discarding terms of O[δt3/2], this may be written in terms ofTandSTasSTσ√
δT/2δST ≤ 12. To the present order of accuracy in δT, this is the same condition that was expressed by equation (7.4), and which came from a seemingly unrelated line of reasoning.
This should of course be of no great surprise:
r
The binomial model is a graphical way of approximating the probability density function of a stock price (or its logarithm).r
This probability density function is a solution of the Kolmogorov backward equation; there- fore the binomial model is a graphical representation of the Kolmogorov equation.r
The explicit difference method was introduced to solve the Black Scholes equation.r
The Kolmogorov and Black Scholes equations are shown in Section A.4(i) of the Appendix to be very closely related.This duality between the explicit finite difference method and the binomial model is also true of the trinomial model which is examined in a later chapter.
8.4 IMPLICIT FINITE DIFFERENCE METHODS
(i) Let us return to the backward difference scheme of Section 8.1(v) which may be written unm−1=(1+2α)unm−α
unm+1+unm−1
um+1n+1
umn+1
um−1n+1
unm−1
unm
unm+1
Figure 8.4 Backward difference and which is represented in Figure 8.4. In this caseunm−1
can be calculated from the adjacentuvalues immedi- ately to the right. Unfortunately, this is an inconvenient way to proceed. We know the values at the left-hand edge of the grid (initial conditions) and the values at the top and bottom edges (boundary conditions); the solution of the problem is the series of values at the right-hand edge. In order to find these right-hand edge solutions, we need to solve a large array of linear si- multaneous equations for all the unm; these are not
given explicitly in terms of known quantities – hence the nameimplicit methods.
(ii) As well as producing awkward simultaneous equations to solve, the implicit difference intro- duces difficult boundary conditions. Compare Figures 8.3 and 8.5, showing boundary con- ditions for the two methods. The simple nature of the explicit difference method meant that we could ignore all values outside the shaded area, including boundary values. But with the implicit method, boundary values are important.
For a European call option the boundary conditions are
S0lim→∞ f0(S0,T)→S0e−q T −Xe−r T
Slim0→0 f0(S0,T)=0
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8 Numerical Solutions of the Black Scholes Equation
In terms ofx,tandkas defined in equation (8.2) this may be writtenxlim→∞u(x,t)→er Tekx+k2t(ex−Xe−r T); lim
x→−∞u(x,t)→0
The boundary conditions are set atx= ±∞, so this would imply that the grid should stretch between these limits. But this would give an infinite number of simultaneous equations to solve!
initial conditions answers
boundary conditions
boundary conditions
x= +∞
x= +∞
M− M+
Figure 8.5 Boundary conditions
Consider the graph of a European call option shown in Figure 8.6. The upper boundary condition is that f0(S0,T)→S0e−q T−Xe−r T. However this condition does not really need to be applied atS0 = ∞; without appreciable loss of accuracy, it can be applied atS0=U3, orU2or evenU1; but if we apply the boundary condition atS0=V, we start introducing an appreciable error. The same principle applies when we seek a practical implementation of the lower boundary condition.
X e-r t V U1 U2 U3
L1
L2
Figure 8.6 Effective boundaries for call option
In terms of the boundary conditions in Figure 8.5, we choose a large positive and a large negativexvalue,M+∞andM−∞beyond which we do not extend the grid. The values that we insert at these edges are the effective boundary conditions. Of course, this begs an important question: how do we know that we have chosenM+∞andM−∞far enough out that we have not introduced an appreciable error, but not so far that we are doing a lot of redundant computing?
The answer is to set up the model on a computer and shiftM+∞andM−∞about a bit; if the answers do not change much, we are in a safe area.
8.4 IMPLICIT FINITE DIFFERENCE METHODS
(iii) At this point, the reader might be wondering why anyone should burden himself with the implicit difference method, when the explicit method is so much easier to solve. The explicit method, cast in the form of the binomial model, is indeed much more popular than implicit methods. After all, for every person who knows how to get finite difference solutions to a partial differential equation, there are 100 guys who can stick numbers into a tree. On the other hand, explicit methods do show an unfortunate tendency to be unstable, while stability is assured over a much wider range by the implicit method.
Recall from Section 8.1 that the forward and backward finite difference schemes are not well centered compared with the Crank Nicolson or Douglas schemes. But these latter two, more stable and accurate schemes are just as easy to implement as the simple implicit method, so they are normally the preferred route if an implicit scheme is used at all. A comparison of the methods is given in Section 8.5.
(iv) The interesting discretization methods laid out in Section 8.1 can be combined into a single formula:
unm+1−unm= 12αδˆ2x
θunm+1+(1−θ)unm
Explicit: θ=0
Implicit: θ=1 Crank Nicolson: θ= 12 Douglas: θ= 12
1−6α1)
Trinomial: as Douglas withα= 16 Written out fully, this formula is
(1+2αθ)unm+1−αθ
unm++11 +unm+−11
=(1−2α(1−θ))unm+α(1−θ)
unm+1+unm−1 (8.3) In the following analysis, we use this in the form
−bunm++11 +aunm+1−bunm+−11 =eunm+1+cunm+eunm−1
This equation is easily expressed in matrix form. A little care is needed with the first and last terms in the sequence (the termunm−1is undefined whenm=1). Taking these edge effects into account, the above equation may be written as
a −b 0 0
−b a −b 0
0 −b a −b
. ..
a
unM+−11 unM+−21
... un−+1M+2 un−+1M+1
−
bunM+1
0 ... 0 bun−+1M
=
c e 0 0 e c e 0
0 e c e
. ..
c
unM−1 unM−2
... un−M+2 un−M+1
+
eunM
0 ... 0 eun−M
or
Apn+1=Bpn+bqn+1+eqn (8.4)
The square matrices have dimension (M−2)×(M−2) and the vectors haveM−2 elements.
(v) We start off knowing the values at the left-hand edge of the grid (initial valuesu0m). From the boundary conditions we also know the values at the top and bottom edges of the grid,
95
8 Numerical Solutions of the Black Scholes Equation
0
uM … uMn
0
u-M+1 u-M+11
…
… …
…
initial conditions boundary conditions
solve for these values
1
uM 0
uM-1 uM-11
0
u-M 1
u-M n
u-M
Figure 8.7 Solution of implicit method
i.e. we knowuiM andui−M. We can therefore calculate the right-hand side of equation (8.4) since we also know the elements of the matrixB; this will be designated by the vectors0. The second column in the grid can therefore be obtained by using the equationAp1 =s0. And so the process can be repeated across the grid, merely by solving the equationsApn+1=sn. This process is illustrated in Figure 8.7.
The trouble is that inverting a 200×200 matrix is more than a question of “merely”. However, the matrixAhas a special tridiagonal form which makes the problem fairly easy to solve by using one of several possible tricks; the simplest of these, known as the LU decomposition, is described in Appendix A.10.
Finally, we note that ifθ=0, then the matrixAbecomes the unit matrix and we have the trivially simple explicit solution explained in Section 8.3.
(vi)Discretization of the Full Black Scholes Model:We finish this section with an observation rather than a new method or technique. By a simple change of variables, we can transform the Black Scholes equation into the simple heat equation (8.2); this simplifies the algebra and makes the theory more easily intelligible. However, there is nothing to prevent us from discretizing equation (8.1) directly.
As before we put
∂f
∂S → 1 2δS
fmn+1− fmn−1
; ∂2f
∂S2 → 1 (δS)2
fmn+1+ fmn−1−2fmn
∂f
∂T →
1 δt
fmn+1− fmn
: forward difference 1
δt
fmn− fmn−1
: backward difference The Black Scholes equation becomes:
(A) Forward Difference 1
δt
fmn+1− fmn
=12m(r−q)
fmn+1− fmn−1
+12σ2m2
fmn+1+ fmn−1−2fmn
−r fmn (B) Backward Difference
1 δt
fmn− fmn−1
=1m(r−q)
fmn+1− fmn−1
+1σ2m2
fmn+1+ fmn−1−2fmn
−r fmn