Most real-world barrier options have no analytic solutions, either because the barrier structure is complex or because of volatility skews in the market. Numerical solutions are a necessity. But options with barriers are notoriously difficult to value numerically on binomial or multinomial trees, or on finite-difference lattices. Their values converge very slowly as the number of tree or lattice levels increase, often requiring unattainably large comput- ing times for even a modest accuracy.
In this paper we analyze the biases implicit in valuing options with barriers on a lattice. We then sug- gest a method for enhancing the numerical solution of boundary value problems on a lattice that helps to correct these biases. It seems to work well in practice.
As derivative markets have matured, options with barriers1 have become increasingly popular because of the greater precision with which they allow investors to obtain or avoid exposure. The value of a knockin stock (or index) option depends sensitively on the risk-neutral probability of the stock being in-the-money and beyond the barrier. Similarly, the value of a knockout option depends on the probability of the stock being in-the- money but not beyond the barrier. The analytic solution for these probabil- ities, and for the value of a European-style knockout option on stock under the standard Black-Scholes assumptions, was published by Merton (1973). This analytic solution provides rapidly computed, accurate values and hedge ratios, so important for managing the risk of large books of exotic and standard options.
Many of the currently traded barrier-style derivatives have no analytic solutions. The analytic method works only for simple barriers at a fixed or exponentially rising level, assuming lognormal stock price evolution and European-style exercise. There are now over-the-counter markets in options whose barriers may have arbitrary time dependence, whose implied volatilities exhibit a skew that corresponds to non-lognormal evo- lution of the underlying stock price2, or whose exercise may be American- style. In most of these cases there exists no general analytic solution for the value of the barrier option, and so a numerical solution is unavoidable. The most common numerical techniques involve solving the differential equation on a binomial lattice (Cox, Ross and Rubinstein 1979), using more general (explicit or implicit) finite difference methods, or using the Monte Carlo method of integral evaluation (Boyle 1977). The numerical accuracy of these methods becomes an important issue.
The binomial method for standard European-style options converges fairly rapidly as the number of levels on the binomial tree increases. Fig- ure 1 shows the variation in value with level number for a typical case. You can see that the answer is accurate to better than 0.4% for binomial trees of greater than 40 levels; the values oscillate about the analytic value of 12.99 as you increment the number of levels, and approach the correct analytic value asymptotically.