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˜.

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˜.