Chapter 1
The method of lines (MOL) for
the diffusion equation
The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just one of the
independent variables. The assumption is the ordinary differential equations
are easier to analyze and solve than the partial differential equations. The
approximation can be based on finite differences, finite elements, collocation
or Fourier series like expansions. A method of lines obtained with finite
differences seems easiest to apply and will be used exclusively in this text.
In order to introduce the basic ideas used later for pricing options let us
consider the diffusion equation
a(x, t)uxx + b(x, t)ux − c(x, t)u − d(x, t)ut = f (x, t)
(1.1)
and the boundary and initial conditions
u(0, t) = α(t),
u(x, 0) = u0 (x).
u(L, t) = β(t)
Options, bonds and their Greeks are described by equations like (1.1).
Consistent with these applications we shall make the assumption that all
coefficients in (1.1) are continuous in x and t and that
a(x, t) > 0
d(x, t) > 0
for 0 < x < L and all t,
for 0 < x < L and all t.
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CHAPTER 1.
The coefficients b and c may change sign in specific problems.
The two independent variables are x and t. In later applications the
variable x will stand for an asset price or an interest rate. We shall call x the
“spatial” variable. t stands for “time”, usually time to expiry of the option
or bond.
It is our choice whether to approximate this problem with ordinary differential equations in x or t. In subsequent chapters we shall always retain
x as the continuous variable and discretize t. However, in the engineering
literature the term MOL invariably refers to ordinary differential equations
in t. This version is sometimes called the vertical method of lines []. For
completeness, we shall give first a brief exposition of a method of lines approximation of (1.1) valid for all t and at discrete values of x, and then turn
to the MOL in x.
1.1
The method of lines with continuous time
(the vertical MOL)
We define a mesh
0 = x0 < · · · < xP = L
with
∆x = L/M
and
x = i∆x,
i = 0, 1, . . . , P,
and approximate (1.1) along the line x = xi with the ordinary differential
equation
ui+1 (t) + ui−1 (t) − 2ui(t)
∆x2
ui+1 (t) − ui−1 (t)
+ b(xi , t)
− c(xi , t)ui(t) − u′i (t) = f (xi , t)
2∆x
a(xi , t)
for i = 1, . . . , P − 1, where u0 (t) = α(t) and uM (t) = β(t), and where
ui(0) = u0 (xi ).
It is possible to write these P −1 ordinary differential equations for the vector
~u(t) = (u1 (t), . . . , uP −1(t)) in matrix form
~u ′ (t) = A(t)~u + ~b(t)
~u(0) = ~u0
2
(1.2)
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where A(t) is a tridiagonal matrix and the vector ~b(t) is determined by the
source term f (x, t) and the boundary conditions {α(t), β(t)}. The initial
condition ~u0 = (u0,1, . . . u0,P −1) is defined by
u0i = u0 (xi ).
Taylor expansions can be used to show that (1.2) is a consistent approximation of (1.1). Moreover, it can be shown that under reasonable hypotheses
on the data of problem (1.1) the analytic solution of (1.2) satisfies
|u(xi , t) − ui (t)| ≤ K∆x2 ,
i = 1, . . . , P − 1
where u(x, t) is the analytic solution of (1.1) and K depends on the smoothness of u and the interval of integration [0, T ] [].
Of course, in general an analytic solution of (1.2) is not available. Then
the system must be solved numerically. It is possible to define a numerical
integrator for (1.2), for example a backward Euler method or the trapezoidal
rule, so that the resulting algebraic equations are identical to standard finite
difference approximations to (1.1). In this case nothing has been gained by
introducing the method of lines. The attraction of (1.2) is due to the fact that
the theory of ordinary differential equations yields insight into the analytic
solution of (1.2), and that many efficient adaptive black box numerical codes
exist for the approximate solution of initial value problems for linear and nonlinear ordinary differential equations. Thus, in principle, a highly accurate
numerical of solution of (1.2) is obtainable. Coupled with the fact that the
MOL discretization is readily extended to nonlinear problems and to other
classes of equations, it is natural that this two step solution approach to time
dependent problems has become commonplace and the basis of software for
the numerical solution of time dependent partial differential equations [].
For a discussion of the method of lines leading to initial value problems
for ordinary differential equations, and some caveats against its thoughtless
application we refer the reader to [].
The transformation of a so-called parabolic initial/boundary value problem like (1.1) to (1.2) is not so straightforward when the domain of the
equation changes with time, e.g. when the boundary conditions are given in
the form
u(s(t), t) = α(t),
u(S(t), t) = β(t)
(1.3)
where x = s(t) and x = S(t) describe non-constant boundaries. The problem
is exacerbated when s(t) or S(t) is not known a priori but a free boundary
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CHAPTER 1.
like an early exercise boundary in an American option. It is possible to define an analog to (1.2) where {ui(t)} is an approximate solution along curves
{xi (t)} (see the method of arbitrary lines [Xanthis]). Alternatively, one can
map the irregular domain into a rectangle at the expense of complicating
the differential equation (1.1). For this approach in a financial setting see
[Tavella, Wu] where the resulting equations are solved with finite differences.
We have no experience with the time-continuous MOL for financial applications and can give no insight into how well it solves, for example, American
option problems. In this book, our goal is to work as much as possible with
the original equation in the natural variables of the application. We believe
that we can reach this goal readily by discretizing t instead of x.
1.2
The method of lines with continuous x
(the horizontal MOL)
For options with discontinuous pay-off, barriers, early exercise features or
jumps, the solution of the Black Scholes equation can vary strongly with x
at any given time. In this setting it is useful to base the method of lines on
discretizing time and solving the resulting ordinary differential equation in
x to a high degree of accuracy. This approach will be followed consistently
throughout the remainder of this book.
Assuming that problem (1.1) is to be solved over the time interval [0, T ],
where T is arbitrary but fixed (usually the time to expiry of the option), we
introduce a partition in time
0 = t0 < t1 < · · · < tN = T
where
∆tn = tn − tn−1
is usually, but not necessarily, constant with respect to n.
For a method of lines approximation of (1.1) at t = tn we simply replace
ut by a time implicit (i.e. backward) difference quotient. Two approximations
are used repeatedly. If un (x) denotes the approximation of the solution
u(x, tn ) then we employ either the backward Euler approximation
un (x) − un−1 (x)
ut (x, tn ) ∼
= Dun (x) =
∆tn
4
(1.4)
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CHAPTER 1.
or the three-level difference quotient
un (x) − un−1(x)
un−1 (x) − un−2 (x)
ut (x, tn ) ∼
+ dn
.
= Dun (x) = cn
∆tn
∆tn−1
(1.5)
A Taylor series expansion of the right hand side of (1.5) shows that the
weights {cn , dn } should be chosen such that
1
1
cn
1
=
.
∆tn−1 ∆tn 2∆tn−1 ∆tn + ∆t2n−1
dn
0
In most applications the time step is constant (i.e. ∆tn = ∆tn−1 ). For this
case the weights are
3
1
cn = ,
dn = − .
2
2
In general it follows from Taylor expansions applied to (1.4) that for any
smooth function φ(x, t)
|φt (x, tn ) − Dφ(x, tn )| = K1 ∆tn
so that the approximation of φt is “of order ∆t”, or of “first order”.
Similarly it follows from (1.5) that
|φt (x, t) − Dφ(x, tn )| = K2 max{∆t2n , ∆t2n−1 }
so that the approximation of φt is of order ∆t2 , i.e. of second order. Here K1
and K2 are constants which depend only on the smoothness of φ.
The method of lines approximation of (1.1) at time t = tn is given by the
two point boundary value problem
a(x, tn )u′′n (x)+b(x, tn )u′n (x)−c(x, tn )un (x)−d(x, tn )Dun (x) = f (x, tn ) (1.6)
subject to
un (0) = α(tn ),
un (L) = β(tn ).
u0 (x) = g(x) is the given initial condition. It is well known that standard
finite difference methods based on a backward Euler or an implicit three-level
time discretization are unconditionally stable. As the spatial mesh parameter
in the finite difference approximation decreases to zero the equation (1.6)
results. Hence we may infer that both (1.4) and (1.5) yield a stable numerical
method for the solution of the diffusion equation (1.1). A direct proof of
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stability is provided in Appendix 1.1. We note that at the first time level
t = t1 the expression is only defined for the backward Euler quotient (1.4).
Once u1 (x) is found then for n ≥ 2 the equation (1.6) is defined for either
difference quotient Dun (x).
If equation (1.1) is to be solved subject to boundary data like (1.3) then
the boundary conditions for (1.6) simply become
un (s(tn )) = α(tn ),
un (S(tn )) = β(tn ).
However, if the interval [s(tn−1 ), S(tn−1 )] does not contain the interval [s(tn ),
S(tn )] then un−1 (x) must be extended beyond its boundary points in order
to be able to define Dun (x) on [s(tn ), S(tn )]. For example, if S(tn ) > S(tn−1 )
then we would use the (smooth pasting) linear extension
un−1 (x) = un−1 (S(tn−1 ) + u′n−1 (S(tn−1 )(x − S(tn−1 )),
x > S(tn−1 ).
Analogous extensions apply to un−2 (x) and at the lower boundary s(t).
The accuracy of the approximation of ut generally implies the same accuracy for the solution of (1.1), meaning that for an analytic solution of (1.6)
we can assert that
|u(x, tn ) − un (x)| ≤ K max ∆t1k or 2
k
where K depends solely on the smoothness of the analytic solution u(x, t)
of (1.1). A second order method is preferable, and in fact essential for an
efficient MOL code for problems in finance. It comes basically for free because numerical methods for (1.6) differ little whether (1.4) or (1.5) is used.
Numerical experiments verify that the performance of the method of lines
is greatly improved by switching over to a second order method for n ≥ 3.
In applications we shall routinely approximate ut (x, t) with the backward
quotient (1.4) for n = 1 and n = 2, and then change over to the three level
quotient (1.5) for n ≥ 3.
The reader familiar with the Crank-Nicolson method for the diffusion
equation will be aware that one could equally well define an MOL approximation involving the spatial differential operator at times tn−1 and tn . However, the Crank-Nicolson approximation requires care when initial data are
non-smooth (typical for options) or the boundary and initial data are discontinuous (typical for barrier options). In fact, even though self-starting in
principle, it often has to be combined with a few implicit Euler steps (sometimes referred to as Rannacher’s method) to suppress unwarranted initial
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1.3. MOL FOR MULTI-DIMENSIONAL PROBLEMS
oscillations. Hence the self-starting feature is lost. The three level scheme
appears simpler to apply and to give comparable results. For this reason a
Crank-Nicolson based MOL will not be considered here.
1.3
The method of lines with continuous x for
multi-dimensional problems
Asian and basket options, stochastic volatility models and multi-factor interest rate bond options lead to multi-dimensional diffusion equations. For such
equations the simplicity of the MOL described in Section 1.2 is retained, in
principle, if we combine the MOL with an analog of standard line iterative
methods known for algebraic approximations of multi-dimensional diffusion
equations. In practice, such approach is limited to very low-dimensional
problems, and we shall treat applications which involve only two “spatial”
variables x and y.
We shall consider problems of the form
a11 (x, y, t)uxx + a12 (x, y, t)uxy + a22 (x, y, t)uyy
+ b1 (x, y, t)ux + b2 (x, y, t)uy − c(x, y, t)u
− d(x, y, t)ut = f (x, y, t)
(1.7)
on a domain D(t) = {(x, y) : s(y, t) < x < S(y, t), y0 < y < Y }, t ≥ 0 where
y0 and Y are given numbers and s and S are smooth curves in y and t. The
boundary of D(t) will be denoted by ∂D(t). A schematic of D(t) is given in
Fig. ??
In applications we impose boundary conditions on u(x, y, t) on all or part
of the boundary ∂D(t), and an initial condition on u(x, y, 0). For definiteness
we shall use so-called Dirichlet boundary data
u(x, y, t) = α(x, y, t),
(x, y) ∈ D(t)
and the initial condition
u(x, y, 0) = u0 (x, y)
where α and u0 are given functions.
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CHAPTER 1.
As in the preceding section we begin by discretizing time in (1.7). Let
un (x, y) denote an approximation to u(x, y, tn ) at time tn . We employ the
difference quotients introduced in Section 1.2
(
un −un−1
for n = 1, 2
ut (x, y, tn ) ≃ Dun (x, y) = 3 un∆t
−un−1
1 un−1 −un−2
−2
for n = 3, . . . , N
2
∆t
∆t
where ∆t = tn − tn−1 is assumed to be constant for all n. For notational
convenience the argument (x, y) of un will generally be suppressed. Moreover,
if the time index n does not appear explicitly it is assumed to be the index
n of the latest time step so that
u(x, y) ≡ un (x, y).
The time index will be written only when it is needed for clarity.
The (parabolic) initial/boundary value problem for (1.7) at time tn is
approximated by the sequence of time discrete (elliptic) problems
a11 (x, y, tn )uxx + a12 (x, y, tn )uxy + a22 (x, y, tn )uyy + b1 (x, y, tn )ux
+ b2 (x, y, tn )uy − ĉ(x, y, tn )u = fˆ(x, y, tn )
(1.8)
where
(
1
d(x, y, tn ) ∆t
ĉ(x, y, tn ) = c(x, y, tn ) +
3
d(x, y, tn ) 2∆t
for n = 1, 2
for n = 3, . . . , N
and
fˆ(x, y, tn ) = f (x, y, tn )
(
d(x, y, tn ) uhn−1∆t(x,y)
−
(x,y)
d(x, y, tn ) 3un−1
+
2∆t
un−1 (x,y)−un−2 (x,y)
2∆t
i for n = 1, 2
for n = 3, . . . , N
u0 (x, y) is given and the boundary data are evaluated at tn .
In order to approximate (1.8) with ordinary differential equations in x we
impose a uniform partition
y0 < y1 < · · · < yM = Y
The points {ym } of the partition will always be indexed by m. Next we
replace derivatives with respect to y in (1.8) by central difference quotients.
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If um (x) (≡ um,n (x)) denotes an approximation to u(x, ym, tn ) then with
∆y = ym − ym−1 we write
um+1 (x) − um−1 (x)
uy (x, ym , tn ) ∼
=
2∆y
um+1 (x) + um−1 (x) − 2um (x)
uyy (x, ym , tn ) ∼
=
∆y 2
′
′
u
(x) − um−1 (x)
uxy (x, ym , tn ) = m+1
2∆y
The method of lines approximation to (1.7) with continuous x at discrete
time tn along the line y = ym for m = 1, . . . , M − 1 takes on the form
Lu(x) = a11 (x, ym , tn )u′′m (x) + b1 (x, ym , tn )u′m (x) − c̃(x, ym , tn )um (x) (1.9)
= F (x, ym , tn , um−1 (x), um+1 (x), u′m−1 (x), u′m+1 (x),
um,n−1 (x), um,n−2 (x))
where
c̃(x, ym , tn ) = ĉ(x, ym , tn ) + a22 (x, ym , tn )
2
∆y 2
and
ˆ ym , tn ) − a22 (x, ym , tn ) um+1 (x) + um−1 (x)
F (x, ym , tn ) = f(x,
∆y 2
um+1 (x) − um−1 (x)
− b2 (x, ym , tn )
2∆y
u′m+1 (x) − u′m−1 (x)
− a12 (x, ym , tn )
2∆y
(We recall the notation: um ∼
= u(x, ym, tn ), um,n−1(x) ∼
= u(x, ym , tn−1 ).)
The initial condition yields {um,0 (x)}M
and
the
boundary
conditions give
m=0
{um (s(ym , tn ))} and {um (S(ym , tn ))}, u0 (x), uM (x) and their derivatives.
Equations (1.9) represent a boundary value problem for a system of M −1
coupled ordinary differential equations. If s(y, tn) or S(y, tn) is not constant
with respect to y then the equations are defined over different x-intervals
and the boundary value problem is a so-called multipoint boundary value
problem. Its numerical solution is often as complicated as the numerical
solution of the partial differential equation (1.8). However, for problems
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CHAPTER 1.
arising in finance it generally is possible to solve the system (1.9) iteratively
as a sequence of scalar second order equations like (1.6).
We introduce a new index k for the iteration count. It appears as a superscript of u so that ukm (x) stands for the kth iterate approximating u(x, ym , tn ).
−1
At time tn let {u0m(x)}M
m=1 denote an initial guess for the solution {um (x)}
of (1.9). (Typically one would choose the solution from the preceding time
step
u0m (x) = um,n−1 (x).)
In the kth iteration for k ≥ 1 we compute a solution {ukm (x)} for m =
1, . . . , M − 1 by solving
k
k−1
′
′
Lm ukm (x) = F (x, ym , tn , ukm−1 , uk−1
m+1 , u m−1 , u m+1 , um,n−1 , um,n−2 ).
(1.10)
Note that at any stage the right hand side F is a known source term.
Assuming that the sequence {ukm (x)} converges as k → ∞ we obtain
um,n (x) = lim ukm (x)
k→∞
for m = 1, . . . , M − 1.
The reader familiar with iterative methods for linear algebraic systems will
recognize that our iterative method is a line Gauss-Seidel iteration, except
that along each line y = ym a two point boundary value problem for a second
order ordinary differential equation instead of a matrix problem has to be
solved. In practice we set
um,n (x) = uK
m (x)
where K is an integer such that
K−1
max |uK
m (x) − um (x)| ≤ ǫ.
m
The choice of the convergence tolerance ǫ is dictated by the accuracy required
by the application and the computer time it takes for convergence.
If a a12 (x, y, t) ≡ 0 then it is possible to prove that the line Gauss Seidel method will indeed converge to a solution of the multipoint problem at
each time level []. Numerical experience suggests that the iteration will also
converge for problems in finance where a12 6= 0.
On occasion spurious oscillatory solutions can arise from the central difference approximation of uy (x, y, t). They can be suppressed by either decreasing ∆y, or if that is infeasible, by replacing
b2 (x, ym , tn )uy (x, ym , tn )
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with a lower order one-sided difference quotient. For example, if b2 ≥ 0 we
would choose
um+1 − u
uy (x, ym , tn ) =
,
∆y
if b2 < 0 we would use
uy (x, ym , tn ) =
u − um−1
.
∆y
These first order quotients represent an “upwinding” of the convection term
b2 uy in (1.7). Upwinding is a well understood and important tool in the
theory of numerical methods for partial differential equations [].
We also remark that if ∆t is very large then the convergence of the line
Gauss Seidel iteration may be unacceptably slow. In this case a line SOR
modification of the Gauss Seidel method may prove helpful. We consider the
solution of (1.10) an intermediate solution and denote it by ũ. The desired
solution ukm is found from
k−1
ukm = uk−1
m + ω[ũm − um ]
for some ω ∈ [1, 2]. The optimum relaxation factor ω is not known a priori
but can be found by trial and error. ω = 1 yields again the Gauss Seidel
iteration.
Finally, we remark that it is straightforward to generalize this approach
to multi-dimensional equations like
p
X
i,j=1
aij uxixj +
p
X
bi uxi − cu − dut = f (~x, t)
i=1
where ~x = (x1 , . . . , xp ). Such equation would arise, for example, in connection with basket options depending on p assets. We discretize t and then
x2 , . . . , xp and replace all derivates with respect to these variables by appropriate finite difference quotients. A system of the order of M p−1 coupled
second order ordinary differential equations in the independent variable x1
results to which a line Gauss Seidel iteration can be applied. Some parallelization of the line Gauss Seidel method is possible, but we have no practical
experience with financial applications for p > 2.
In summary then, the method of lines appximation for the parabolic
equation describing financial derivatives is reduced at each time level to a
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single linear second order differential equation (1.6), or to a sequence of such
equations. This is a sensible approach only if such second order equation
can be solved accurately and efficiently. As stated in Section 1.1, a standard
approach would be to replace the derivatives in (1.6) by their finite difference
analogs and to set up a matrix system. The resulting algorithm is again
a standard finite difference algorithm for the diffusion equation found in
every textbook on numerical methods for partial differential equations. Such
method would not merit the detour via the method of lines approximation.
Here we shall employ a less familiar algorithm for the solution of (1.6),
but one which is readily adapted to the class of problems associated with
American options. This algorithm is the subject of the next chapter.
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