PDE, Linear algebra, Probability.

Partial di?erential equations
1. Consider the Black-Scholes problem
@V
@t
+ 12
“2 S2 @2V
@S2 + r S
@V
@S – r V = 0, S>0, t<T,
V (S, T) = f(S),
(1)
where ” > 0, r 2 R and T > 0 are constants and f : R+ ! R is a given function.
(i) Given that V (S, t), the solution of (1), is infinitely di?erentiable in S > 0 and
t < T, show that
V1(S, t) = S
@V
@S
(S, t)
also satisfies the partial di?erential equation in (1).
(ii) Assume that V (S, t) and all its t and S-partial derivatives are di?erentiable with
respect to the parameter r, for all S > 0 and t < T, and any value of r. Deduce
that
?(S, t) =
@V
@r
(S, t)
satisfies the problem
@?
@t
+ 12
“2 S2 @2?
@S2 + r S
@?
@S – r ? = V – S
@V
@S
, S>0, t<T,
?(S, T) = 0.
(2)
(iii) Given that (2) uniquely determines ?(s, t) and assuming that both V and
S @V/@S are order o
!
1/(T – t)
”
as t ! T-, show that for S > 0 and t ? T
?(S, t) = (T – t)
?
S
@V
@S
(S, t) – V (S, t)
?
.
(iv) You may assume that if K >0 is a constant and
d =
log(S/K) + (r – 12
“2)(T – t) p
“2(T – t)
, “(x) =
1
p2?
Z x
-1
e-p2/2 dp,
then
V (S, t) = e-r(T-t)”(d)
is a solution of the Black-Scholes equation in (1), for S > 0, t < T, ” > 0. For
this particular V , and assuming S > 0 and t < T, find
f(S) = lim
t!T

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