#### Question Details

Math 294, Winter 2016

Homework 5 ? Due March 23

In problems 1 and 2, the notation is that used in class for the Blach-Scholes model, as found

in section 4.2 of the text (pages 57?61).

1. In each case, for the given ?t (holdings at time t in the stock) and for a given real number c,

?nd ?t such that the trading strategy ? = (?t , ?t )0?t?T is self-?nancing with V0 (?) = c.

(a) ?t = 1, 0 ? t ? T .

(b) ?t = St , 0 ? t ? T .

t

(c) ?t = 0 Su du, 0 ? t ? T .

2. The time-t value of a European call option X = (ST ? K)+ can be expressed as Vt = F (St , t).

Under the Equivalent Martingale Measure P? , the discounted value process Vt? := e?rt Vt is a

martingale. Observe that

?

Vt? = G(St , t),

where

G(x, t) = e?rt F (xert , t),

for x &gt; 0 and 0 ? t ? T . Also,

t

? ?

Su dWu ,

?

?

St = S0 + ?

0

?

with W a Brownian motion under P? . We have seen in class that G satis?es the partial di?erential

equation (PDE)

?G x2 ? 2 ? 2 G

+

= 0,

?t

2 ?x2

on (0, ?) ? [0, T ). Using this show that F satis?es the PDE

?F

x2 ? 2 ? 2 F

?F

+

+ rx

? rF = 0,

2

?t

2 ?x

?x

on (0, ?) ? [0, T ), and that F (x, T ) = (x ? K)+ , the so-called Black-Scholes equation.

3. [In this exercise, (Wt )t?0 is a (standard) Brownian motion de?ned on some ?ltered probability

space (?, F, (Ft ), P).] Fix T &gt; 0, and consider the random variable X := eWT . The goal of this

T

exercise is to ?nd a real number ? and an integrand H = Hs (?) ? L such that X = ? + 0 Hs dWs .

(a) The martingale Mt := E[X|Ft ], 0 ? t ? T , takes the form f (Wt , t) for some smooth

function f : R ? [0, T ] ? (0, ?). Find f .

2

(b) Check that ?f + 1 ? f = 0 for (x, t) ? R ? [0, T ], and f (x, T ) = ex .

?t

2 ?x2

(c) Now expand f (Wt , t) using It??s formula, to obtain the desired representation of X(=

o

MT ). Notice that ? = f (0, 0).

4. Exercise 5.8.2, page 121 of the text.

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