Malliavin Greeks without Malliavin calculus. (English) Zbl 1133.60030
Consider the expected value of a contingent claim
\[
u(x)=E[\Phi(X_{t_1},\ldots,X_{t_m})],
\]
where \(X_t\) is the solution of a stochastic differential equation
\[
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t,\qquad X_0=x.
\]
The problem of “Greeks” for such a claim consists of computing the derivative of \(u(x)\) with respect to various perturbations: a change of the initial value \(x\) (delta), a perturbation of the diffusion coefficient \(\sigma\) (vega), or a perturbation of the drift coefficient \(\mu\) (rho).
In this work, a time discretization is applied on the equation by means of the Euler scheme, and unbiased estimators of the Greeks for the discretized equation are given. Then, letting the discretization step tend to 0, the consistency of these estimators and of the corresponding continuous time estimators (based on Malliavin’s calculus) is proved.
In this work, a time discretization is applied on the equation by means of the Euler scheme, and unbiased estimators of the Greeks for the discretized equation are given. Then, letting the discretization step tend to 0, the consistency of these estimators and of the corresponding continuous time estimators (based on Malliavin’s calculus) is proved.
Reviewer: Jean Picard (Aubière)
MSC:
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
Keywords:
Monte Carlo simulation; likelihood ratio method; pathwise derivative method; Malliavin calculus; weak convergenceReferences:
| [1] | Benhamou, E., Optimal Malliavin weighting function for the computation of Greeks, Mathematical Finance, 13, 37-53 (2003) · Zbl 1049.91058 |
| [2] | Bermin, H.-P.; Kohatsu-Higa, A.; Montero, M., Local vega index and variance reduction methods, Mathematical Finance, 13, 85-97 (2003) · Zbl 1049.91062 |
| [3] | Cvitanic, J.; Ma, J.; Zhang, J., Efficient computation of hedging portfolios for options with discontinuous payoffs, Mathematical Finance, 13, 135-151 (2003) · Zbl 1060.91058 |
| [4] | Davis, M.; Johansson, M., Malliavin Monte Carlo Greeks for jump diffusions, Stochastic Processes and their Applications, 116, 101-129 (2006) · Zbl 1081.60040 |
| [5] | Durrett, R., Probability: Theory and Examples (1996), Duxbury Press |
| [6] | Evans, L., (Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (1998), American Mathematical Society) · Zbl 0902.35002 |
| [7] | Fournié, E.; Lasry, J.-M.; Lebuchoux, J.; Lions, P.-L.; Touzi, N., Applications of Malliavin calculus to Monte Carlo methods in finance, Finance and Stochastics, 3, 391-412 (1999) · Zbl 0947.60066 |
| [8] | Fournié, E.; Lasry, J.-M.; Lebuchoux, J.; Lions, P.-L., Applications of Malliavin calculus to Monte Carlo methods in finance. II, Finance and Stochastics, 5, 201-236 (2001) · Zbl 0973.60061 |
| [9] | Glasserman, P., Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1038.91045 |
| [10] | Gobet, E.; Kohatsu-Higa, A., Computation of Greeks for barrier and lookback options using Malliavin calculus, Electronic Communications in Probability, 8, 51-62 (2003) · Zbl 1061.60054 |
| [11] | Jacod, J.; Shiryayev, A. N., Limit Theorems for Stochastic Processes (2003), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1018.60002 |
| [12] | Kohatsu-Higa, A.; Montero, M., Malliavin calculus in finance, (Handbook of Computational and Numerical Methods in Finance (2004), Birkhauser), 111-174 · Zbl 1138.91454 |
| [13] | Kurtz, T. G.; Protter, P., Weak limit theorems for stochastic integrals and stochastic differential equations, Annals of Probability, 19, 3, 1035-1070 (1991) · Zbl 0742.60053 |
| [14] | Kurtz, T. G.; Protter, P., Weak convergence of stochastic integrals and differential equations, Lecture Notes in Mathematics, 1627, 1-41 (1995) · Zbl 0862.60041 |
| [15] | Nualart, D., Malliavin Calculus and Related Topics (1995), Springer-Verlag · Zbl 0837.60050 |
| [16] | Protter, P., Stochastic Integration and Differential Equations (1990), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0694.60047 |
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