Uniform approximation of the Cox-Ingersoll-Ross process. (English) Zbl 1335.65011

Milstein, Grigori N.; Schoenmakers, John

Adv. Appl. Probab. 47, No. 4, 1132-1156 (2015).

A simulation algorithm is developed for uniformly approximating the trajectories of the Cox-Ingersoll-Ross process \(V(t)\) defined as the solution of the stochastic differential equation \[ dV(t)=k(\lambda-V(t))dt+\sigma\sqrt{V(t)}dw,\quad V(t_0)=V_0, \] where \(w\) is a scalar Brownian motion and \(k, \lambda,\sigma\) are positive constants. It is proved that the algorithm converges on any trajectory of \(V(t)\) that is positive on \([t_0,t_0+T]\). A modification of the algorithm is devised to deal with trajectories that get close to zero, and its convergence is proved. Numerical implementation is briefly discussed with emphasis on examples from Finance.

Reviewer: Melvin D. Lax (Long Beach)

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
91G60	Numerical methods (including Monte Carlo methods)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness
60J65	Brownian motion
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C15	Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)

Keywords:

Cox-Ingersoll-Ross process; Doss-Sussmann formalism; Bessel function; confluent hypergeometric equation; numerical examples; stochastic differential equation; Brownian motion; algorithm; convergence

References:

[1]	Alfonsi, A. (2005). On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11 , 355-384. · Zbl 1100.65007
[2]	Alfonsi, A. (2010). High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79 , 209-237. · Zbl 1198.60030
[3]	Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11 , 1-42.
[4]	Bateman, H. and Erdélyi, A. (1953). Higher Transcendental Functions . McGraw-Hill, New York. · Zbl 0143.29202
[5]	Broadie, M. and Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operat. Res. 54 , 217-231. · Zbl 1167.91363
[6]	Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 , 385-407. · Zbl 1274.91447
[7]	Dereich, S., Neuenkirch, A. and Szpruch, L. (2012). An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process. Proc. R. Soc. London A 468 , 1105-1115. · Zbl 1364.65013
[8]	Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.) 13 , 99-125. · Zbl 0359.60087
[9]	Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering . Springer, New York. · Zbl 1038.91045
[10]	Göing-Jaeschke, A. and Yor, M. (2003). A survey on some generalizations of Bessel processes. Bernoulli 9 , 313-349. · Zbl 1038.60079
[11]	Hartman, P. (1964). Ordinary Differential Equations . John Wiley, New York. · Zbl 0125.32102
[12]	Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6 , 327-343. · Zbl 1384.35131
[13]	Higham, D. J. and Mao, X. (2005). Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comp. Finance 8 , 35-61.
[14]	Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40 , 1041-1063. · Zbl 1026.65003
[15]	Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes . North-Holland, Amsterdam. · Zbl 0495.60005
[16]	Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd edn. Springer, New York. · Zbl 0734.60060
[17]	Milstein, G. N. and Schoenmakers, J. G. M. (2015). Uniform approximation of the CIR process via exact simulation at random times. Preprint. WIAS preprint no. 2113, ISSN 2198-9855. · Zbl 1362.65016
[18]	Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics . Springer, Berlin. · Zbl 1085.60004
[19]	Milstein, G. N. and Tretyakov, M. V. (2005). Numerical analysis of Monte Carlo evaluation of Greeks by finite differences. J. Comp. Finance 8 , 1-34.
[20]	Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002
[21]	Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus . John Wiley, New York. · Zbl 0977.60005
[22]	Sussmann, H. J. (1978). On the gap between deterministic and stochastic ordinary differential equations. Ann. Prob. 6 , 19-41. · Zbl 0391.60056