PDE for Finance Notes, Spring 2011 – Section 6
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences.
For use only in
connection with the NYU course PDE for Finance, G63.2706.
Prepared in 2003, minor
updates made in 2011. Revised and extended 4/3/2011.
Optimal stopping and American options.
Optimal stopping refers to a special class
of stochastic control problems where the only decision to be made is “when to stop.” The
decision
when to sell an asset
is one such problem. The decision
when to exercise an Amer
ican option
is another. Mathematically, such a problem involves optimizing the expected
payoff over a suitable class of stopping times. The value function satisfies a “free boundary
problem” for the backward Kolmogorov equation.
We shall concentrate on some simple yet representative examples which display the main
ideas, namely: (a) a specific optimal stopping problem for Brownian motion; (b) when to
sell a stock which undergoes lognormal price dynamics; and (c) the pricing of a perpetual
American option.
At the end we discuss how the same ideas apply to the pricing of an
American option with a specified maturity. My discussion of (a) borrows from notes Raghu
Varadhan prepared when he taught PDE for Finance; my treatment of (b) is a dumbed
down version of one in Oksendal’s book “Stochastic Differential Equations;” the perpetual
American put is discussed in many places. My main suggestion for further reading in this
area is however FR Chang’s excellent book “Stochastic Optimization in Continuous Time”
(Cambridge Univ Press), which is on reserve in the CIMS library. It includes many examples
of stochastic control, some close to the ones considered here and others quite different.
***********************
Optimal stopping for 1D Brownian motion.
Let
y
(
t
) be 1D Brownian motion starting
from
y
(0) =
x
. For any function
f
, we can consider the simple optimal stopping problem
u
(
x
) = max
τ
E
y
(0)=
x
e

τ
f
(
y
(
τ
)
.
Here
τ
varies over all
stopping times
. We have set the discount rate to 1 for simplicity. We
first discuss some general principles then obtain an explicit solution when
f
(
x
) =
x
2
.
What do we expect? The
x
axis should be divided into two sets, one where it is best to
stop immediately
, the other where it is best to
stop later
.
For
x
in the stopimmediately
region the value function is
u
(
x
) =
f
(
x
) and the optimal stopping time is
τ
= 0. For
x
in
the stoplater region the value function solves a PDE. Indeed, for Δ
t
sufficiently small (and
assuming the optimal stopping time is larger than Δ
t
)
u
(
x
)
≈
e

Δ
t
E
y
(0)=
x
[
u
(
y
(Δ
t
))]
.
By Ito’s formula
E
y
(0)=
x
[
u
(
y
(
t
)] =
u
(
x
) +
Z
t
0
1
2
u
xx
(
y
(
s
))
ds.
Applying this with
t
= Δ
t
and approximating the integral by
1
2
u
xx
(
x
)Δ
t
we conclude that
u
(
x
)
≈
e

Δ
t
(
u
(
x
) +
1
2
u
xx
Δ
t
). As Δ
t
→
0 this gives the PDE in the stoplater region:
1
2
u
xx

u
= 0
.
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 Fall '11
 Bayou
 Finance, Dynamic Programming, Mathematical finance, free boundary, highorder contact condition

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