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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