8 Numerical Solutions of the Black Scholes Equation
And finally, using Section A.10(iv) leads to
1.500 −0.25 0 0 0
0 1.458 −0.25 0 0
0 0 1.457 −0.25 0
0 0 0 1.457 −0.25
0 0 0 0 1.457
u12 u11 u10 u1−1 u1−2
=
3092.14 1499.06 487.77
83.69 14.36
or
u12 u11 u10 u1−1 u1−2
=
2242.60 1087.05 344.88
59.12 9.85
This whole process is now repeated for the next two columns in Table 8.1. The option prices are obtained from the valuesu3min the final column; for example
f(S0=100,T =6 months)=e−r T(e−kx−k2tu(x,T))
=e−0.1×0.5(e−4.61−0.01×722.67)=6.81
This is 3% away from the Black Scholes result, which for an absurdly small number of steps is quite surprising. The performance of these finite difference methods as a function of step numbers is examined in the next section.
(iv)American Options:The treatment of American options follows the method that was described for the binomial model. We start by recognizing that the value of an American option is only a solution of the Black Scholes equation when the stock price is above the exercise boundary;
below this level, the value is simply the exercise value (see Section 6.1). At each grid point, we therefore compare the valueunm with a valueνmn =er Tekx+k2tE, whereEis the exercise value of the option; we adopt the greater ofνmn andunm.
(v)Discrete Dividends:Suppose that instead of a continuous dividend, a single dividend is paid between time grid pointsnandn+1. The standard way of handling this is to make the stock pricesSmn+1drop by an amountQ. We assume that the dividend is paid an instant beforetn+1, so that each of the pointsxmn+1drop by an amountδx=lnSmn −ln(Smn −Q)=ln[Smn/(Smn −Q)].
This is a function ofSmn+1so that the grid would no longer maintain uniform spacing in the x-direction; practical calculations would become very cumbersome.
The same problem was encountered with the binomial model and was solved by assuming that the dividend is proportional to the stock price, i.e. Qnm+1=k Smn+1; the grid points then drop by an amountδx=ln[1/(1−k)]=constant. The whole grid is simply dislocated at the dividend point while retaining the same spacing as before.
8.6 COMPARISON OF METHODS
with the following parameters: S0=100; X =110; T =1 year; r =10%;q =4%; σ = 20%.
The graphs that follow show the calculated price of this option plotted against the number of time steps. In each case we have used twice as many grid points in thex-direction as in the t-direction, and unless otherwise stated, α= 12. In practice, for large values ofN this leads to an unnecessarily large spread ofxvalues, which can be truncated without loss of accuracy.
The Black Scholes value of this option is superimposed on the following graphs. The inside (darker) band denotes ±0.1% of the Black Scholes price (6.185), while the outer band is ±0.5%. When translated into volatility spreads, these levels of accuracy corre- spond to volatilities of 20.000±0.016% and 20.000±0.081%. Any practitioner will re- alize that even the broader band is well within the tolerances encountered in the options markets.
(ii)Binomial Model:This is the simplest model to apply. By definition it hasα=12 (otherwise it becomes atrinomial model); but we have seen that an explicit finite difference method is only stable ifα≤ 12, i.e. this model hovers uncomfortably at the edge of instability. This is reflected in the zig-zag pattern of Figure 8.8.
Number of Steps Calculated Price
6.05 6.1 6.15 6.2 6.25 6.3
0 20 40 60 80 100 120 140 160 180 200
Figure 8.8 Binomial method (explicit finite difference method:α=0.50)
(iii)Instability: Figures 8.9 and 8.10 illustrate just how sharp the edge between stability and instability really is. Results are shown forα=0.51 and α=0.49 which are immediately adjacent to, but on either side of the stability boundary. Remember that with these values for α, the model is no longer binomial. These very slight differences inαmake the difference between a wildly unstable model and one which converges fairly quickly.
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8 Numerical Solutions of the Black Scholes Equation
Number of Steps Calculated Price
-10.0 -5.0 0.0 5.0 10.0 15.0 20.0
20
0 40 60 80 100 120 140 160 180 200
Figure 8.9 Explicit finite difference method:α=0.51
Number of Steps Calculated Price
6.05 6.1 6.15 6.2 6.25 6.3
0 20 40 60 80 100 120 140 160 180 200
Figure 8.10 Explicit finite difference method:α=0.49
(iv)Trinomial:The conclusion of the last subsection is that we are more likely to get reliable results by using a trinomial model than with a binomial scheme. In the next subsection it will be seen that there are good reasons for using a trinomial rather than a binomial tree, other than considerations of stability. A popular scheme usesα=1/6 and the results are shown in Figure 8.11.
8.6 COMPARISON OF METHODS
Number of Steps Calculated Price
6.05 6.1 6.15 6.2 6.25 6.3
0 20 40 60 80 100 120 140 160 180 200
Figure 8.11 Trinomial tree
Number of Steps Calculated Price
6.05 6.1 6.15 6.2 6.25 6.3
0 20 40 60 80 100 120 140 160 180 200
Figure 8.12 Crank Nicolson
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8 Numerical Solutions of the Black Scholes Equation
(v)Crank Nicolson: This implicit scheme is illustrated in Figure 8.12. It is clearly the most consistent method illustrated so far. Two other schemes which might be of interest to the reader are not illustrated: the simple implicit method, which in theory should be somewhat less accurate and Douglas, which should be more accurate. But for the option which we have chosen as an example, there is very little difference from the Crank Nicolson result.
(vi)Average of Successive Binomial:It was seen in Chapter 7 that we can make a more consistent result from the binomial method by averaging the results obtained usingNandN +1 steps.
The results are shown in Figure 8.13 and turn out to be surprisingly close in form to the Crank Nicolson results. In a way, this should not be surprising if we look back to Section 8.1(viii) and remember that Crank Nicolson implies an averaging between values at stepsNandN+1.
Number of Steps Calculated Price
6.05 6.1 6.15 6.2 6.25 6.3
0 20 40 60 80 100 120 140 160 180 200
Figure 8.13 Average of successive binomial steps