Answered: - Could you please help me with these financial math problems?

Could you please help me with these financial math problems?

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 > 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 > 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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