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September  2015, 35(9): 3965-3988. doi: 10.3934/dcds.2015.35.3965

## Large deviations for some fast stochastic volatility models by viscosity methods

 1 Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35133 Padova, Italy, Italy, Italy

Received  May 2014 Revised  September 2014 Published  April 2015

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.
Citation: Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
##### References:
 [1] O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control,, SIAM J. Control Optim., 40 (): 1159.  doi: 10.1137/S0363012900366741. [2] O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61. doi: 10.1007/s00205-003-0266-5. [3] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. doi: 10.1090/S0065-9266-09-00588-2. [4] O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027. [5] M. Avellaneda, D. Boyer-Olson, J. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C.R. Math. Acad. Sci. Paris, 336 (2003), 263-266. doi: 10.1016/S1631-073X(03)00032-3. [6] M. Arisawa and P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413. [7] S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione, Master thesis, University of Padova, 2012. [8] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [9] M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265. doi: 10.1137/090748147. [10] M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach, Eur. J. Control, 17 (2011), 30-45. doi: 10.3166/ejc.17.30-45. [11] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, (). [12] G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44. doi: 10.1007/BF01445155. [13] F. Camilli, A. Cesaroni and C. Marchi, Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates, Adv. Nonlinear Stud., 11 (2011), 405-428. [14] F. Da Lio and O. Ley, Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications, SIAM J. Control Optim., 45 (2006), 74-106. doi: 10.1137/S0363012904440897. [15] G. Dal Maso and H. Frankowska, Value functions for Boltza problems with discontinuous lagrangian and Hamilton-Jacobi inequality, ESAIM Control Optim. Calc. Var., 5 (2000), 369-393. doi: 10.1051/cocv:2000114. [16] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4. [17] P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods, Stoch. Process. Appl., 122 (2012), 1947-1987. doi: 10.1016/j.spa.2011.12.006. [18] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [19] L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 1-20. [20] J. Feng, M. Forde and J.-P. Fouque, Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, SIAM J. Financial Math., 1 (2010), 126-141. doi: 10.1137/090745465. [21] J. Feng, J.-P. Fouque and R. Kumar, Small time asymptotic for fast mean-reverting stochstic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575. doi: 10.1214/11-AAP801. [22] J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131. [23] W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions, Springer, New York, 2006. [24] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge university press, Cambridge, 2000. [25] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), 1648-1665. doi: 10.1137/S0036139902401550. [26] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), 22-42. doi: 10.1137/030600291. [27] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9781139020534. [28] D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., (). [29] Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-13242-5. [30] H. Kaise and S. Sheu, On the structure of solution of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2006), 284-320. doi: 10.1214/009117905000000431. [31] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston 1990. doi: 10.1007/978-1-4612-4482-0. [32] H.J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322. doi: 10.1007/s00245-010-9104-y. [33] R. Lipster, Large deviations for two scaled diffusions, Probab. Theory Relat. Fields, 106 (1996), 71-104. doi: 10.1007/s004400050058. [34] K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion, Appl Math Optim, 67 (2013), 123-161. doi: 10.1007/s00245-012-9183-z. [35] A. Takahashi and K. Yamamoto, A remark on a singular perturbation method for option pricing under a stochastic volatility, Asia-Pacific Financial Markets, 16 (2009), 333-345. [36] A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging, Stochastic Process. Appl., 89 (2000), 69-79. doi: 10.1016/S0304-4149(00)00013-2. [37] D. Williams, Probability with Martingales, Cambridge University Press, 1991. doi: 10.1017/CBO9780511813658.

show all references

##### References:
 [1] O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control,, SIAM J. Control Optim., 40 (): 1159.  doi: 10.1137/S0363012900366741. [2] O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61. doi: 10.1007/s00205-003-0266-5. [3] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. doi: 10.1090/S0065-9266-09-00588-2. [4] O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027. [5] M. Avellaneda, D. Boyer-Olson, J. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C.R. Math. Acad. Sci. Paris, 336 (2003), 263-266. doi: 10.1016/S1631-073X(03)00032-3. [6] M. Arisawa and P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413. [7] S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione, Master thesis, University of Padova, 2012. [8] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. [9] M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265. doi: 10.1137/090748147. [10] M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach, Eur. J. Control, 17 (2011), 30-45. doi: 10.3166/ejc.17.30-45. [11] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, (). [12] G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44. doi: 10.1007/BF01445155. [13] F. Camilli, A. Cesaroni and C. Marchi, Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates, Adv. Nonlinear Stud., 11 (2011), 405-428. [14] F. Da Lio and O. Ley, Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications, SIAM J. Control Optim., 45 (2006), 74-106. doi: 10.1137/S0363012904440897. [15] G. Dal Maso and H. Frankowska, Value functions for Boltza problems with discontinuous lagrangian and Hamilton-Jacobi inequality, ESAIM Control Optim. Calc. Var., 5 (2000), 369-393. doi: 10.1051/cocv:2000114. [16] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4. [17] P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods, Stoch. Process. Appl., 122 (2012), 1947-1987. doi: 10.1016/j.spa.2011.12.006. [18] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [19] L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 1-20. [20] J. Feng, M. Forde and J.-P. Fouque, Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, SIAM J. Financial Math., 1 (2010), 126-141. doi: 10.1137/090745465. [21] J. Feng, J.-P. Fouque and R. Kumar, Small time asymptotic for fast mean-reverting stochstic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575. doi: 10.1214/11-AAP801. [22] J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131. [23] W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions, Springer, New York, 2006. [24] J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge university press, Cambridge, 2000. [25] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), 1648-1665. doi: 10.1137/S0036139902401550. [26] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), 22-42. doi: 10.1137/030600291. [27] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9781139020534. [28] D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., (). [29] Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-13242-5. [30] H. Kaise and S. Sheu, On the structure of solution of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2006), 284-320. doi: 10.1214/009117905000000431. [31] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston 1990. doi: 10.1007/978-1-4612-4482-0. [32] H.J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322. doi: 10.1007/s00245-010-9104-y. [33] R. Lipster, Large deviations for two scaled diffusions, Probab. Theory Relat. Fields, 106 (1996), 71-104. doi: 10.1007/s004400050058. [34] K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion, Appl Math Optim, 67 (2013), 123-161. doi: 10.1007/s00245-012-9183-z. [35] A. Takahashi and K. Yamamoto, A remark on a singular perturbation method for option pricing under a stochastic volatility, Asia-Pacific Financial Markets, 16 (2009), 333-345. [36] A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging, Stochastic Process. Appl., 89 (2000), 69-79. doi: 10.1016/S0304-4149(00)00013-2. [37] D. Williams, Probability with Martingales, Cambridge University Press, 1991. doi: 10.1017/CBO9780511813658.
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