A note on the numerical approximation of Greeks for American-style options (2401.13361v1)
Abstract: In this note we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank-Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed. In this note we study for the temporal discretization an interesting family of diagonally implicit Runge-Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter $\theta$ is preferable.
- Transient simulation of silicon devices and circuits. IEEE Trans. Comput.-Aided Design Int. Circ. Sys., 4:436–451, 1985.
- The valuation of American put options. J. Finance, 32:449–462, 1977.
- J. R. Cash. Two new finite difference schemes for parabolic equations. SIAM J. Numer. Anal., 21:433–446, 1984.
- Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comp., 23:2095–2122, 2002.
- E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer, 1991.
- Analysis and implementation of TR-BDF2. Appl. Numer. Math., 20:21–37, 1996.
- W. Hundsdorfer and J. G. Verwer. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, 2003.
- S. Ikonen and J. Toivanen. Pricing American options using LU decomposition. Appl. Math. Sc., 1:2529–2551, 2007.
- S. Ikonen and J. Toivanen. Operator splitting methods for pricing American options under stochastic volatility. Numer. Math., 113:299–324, 2009.
- Efficient numerical valuation of European options under the two-asset Kou jump-diffusion model. J. Comp. Finan., 26:101–137, 2023.
- K. J. in ’t Hout and B. D. Welfert. Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms. Appl. Numer. Math., 57:19–35, 2007.
- K. J. in ’t Hout and B. D. Welfert. Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms. Appl. Numer. Math., 59:677–692, 2009.
- A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach. J. Bank Finan., 30:489–502, 2006.
- Pricing exotic options with L𝐿Litalic_L-stable Padé schemes. J. Bank Finan., 31:3438–3461, 2007.
- F. Le Floc’h. TR-BDF2 for fast stable American option pricing. J. Comp. Finan., 17:31–56, 2014.
- C. Reisinger and A. Whitley. The impact of a natural time change on the convergence of the Crank–Nicolson scheme. IMA J. Numer. Anal., 34:1156–1192, 2014.
- M. Yousuf and A. Q. M. Khaliq. Partial differential integral equation model for pricing American option under multi state regime switching with jumps. Numer. Meth. Part. Diff. Eq., 39:890–912, 2023.
- The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost. Int. J. Comp. Math., 89:1239–1254, 2012.
- Pricing American options under multi-state regime switching with an efficient L𝐿Litalic_L-stable method. Int. J. Comp. Math., 92:2530–2550, 2015.
- Penalty methods for American options with stochastic volatility. J. Comp. Appl. Math., 91:199–218, 1998.
- A finite volume approach for contingent claims valuation. IMA J. Numer. Anal., 21:703–731, 2001.
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