Black-Scholes PDE - mimuw

Lecture 6

Black-Scholes PDE

Lecture Notes by Andrzej Palczewski

Computational Finance ? p. 1

Pricing function

Let the dynamics of underlining St be given in the risk-neutral measure Q by

dSt = rStdt + StdWt. If the contingent claim X equals

X = h(ST ) for some function h, then the price of X at time t is given by

Vt = V (St, t), where V (s, t) is given by the formula

V (s, t) = e-r(T -t)EQ h(ST )|St = s .

Computational Finance ? p. 2

Replication strategy

We call (, ) the replication strategy for contingent claim X, if the value process is given by

Vt = tBt + tSt, where Bt is the bank account.

Computational Finance ? p. 3

Goal

Find the function V (s, t) : (0, ) ? [0, T ] [0, ).

With the function V (s, t) we can: compute the price of the contingent claim: at t it equals

V (St, t).

find the replicating strategy

t

=

V s

(St, t),

t = e-rt V (St, t) - tSt .

Computational Finance ? p. 4

The Black-Scholes PDE

Theorem. If X = h(ST ) then there exists a function V : (0, ) ? [0, T ] R such that

Vt = V (St, t).

This function is a solution to the Black-Scholes partial differential equation

V (s, t

t)

+

V (s, t) rs s

+

1 2

2s2

2V (s, s2

t)

-

rV

(s,

t)

=

0

with the terminal condition

V (s, T ) = h(s)

for any s > 0 and t [0, T ].

Computational Finance ? p. 5

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