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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

Martino Bardi 1, , Annalisa Cesaroni 1, and  Daria Ghilli 1,

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.
Keywords: viscosity solutions, Singular perturbations, stochastic volatility, large deviations..
Mathematics Subject Classification: Primary: 93C70, 60F10, 49L25, 91G20, 35B2.
Citation: Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result,, Arch. Ration. Mech. Anal., 170 (2003), 17.  doi: 10.1007/s00205-003-0266-5.  Google Scholar

[3]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations,, Mem. Amer. Math. Soc., 204 (2010).  doi: 10.1090/S0065-9266-09-00588-2.  Google Scholar

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O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations,, J. Differential Equations, 243 (2007), 349.  doi: 10.1016/j.jde.2007.05.027.  Google Scholar

[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.  doi: 10.1016/S1631-073X(03)00032-3.  Google Scholar

[6]

M. Arisawa and P.-L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187.  doi: 10.1080/03605309808821413.  Google Scholar

[7]

S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione,, Master thesis, (2012).   Google Scholar

[8]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhäuser Boston, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[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.  doi: 10.1137/090748147.  Google Scholar

[10]

M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach,, Eur. J. Control, 17 (2011), 30.  doi: 10.3166/ejc.17.30-45.  Google Scholar

[11]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, ().   Google Scholar

[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.  doi: 10.1007/BF01445155.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0363012904440897.  Google Scholar

[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.  doi: 10.1051/cocv:2000114.  Google Scholar

[16]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications,, Springer, (1998).  doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[17]

P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods,, Stoch. Process. Appl., 122 (2012), 1947.  doi: 10.1016/j.spa.2011.12.006.  Google Scholar

[18]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/090745465.  Google Scholar

[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.  doi: 10.1214/11-AAP801.  Google Scholar

[22]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes,, American Mathematical Society, (2006).  doi: 10.1090/surv/131.  Google Scholar

[23]

W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions,, Springer, (2006).   Google Scholar

[24]

J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility,, Cambridge university press, (2000).   Google Scholar

[25]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing,, SIAM J. Appl. Math., 63 (2003), 1648.  doi: 10.1137/S0036139902401550.  Google Scholar

[26]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics,, Multiscale Model. Simul., 2 (2003), 22.  doi: 10.1137/030600291.  Google Scholar

[27]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives,, Cambridge University Press, (2011).  doi: 10.1017/CBO9781139020534.  Google Scholar

[28]

D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., ().   Google Scholar

[29]

Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control,, Springer-Verlag, (2003).  doi: 10.1007/978-3-662-13242-5.  Google Scholar

[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.  doi: 10.1214/009117905000000431.  Google Scholar

[31]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems,, Birkhäuser, (1990).  doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[32]

H.J. Kushner, Large deviations for two-time-scale diffusions, with delays,, Appl. Math. Optim., 62 (2010), 295.  doi: 10.1007/s00245-010-9104-y.  Google Scholar

[33]

R. Lipster, Large deviations for two scaled diffusions,, Probab. Theory Relat. Fields, 106 (1996), 71.  doi: 10.1007/s004400050058.  Google Scholar

[34]

K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion,, Appl Math Optim, 67 (2013), 123.  doi: 10.1007/s00245-012-9183-z.  Google Scholar

[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.   Google Scholar

[36]

A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging,, Stochastic Process. Appl., 89 (2000), 69.  doi: 10.1016/S0304-4149(00)00013-2.  Google Scholar

[37]

D. Williams, Probability with Martingales,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511813658.  Google Scholar

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.  Google Scholar

[2]

O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result,, Arch. Ration. Mech. Anal., 170 (2003), 17.  doi: 10.1007/s00205-003-0266-5.  Google Scholar

[3]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations,, Mem. Amer. Math. Soc., 204 (2010).  doi: 10.1090/S0065-9266-09-00588-2.  Google Scholar

[4]

O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations,, J. Differential Equations, 243 (2007), 349.  doi: 10.1016/j.jde.2007.05.027.  Google Scholar

[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.  doi: 10.1016/S1631-073X(03)00032-3.  Google Scholar

[6]

M. Arisawa and P.-L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187.  doi: 10.1080/03605309808821413.  Google Scholar

[7]

S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione,, Master thesis, (2012).   Google Scholar

[8]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhäuser Boston, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[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.  doi: 10.1137/090748147.  Google Scholar

[10]

M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach,, Eur. J. Control, 17 (2011), 30.  doi: 10.3166/ejc.17.30-45.  Google Scholar

[11]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, ().   Google Scholar

[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.  doi: 10.1007/BF01445155.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0363012904440897.  Google Scholar

[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.  doi: 10.1051/cocv:2000114.  Google Scholar

[16]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications,, Springer, (1998).  doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[17]

P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods,, Stoch. Process. Appl., 122 (2012), 1947.  doi: 10.1016/j.spa.2011.12.006.  Google Scholar

[18]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/090745465.  Google Scholar

[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.  doi: 10.1214/11-AAP801.  Google Scholar

[22]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes,, American Mathematical Society, (2006).  doi: 10.1090/surv/131.  Google Scholar

[23]

W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions,, Springer, (2006).   Google Scholar

[24]

J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility,, Cambridge university press, (2000).   Google Scholar

[25]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing,, SIAM J. Appl. Math., 63 (2003), 1648.  doi: 10.1137/S0036139902401550.  Google Scholar

[26]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics,, Multiscale Model. Simul., 2 (2003), 22.  doi: 10.1137/030600291.  Google Scholar

[27]

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives,, Cambridge University Press, (2011).  doi: 10.1017/CBO9781139020534.  Google Scholar

[28]

D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., ().   Google Scholar

[29]

Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control,, Springer-Verlag, (2003).  doi: 10.1007/978-3-662-13242-5.  Google Scholar

[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.  doi: 10.1214/009117905000000431.  Google Scholar

[31]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems,, Birkhäuser, (1990).  doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[32]

H.J. Kushner, Large deviations for two-time-scale diffusions, with delays,, Appl. Math. Optim., 62 (2010), 295.  doi: 10.1007/s00245-010-9104-y.  Google Scholar

[33]

R. Lipster, Large deviations for two scaled diffusions,, Probab. Theory Relat. Fields, 106 (1996), 71.  doi: 10.1007/s004400050058.  Google Scholar

[34]

K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion,, Appl Math Optim, 67 (2013), 123.  doi: 10.1007/s00245-012-9183-z.  Google Scholar

[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.   Google Scholar

[36]

A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging,, Stochastic Process. Appl., 89 (2000), 69.  doi: 10.1016/S0304-4149(00)00013-2.  Google Scholar

[37]

D. Williams, Probability with Martingales,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511813658.  Google Scholar

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