Details

In this detailed description, the symbols have the following meanings:

is the current time

is the maturity date

is the stock price at time

is the strike price

is the stock dividend yield

is the risk-free interest rate

is the stock volatility

is the cumulative distribution function of the standard normal distribution

is the moving free boundary

is the optimal boundary.

Consider the class of contracts consisting of a European call option and a sure flow of payments that are paid at the rate

for ,

where

,

,

and is a non-negative continuous function of time. Each member of the class of contracts is parametrized by . The value of the contract at time is

,

where denotes the value at time of a European call option on with strike price and maturity . The optimal exercise boundary for the American call option is obtained by solving the value matching condition:

,

for for all .

The value of the American call option is then given by .

Subject to the value matching condition, the critical asset price at time can be numerically approximated by a computationally intensive recursive procedure. This method requires solving integral equations, where is the number of time steps. Each time the integral equation is solved, either the trapezoidal rule (Wolfram MathWorld) or Simpson's rule (Wolfram MathWorld) can be employed to approximate the integral.

For the mesh relocation process, the following notation is used:

: an invertible mesh function that maps the arbitrary set in the computational space , to a set in the physical time domain , where and .

: an invertible mesh function that maps the arbitrary set to a set , .

: a scalar function that uses Kim's method to approximate the American call value at , over the mesh mapping function .

: the theoretical value of the American call at .

: the impact of the element's local refinement on the solution. Every element's impact on the solution is evaluated as it gets divided into two equal subintervals, while all other elements remain unchanged.

: the absolute impact of the element's local refinement on the solution. A small positive real is added to ensure that .

: the adjoint functional derives as the sum of absolute impacts and serves as the dual-objective quantity under minimization.

The minimization of requires that , . Hence, the element should be resized according to the ratio . In order to run a smooth element resizing process, a weight is introduced:

where is a restrictive parameter that controls the variability in , and is the iteration index (this Demonstration uses , and ). Through every iteration, all mesh points are relocated simultaneously; the new position of the temporal point is obtained by , where is an adjusting coefficient to ensure that . The refinement process is terminated when or .

References

[1] I. J. Kim, "The Analytic Valuation of American Options," The Review of Financial Studies, 3(4), 1990 pp. 547–572. www.jstor.org/stable/2962115.

[2] A. Ern, S. Villeneuve and A. Zanette, "Adaptive Finite Element Methods for Local Volatility European Option Pricing," International Journal of Theoretical and Applied Finance, 7(6), 2004 pp. 659–684. doi:10.1142/S0219024904002669.

[3] C. Goll, R. Rannacher and W. Wollner, "The Damped Crank–Nicolson Time-Marching Scheme for the Adaptive Solution of the Black–Scholes Equation," Journal of Computational Finance, 18(4), 2015 pp. 1–37. doi:10.21314/JCF.2015.301.

[4] J. T. Oden and S. Prudhomme, "Goal-Oriented Error Estimation and Adaptivity for the Finite Element Method," Computers and Mathematics with Applications, 41(5–6), 2001 pp. 735–756. doi:10.1016/S0898-1221(00)00317-5.

[5] K. G. van der Zee, E. H. van Brummelen and R. de Borst, "Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Domain-Map Linearization Approach," SIAM Journal on Scientific Computing, 32(2), 2010 pp. 1064–1092. doi:10.1137/080741227.

[6] P. Luchini, F. Giannetti and V. Citro, "Error Sensitivity to Refinement: A Criterion for Optimal Grid Adaptation," Theoretical and Computational Fluid Dynamics, 2016. doi:10.1007/s00162-016-0413-x.

[7] M. Broadie and J. Detemple, "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, 9(4), 1996 pp. 1211–1250. doi:10.1093/rfs/9.4.1211.

[8] S.-L. Chung, M.-W. Hung, and J.-Y. Wang, "Tight Bounds on American Option Prices," Journal of Banking and Finance, 34(1), 2010 pp. 77–89. doi:10.1016/j.jbankfin.2009.07.004.