FIGURE 5 Profile of the free boundary points obtained by various methods.
0 0.01 0.02 0.03 0.04 0.05
Time (τ )
Free boundary position
Asymptotic solution BWM
Integral equation Binomial Trinomial
Adaptive finite difference Nonuniform finite difference
9.1 9.4 9.7 10.0
boundary at each timestep obtained by various methods versus time. It is evident that the adaptive technique (adaptive finite difference) captures the free boundary locations quite well as time evolves, and certainly much better than the nonuniform methods (nonuniform finite difference). Note that, for both the binomial and trinomial methods, a depth of 1000 subdivisions was used, and these results are considered as reference solutions in Stamicar et al (1999). The free boundary locations captured by the adaptive technique follow closely those obtained by the tree methods and the integral equation method. The profiles generated by adaptive mesh methods are smooth, while those generated by nonuniform methods are highly nonsmooth like a step function, and hence do not capture the movement of the moving free boundary properly. The data used for the plot is taken from Table 2 of Stamicar et al (1999), except for the adaptive finite difference and nonuniform finite difference methods data, which is generated by the present authors’ implementation.
7 CONCLUSIONS AND EXTENSIONS
We have considered a PDE approach to pricing American options written on a single asset. We have formulated several highly accurate and efficient methods for pricing American options. These methods are built upon second-order centered finite dif-ference or optimal fourth-order QSC methods for the spatial discretization, and are integrated with adaptive mesh PDE methods, which rely on grading and monitor functions to determine the distribution of the error along the spatial dimension and, from that, the location of the spatial grid points. At certain timesteps, the adaptive techniques relocate the nodes to equidistribute the error in some chosen norm among the subintervals of the partition. For the solution of the LCP at each timestep we considered a discrete penalty method. The results show that adaptive PDE methods are effective on the American option pricing problem and, in particular, that they have a better ratio of accuracy over computational cost compared with their nonadaptive (still nonuniform) counterparts, and allow for more accurate tracking of the moving boundary. Furthermore, high-order spatial discretization methods have a better ratio of accuracy over computational cost compared with their standard second-order counter-parts, and they provide highly accurate option prices, as well as highly accurate values of the Greeks delta and gamma. The combination of high-order and adaptive mesh methods gives the best results regarding ratio of accuracy over computational cost.
We conclude by mentioning some extensions of this work. It would be desirable to have a theoretical analysis of the boundedness of (4.4) and the convergence of the penalty iteration in the context of the QSC methods that we have observed in the experiments. It would also be interesting to extend the pricing methods considered in this paper to other American-style options, such as American-style Asian options or the pricing of convertible bonds with early exercise features. In addition, an
appli-cation of adaptive techniques to other exotic options, such as barrier options, is of much interest. The fact that we obtained high accuracy over cost by using a uniform grid to start the adaptive technique and letting the adaptive technique take over the
“study” of the solution of the American option pricing problem is an indication that the adaptive mesh methods have the potential to be used as a “black box” to determine the behavior of the solutions of other related financial problems as well. Extend-ing the adaptive techniques to multidimensional problems is certainly challengExtend-ing.
In this regard, possible approaches include moving mesh finite difference methods such as those of Huang and Russell (1997, 1999), moving mesh spline collocation methods, and skipped grid spline collocation methods (Ng (2005)). However, such approaches involve considerable overhead, and their effectiveness has not yet been studied extensively, even for simple PDE problems. There is a lot more to be done to make the adaptive and/or high-order methods effective and practical for the solution of multidimensional financial problems.
REFERENCES
Achdou, Y., and Pironneau, O. (2005). Computational Methods for Option Pricing. SIAM, Philadelphia, PA.
Barone-Adesi, G., and Whaley, R. (1987). Efficient analytic approximations of American option values. Journal of Finance 42, 301–320.
Carey, G. F., and Dinh, H. T. (1985). Grading functions and mesh redistribution. SIAM Journal on Numerical Analysis 22, 1028–1040.
Carr, P., Jarrow, R., and Myneni, R. (1992). Alternative characterizations of American put options. Mathematical Finance 2, 87–106.
Chen, X., and Chadam, J. (1998). A mathematical analysis of the optimal exercise bound-ary for American put options. SIAM Journal on Mathematical Analysis 38, 1613–1641.
Christara, C. C., and Ng, K. S. (2006a). Adaptive techniques for spline collocation. Com-puting 76, 259–277.
Christara, C. C., and Ng, K. S. (2006b). Optimal quadratic and cubic spline collocation on nonuniform partitions. Computing 76, 227–257.
Christara, C., Chen, T., and Dang, D. M. (2009). Quadratic spline collocation for one-dimensional linear parabolic partial differential equations. Numerical Algorithms 53, 511–
553.
Clarke, N., and Parrott, K. (1999). Multigrid for American option pricing with stochastic volatility. Applied Mathematical Finance 6, 177–195.
Dang, D. M. (2007). Adaptive finite difference methods for valuing American options. Mas-ter’s Thesis, University of Toronto.
De Boor, C. (1974). Good approximation by splines with variable knots. II. In Conference on the Numerical Solution of Differential Equations, Lecture Notes in Mathematics, Vol-ume 363, pp. 12–20. Springer.
Eriksson, K., and Johnson, C. (1991). Adaptive finite element methods for parabolic prob-lems. I. A linear model problem. SIAM Journal on Numerical Analysis 28, 43–77.
Eriksson, K., and Johnson, C. (1995). Adaptive finite element methods for parabolic prob-lems. II. Optimal error estimates in L1L2 and L1L1. SIAM Journal on Numerical Analysis 32, 706–740.
Forsyth, P. A., and Vetzal, K. (2002). Quadratic convergence for valuing American options using a penalty method. SIAM Journal on Scientific Computing 23, 2095–2122.
Fritsch, F. N., and Carlson, R. E. (1990). Monotone piecewise cubic interpolation. SIAM Journal on Numerical Analysis 17, 238–246.
Fu, M. C., Laprise, S. B., Madan, D. B., Su, Y., and Wu, R. (2001). Pricing American options:
a comparison of Monte Carlo simulation approaches. The Journal of Computational Finance 4(3), 39–88.
Hackbush, W. (1993). Iterative Solution of Large Sparse Systems of Equations. Springer.
Houstis, E. N., Christara, C. C., and Rice, J. R. (1988). Quadratic-spline collocation methods for two-point boundary value problems. International Journal for Numerical Methods in Engineering 26, 935–952.
Huang, W., and Russell, R. D. (1997). A high dimensional moving mesh strategy. Applied Numerical Mathematics 26, 63–76.
Huang, W., and Russell, R. D. (1999). Moving mesh strategy based on a gradient flow equation for two-dimensional problems. SIAM Journal on Scientific Computing 20, 998–
1015.
Hull, J. C. (2008). Options, Futures, and Other Derivatives. Prentice Hall, Upper Saddle River, NJ.
Jiang, L., and Dai, M. (2004). Convergence of binomial tree methods for European and American path-dependent options. SIAM Journal on Numerical Analysis 42, 1094–1109.
Johnson, C. (1987). Numerical Solution of Partial Differential Equations by the Finite Ele-ment Method. Cambridge University Press.
Kuske, R., and Keller, J. (2007). Optimal exercise boundary for an American put option.
Applied Mathematical Finance 5, 107–116.
Longstaff, F. A., and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. Review of Financial Study 14, 113–149.
Lötstedt, P., Persson, J., Sydow, L. V., and Tysk, J. (2007). Space–time adaptive finite difference method for European multi-asset options. Computers and Mathematics with Applications 53, 1159–1180.
MacMillan, L. (1986). Analytic approximation for the American put option. Advanced Futures Options Research 1, 119–139.
Ng, K. S. (2005). Spline collocation on adaptive grids and non-rectangular domains. Doc-toral Thesis, Department of Computer Science, University of Toronto.
Nielsen, B., Skavhaug, O., and Tveito, A. (2002). Penalty and front-fixing methods for the numerical solution of American option problems. The Journal of Computational Finance 5(4), 69–97.
Persson, J., and Sydow, L. V. (2007). Pricing European multi-asset options using a space–
time adaptive FD-method. Computing and Visualization in Science 10, 173–183.
Pooley, D. M., Verzal, K. R., and Forsyth, P. A. (2003). Convergence remedies for non-smooth payoffs in option pricing. The Journal of Computational Finance 6(4), 25–40.
Rannacher, R. (1984). Finite element solution of diffusion problems with irregular data.
Numerische Mathematik 43, 309–327.
Stamicar, R., Ševˇcoviˇc, D., and Chadam, J. (1999). The early exercise boundary for the American put near expiry: numerical approximation. Canada Applied Mathematics 7, 427–444.
Tavella, D., and Randall, C. (2000). Pricing Financial Instruments: The Finite Difference Method. John Wiley & Sons.
Toivanen, J. (2008). Numerical valuation of European and American options under Kou’s jump-diffusion model. SIAM Journal on Scientific Computing 30, 1949–1970.
Wang, R., Keast, P., and Muir, P. (2004). A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Applied Numerical Analysis 50, 239–260.
Wilmott, P., Dewynne, J., and Howison, S. (1993). Option Pricing: Mathematical Models and Computation. Oxford Financial Press.
Wilmott, P., Howison, S., and Dewynne, J. (1995). Mathematics of Financial Derivatives.
Cambridge University Press.
Zvan, R., Forsyth, P. A., and Vetzal, K. (1998a). Penalty methods for American options with stochastic volatility. Journal of Computational and Applied Mathematics 91, 199–218.
Zvan, R., Forsyth, P. A., and Vetzal, K. (1998b). Robust numerical methods for PDE models of Asian options. The Journal of Computational Finance 1(2), 39–78.