Accurate pricing of barrier options is a tricky business, and traditional lattice methods are ill-suited to the task. Unless, through care or luck, the correct relationship between barrier position and lattice nodes is obtained, large pricing errors and erratic convergence behavior result. Furthermore, the "correct relationship" depends on features of the contract, whether the barriers are continuously or discretely monitored, for example.
On the other hand, partial-differential equation (PDE) methods, which have been used for decades in the physical sciences and engineering, are ideal for pricing options with multiple, discretely-monitored, or even moving barriers. The numerical solution of PDEs can be accomplished with many methods, among them finite difference, finite element, and pseudo-spectral techniques. This articles focuses on the finite difference method. Here partial derivatives in the governing PDE (for example, the Black-Scholes equation) are replaced with discrete approximations in terms of the option value at each node on a grid of spot prices Si: 0 = < i < = N. The result is a system of coupled linear equations that can be solved using efficient algorithms.
The simplest application of the finite difference method to the pricing of barrier options is that of continuously-monitored fixed knockout barriers. In this case, the extreme nodes of the grid are placed at the barriers to enforce the knockout conditions there. That is, option value equals rebate (discounted rebate, if deferred.) Since the finite difference method yields present option value on the entire price grid, the current spot price need not be a node. The option value there may be accurately obtained by interpolation. Furthermore, the error in the calculated option value varies smoothly with the grid size, diminishing as 1/N2. Using well-known techniques, one can reliably extrapolate to the exact limit, N *ƒ, from several very fast, sparse grid calculations.
Pricing discretely-monitored barrier options is a bit more complex, but nevertheless is efficiently done using the finite difference method. In this case, the option value must be solved at spot prices beyond the barriers. Consider an up-and-out call with barrier B that is monitored weekly, say on Fridays. On a Monday, the option will have a substantial value even for S greater than B because there is a non-zero probability that S will drop back below B by the following Friday. In general, the option value will be substantial for S¾B exp (*ˆtm/2) where * is the volatility and tm is the monitoring period.
Between monitoring dates, the option value "diffuses" beyond the barriers as described. On the monitoring dates, the option value is abruptly reset according to the knockout conditions. The strong gradients (large 's) created by this periodic adjustment can be adequately resolved only with a very fine grid. However, because the solution is smooth away from the barrier, using a uniformly fine grid is very inefficient. Coordinate transformations provide a rigorous method for locally refining the grid in the neighborhood of barriers, and they are a key reason why the finite difference method is ideal for pricing such options. Under a time-independent coordinate transformation, the Black-Scholes equation becomes
where * is a new coordinate, typically uniformly spaced. The effective local grid spacing in the original spot price coordinate S(*) is then S(*) = J(*)*, where J(*) = S/* is the Jacobian of the transformation. A coordinate transformation is then chosen that maximizes the utility of grid points, clustering them near barriers. That is, one that minimizes J near barriers.
The solid curve in Figure A is a finite difference calculation of present value for a weekly-monitored up-and-out call with strike K = 100, barrier B = 120, a term of one year and no rebate. The volatility is * = 0.25 and the risk free rate is r = 0.05. The last monitoring date is at expiration. The calculated value at the barrier, V(B) = 0.1541, is over 15% of the at-the-money value, V(K) = 0.9693. For comparison, the dashed curve is the analytic result for a continuously-monitored barrier. It severely underprices the weekly-monitored option for all possible spot prices. Just N = 150 grid points were used in the finite difference calculation. The coordinate transformation employed here yields an effective grid spacing S, shown in Figure B, that increases linearly in S away from the barrier. The minimum grid spacing is about one fifth that of a uniform grid. Extrapolating to the N *ƒ result from several finite difference calculations yields an at-the-money present option value V(K) = 0.9686. Thus, the very fast finite difference calculation with N = 150 is in error by about 0.07%. Achieving similar accuracy with a uniform grid requires several thousand grid points, with a proportional increase in execution time.
Coordinate transformations may even be time-dependent. In this case the transformed Black-Scholes equation becomes
whereJ(*,t) = S(*,t)/* is the spatial Jacobian, and Jt(*,t) = S(*,t) /t is the time Jacobian. This type of transformation allows the finite difference grid to have concentration points that move in synchrony with moving barriers. The motion of the barriers can be arbitrarily complex.
Additional option features and market characteristics easily handled by the finite difference method include multiple assets or factors, discrete dividends, term structure of interest rates and forward volatility surfaces, stochastic volatility and interest rate models, jump-diffusion models, and complex path dependencies such as payoffs that depend on the number of days that a spot price is between two barriers.
This week's Learning Curve was written by Curt Randall, v.p. at Austin, TX-based SciComp Inc.
