# Mathematics for finance

Mathematics for finance

Math 194 Homework 3

1. Textbook Vol I, Chapter 2, Exercise 2.3.

2. Textbook Vol I, Chapter 2, Exercise 2.4.

3. Textbook Vol I, Chapter 2, Exercise 2.6.

4. Let Mn be a martingale such that M0 = 0. Show that (a) E[Mn ] = 0; (b) Cov(Mn+1 , Mn ) =

E[Mn2 ].

5. (a) Background : The stock (without dividend paying) return µ and its volatility can be

computed as follows. Suppose a sequence of historical prices Si is observed on daily basis.

ui is defined as the continuous compound return ln(Si+1 /Si ) on day i (e.g., Si+1 = Si eui ).

Under the assumption that ui are i.i.d. random variables for all i, the daily return µ =

E[ui ] and the daily volatility = Std[ui ] (e.g., standard deviation). The daily risk-free

continuous compound return is r (e.g., $1 becomes $er one day later).

Show that Var[ln(ST /S0 )] = 2 T (Stock price ST for day T ).

(b) Binomial Tree Construction : Consider the stock price from the time 0 to T (in days).

In the n-periods binomial tree, each period corresponds

to t = T /n

day. We showed in

p

p

eµ t d

t

t satisfy (ignore a

class that the real probability p = u d , u = e

and d = e

higher order term t3/2 )

pS0 u + (1 p)S0 d = S0 eµ t

and

pu2 + (1

p)d2

[pu + (1

p)d]2 =

2

t.

In other words, the return and volatility of the binomial model is matched with the real

data.

r t

Show that the risk neutral probability is p˜ = e u d d and under this risk neutral measure,

the volatility of the binomial model does not change, ignoring a higher order term t3/2

(Hint: use the Taylor expansion).

(c) ST Distribution under p˜: Denote Bi t be the random variable taking 1 when i-th coin

toss H and 1 otherwise. Si t is the stock price at the i-th period

is at the last period).

p(STP

Given the binomial tree in (a), it is obvious that ln(ST /S0 ) =

t nk=1 Bi t . Question

g

(b) proved that the volatility under p and p˜ are the same, meaning Var[ln(S

T /S0 )] =

2

Var[ln(ST /S0 )] = T . Therefore, the central limit theorem (let n ! 1) implies that

ln(ST /S0 ) has the normal distribution N (a, 2 T ) under p˜ for some unknown constant a.

2

˜

Show that a = E[ln(S

T /S0 )] = (r

2 )T . (Hint: use the fact that ln(ST /S0 ) has Gaussian

˜ T ], e.g., the discount stock price is martingale

distribution and the formula S0 = e rT E[S

under p˜.)

2

2

(d) Show that ST = S0 e(r 2 )T + T z with the standard normal random variable z ? N (0, 1)

under p˜. ST is said to satisfy the lognormal distribution under p˜.

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