PDE for Finance Notes, Spring 2011 – Section 2.
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates
made in 2011.
Solution formulas for the linear heat equation. Applications to barrier options.
Section 1 was relatively abstract – we listed many PDE’s but solved just a few of them.
This section has the opposite character: we discuss explicit solution formulas for the linear
heat equation – both the initial value problem in all space and the initialboundaryvalue
problem in a halfspace. This is no mere academic exercise, because the (constantvolatility,
constantinterestrate) BlackScholes PDE can be reduced to the linear heat equation. As
a result, our analysis provides all the essential ingredients for valuing barrier options. The
PDE material here is fairly standard – most of it can be found in John or Evans or Strauss
(among other places). Our discussion of the halfspace problem with initial condition 0
follows the treatment by John. For the ﬁnancial topics (reduction of BlackScholes to the
linear heat equations; valuation of barrier options) see e.g. the “student guide” by Wilmott,
Howison, and Dewynne.
****************************
TheheatequationandtheBlackScholesPDE.
We’ve seen that linear parabolic equa
tions arise as
backward
Kolmogorov equations, determining the expected values of various
payoﬀs. They also arise as
forward
Kolmogorov equations, determinining the probability
distribution of the diﬀusing state. The simplest special cases are the backward and forward
linear heat equations
u
t
+
1
2
σ
2
Δ
u
=0and
p
s

1
2
σ
2
Δ
p
= 0, which are the backward and
forward Kolmogorov equations for
dy
=
σdw
, i.e. for Brownian motion scaled by a factor
of
σ
. From a PDE viewpoint the two equations are equivalent, since
v
(
t, x
)=
u
(
T

t, x
)
solves
v
t

1
2
σ
2
Δ
v
= 0, and ﬁnaltime data for
u
at
t
=
T
determines initialtime data for
v
at
t
=0
.
This basic example has direct ﬁnancial relevance, because the BlackScholes PDE can be
reduced to it by a simple change of variables. Indeed, the BlackScholes PDE is
V
t
+
rsV
s
+
1
2
σ
2
s
2
V
ss

rV
.
It is to be solved for
t<T
, with speciﬁed ﬁnaltime data
V
(
s, T
)=Φ
(
s
). (Don’t be
confused: in the last paragraph
s
was time, but here it is the “spatial variable” of the Black
Scholes PDE, i.e. the stock price.) We claim this is simply the standard heat equation
u
t
=
u
xx
written in special variables.
To see this, consider the preliminary change of
variables (
s, t
)
→
(
x, τ
) deﬁned by
s
=
e
x
,τ
=
1
2
σ
2
(
T

t
)
,
and let
v
(
x, τ
V
(
s, t
). An elementary calculation shows that the BlackScholes equation
becomes
v
τ

v
xx
+(1

k
)
v
x
+
kv
1