in the context of one-dimensional local volatility models (standard quadratic and CEV).

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91B26 Auctions, bargaining, bidding and selling, and other market models
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35K10 Second-order parabolic equations
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
91G60 Numerical methods (including Monte Carlo methods)
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References:

[1] Antonelli F., Scarlatti S., Pricing options under stochastic volatility: a power series approach, Finance Stoch., 2009, 13(2), 269-303 http://dx.doi.org/10.1007/s00780-008-0086-4; · Zbl 1199.91200
[2] Barjaktarevic J.P., Rebonato R., Approximate solutions for the SABR model: improving on the Hagan expansion, Talk at ICBI Global Derivatives Trading and Risk Management Conference, 2010;
[3] Benhamou E., Gobet E., Miri M., Expansion formulas for European options in a local volatility model, Int. J. Theor. Appl. Finance, 2010, 13(4), 603-634 http://dx.doi.org/10.1142/S0219024910005887; · Zbl 1205.91153
[4] Berestycki H., Busca J., Florent I., Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 2004, 57(10), 1352-1373 http://dx.doi.org/10.1002/cpa.20039; · Zbl 1181.91356
[5] Capriotti L., The exponent expansion: an effective approximation of transition probabilities of diffusion processes and pricing kernels of financial derivatives, Int. J. Theor. Appl. Finance, 2006, 9(7), 1179-1199 http://dx.doi.org/10.1142/S0219024906003925; · Zbl 1140.91380
[6] Cheng W., Costanzino N., Liechty J., Mazzucato A., Nistor V., Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing, SIAM J. Financial Math., 2011, 2, 901-934 http://dx.doi.org/10.1137/100796832; · Zbl 1242.35131
[7] Constantinescu R., Costanzino N., Mazzucato A.L., Nistor V., Approximate solutions to second order parabolic equations I: analytic estimates, J. Math. Phys., 2010, 51(10), #103502; · Zbl 1314.35063
[8] Corielli F., Foschi P., Pascucci A., Parametrix approximation of diffusion transition densities, SIAM J. Financial Math., 2010, 1, 833-867 http://dx.doi.org/10.1137/080742336;
[9] Cox J.C., Notes on option pricing I: constant elasticity of variance diffusion, Stanford University, Stanford, 1975, manuscript;
[10] Davydov D., Linetsky V., Pricing and hedging path-dependent options under the CEV process, Management Sci., 2001, 47(7), 949-965 http://dx.doi.org/10.1287/mnsc.47.7.949.9804; · Zbl 1232.91659
[11] Delbaen F., Shirakawa H., A note on option pricing for the constant elasticity of variance model, Financial Engineering and the Japanese Markets, 2002, 9(2), 85-99; · Zbl 1072.91020
[12] Doust P., No arbitrage SABR, 2010, manuscript;
[13] Ekström E., Tysk J., Boundary behaviour of densities for non-negative diffusions, preprint available at www.math.uu.se/ johant/pq.pdf; · Zbl 1140.35494
[14] Foschi P., Pagliarani S., Pascucci A., Black-Scholes formulae for Asian options in local volatility models, preprint available at http://ssrn.com/paper=1898992; · Zbl 1260.91100
[15] Fouque J.-P., Papanicolaou G., Sircar R., Solna K., Singular perturbations in option pricing, SIAM J. Appl. Math., 2003, 63(5), 1648-1665 http://dx.doi.org/10.1137/S0036139902401550; · Zbl 1039.91024
[16] Gatheral J., Hsu E.P., Laurence P., Ouyang C., Wang T.-H., Asymptotics of implied volatility in local volatility models, Math. Finance (in press), DOI: 10.1111/j.1467-9965.2010.00472.x; · Zbl 1270.91093
[17] Gatheral J., Wang T.-H., The heat-kernel most-likely-path approximation, preprint available at http://ssrn.com/paper=1663318; · Zbl 1236.91147
[18] Hagan P.S., Kumar D., Lesniewski A.S., Woodward D.E., Managing smile risk, Wilmott Magazine, 2002, September, 84-108;
[19] Hagan P., Lesniewski A., Woodward D., Probability distribution in the SABR model of stochastic volatility, 2005, preprint available at www.lesniewski.us/papers/working/ProbDistrForSABR.pdf; · Zbl 1418.91517
[20] Hagan P.S., Woodward D.E., Equivalent Black volatilities, Appl. Math. Finance, 1999, 6(3), 147-157 http://dx.doi.org/10.1080/135048699334500; · Zbl 1009.91033
[21] Henry-Labordère P., A geometric approach to the asymptotics of implied volatility, In: Frontiers in Quantitative Finance, Wiley Finance Ser., John Wiley & Sons, Hoboken, 2008, chapter 4, 89-128;
[22] Henry-Labordère P., Analysis, Geometry, and Modeling in Finance, Chapman Hall/CRC Financ. Math. Ser., CRC Press, Boca Raton, 2009; · Zbl 1151.91006
[23] Heston S.L., Loewenstein M., Willard G.A., Options and bubbles, Review of Financial Studies, 2007, 20(2), 359-390 http://dx.doi.org/10.1093/rfs/hhl005;
[24] Howison S., Matched asymptotic expansions in financial engineering, J. Engrg. Math., 2005, 53(3-4), 385-406 http://dx.doi.org/10.1007/s10665-005-7716-z; · Zbl 1099.91061
[25] Janson S., Tysk J., Feynman-Kac formulas for Black-Scholes-type operators, Bull. London Math. Soc., 2006, 38(2), 269-282 http://dx.doi.org/10.1112/S0024609306018194; · Zbl 1110.35021
[26] Kristensen D., Mele A., Adding and subtracting Black-Scholes: a new approach to approximating derivative prices in continuous-time models, Journal of Financial Economics, 2011, 102(2), 390-415 http://dx.doi.org/10.1016/j.jfineco.2011.05.007;
[27] Lesniewski A., Swaption smiles via the WKB method, Mathematical Finance Seminar, Courant Institute of Mathematical Sciences, 2002;
[28] Lindsay A., Brecher D., Results on the CEV process, past and present, preprint available at http://ssrn.com/paper=1567864; · Zbl 1237.91218
[29] Pagliarani S., Pascucci A., Riga C., Expansion formulae for local Lévy models, preprint available at http://ssrn.com/paper=1937149; · Zbl 1285.60084
[30] Pascucci A., PDE and Martingale Methods in Option Pricing, Bocconi Springer Ser., 2, Springer, Milan, 2011; · Zbl 1214.91002
[31] Paulot L., Asymptotic implied volatility at the second order with application to the SABR model, preprint available at http://ssrn.com/paper=1413649; · Zbl 1418.91533
[32] Shaw W.T., Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, 1998; · Zbl 0939.91002
[33] Taylor S., Perturbation and Symmetry Techniques Applied to Finance, PhD thesis, Frankfurt School of Finance & Management, Bankakademie HfB, 2011;
[34] Whalley A.E., Wilmott P., An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Math. Finance, 1997, 7(3), 307-324 http://dx.doi.org/10.1111/1467-9965.00034; · Zbl 0885.90019
[35] Widdicks M., Duck P.W., Andricopoulos A.D., Newton D.P., The Black-Scholes equation revisited: asymptotic expansions and singular perturbations, Math. Finance, 2005, 15(2), 373-391 http://dx.doi.org/10.1111/j.0960-1627.2005.00224.x; · Zbl 1124.91342
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