NOTATION, CONVENTION and FORMULAE
This page and the next contain the notation and convention used in this exam paper and
some formulae for your reference.
Throughout the paper, T denotes the maturity date of contracts. The risk-free interest
rate r (per a

CHAPTER 0
Overview
This course is an introduction to quantitative methods used in pricing financial derivatives. The goal
is to explain the assumptions and concepts behind these mathematical models.
0.1
Prerequisites
I assume you already have basic knowle

MA3245 Mid-Semester Test
6 MARCH 2013.
Time allowed: 60 minutes
INSTRUCTIONS TO CANDIDATES
1. This test contains a total of FOUR(4) questions. Answer ALL questions.
2. Each candidate is allowed to bring ONE(1) piece of A4-sized two-sided help sheet.
3. Wr

MA3245 Test Solution
Q1. Note the put-call parity is violated since
cp =4
> S0 eqT KerT
= 60e0.05 59e0.1 = 3.688
Consider the portfolio 0 initiated today at time 0 which comprises the following:
(1) short one call;
(2) long one put;
(3) long eqT = e0.05 u

MA3245 EXAM DATE AND VENUE
Date : 04 MAY 2013 (SAT)
Time : 1:00 PM
Venue : MPSH 2B
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination. Each student is allowed to bring at most ONE (1)
piece of A4-sized two-sided help sheet into the examinatio

Tutorial 11
MA3245 Financial Mathematics 1
1. Calculate the value of a five-month European put futures price when the futures price is $19 , the
strike price is $20 , the risk-free interest rate is 12% per annum, and the volatility of the futures
price is

Tutorial 8
MA3245 Financial Mathematics 1
1. What is the price of a European put option on a non-dividend paying stock when the stock price is
$69 , the strike price is $70 , the risk-free interest rate is 5% per annum, the volatility is 32%
per annum, an

Tutorial 10
MA3245 Financial Mathematics 1
1. What does it mean to assert that the theta of an option is 0.1 when time is measured in years?
If a trader feels that neither a stock price nor its implied volatility will change, what type of option
position

Tutorial 7
MA3245 Financial Mathematics 1
1. A companys cash position, measured in millions of dollars, follows a generalized Wiener process
with a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position
is 2.0 .
(a) W

Tutorial 9
MA3245 Financial Mathematics 1
1. An index currently stands at 1500 . European call and put options with a strike price of 1400
and time to maturity of six months have market prices of 154.00 and 32.45 respectively. The
continuously compounded

Tutorial 5
MA3245 Financial Mathematics 1
1. A non-dividend paying stock is currently $25 . It is known that in 2 months it will be either
$23 or $27 . The risk-free interest rate is 10% per annum with continuous compounding. Suppose
ST is the stock price

Tutorial 11 Solution
MA3245 Financial Mathematics 1
1. In this case
F0 = 19, K = 20, r = 0.12, = 0.2, T = 0.4167.
The value of the European put futures option is
p = 20N (d )erT 19N (d+ )erT
where
d+ =
ln(19/20) + (0.04/2)0.4167
= 0.3327,
0.2 0.4167
d = d

Tutorial 9 Solution
MA3245 Financial Mathematics 1
1. By put-call parity
cp
154 32.45
q
= St eq Ker
1500e0.5q 1400e0.10.5
=
0.0633.
This is the implied dividend yield per annum.
2. Note that
V
= aeat S 2 ,
t
V
= 2eat S,
S
2V
= 2eat .
S 2
Suppose V satisf

Tutorial 2
MA3245 Financial Mathematics 1
1. Suppose the risk-free interest rate is 4% with continuous compounding and the S & P index
6 -month forward price is $1020 . Suppose the premium on a 6 -month S & P European call
is $109.20 and the premium on a

Tutorial 4
MA3245 Financial Mathematics 1
1. Suppose the current stock price is $50 and the American call and the American put option prices
with the same maturity date T are given by
Strike
50
55
Call premium
Put premium
10
8
6
12.50
Assume that the pres

Tutorial 10 Solution
MA3245 Financial Mathematics 1
1. A theta of 0.1 means that if 4t units of time pass with no change in either the stock price or its
volatility, the value of the option declines by 0.1 4t. A trader who feels that neither stock price
n

Tutorial 3
MA3245 Financial Mathematics 1
1. In the following, c(K1 ) , c(K2 ) , p(K1 ) and p(K2 ) denote respectively the prices of
European calls and puts with strike prices K1 and K2 where K1 < K2 . Note that all these
European options are on the same

Tutorial 1
MA3245 Financial Mathematics 1
1. Use arbitrage to show that if we know with certainty that two self-financing portfolios will have
precisely the same value at some time T in the future, then they must have the same value now at
time 0 . Assume

Tutorial 6 Solution
MA3245 Financial Mathematics 1
1. In this case,
a = e(rrf )4t = e(0.050.08)1/12
= 0.9975
4t
= e0.12 1/12 = 1.0352
1
d = = 0.9660
u
ad
0.9975 0.9660
p=
=
= 0.455.
ud
1.0352 0.9660
u = e
2.
(a) Let c(K) denote the European call price wit

Tutorial 2 Solution
MA3245 Financial Mathematics 1
1. This is a direct application of the put-call parity:
c p = (F0,T K)erT .
Thus,
109.20 60.18
49.02
49.02e0.040.5
K
=
=
=
=
(1020 K)er(T 0)
(1020 K)e0.040.5
1020 K
970.
(1)
(2)
(3)
(4)
2.
(a) Let ST be t

Tutorial 7 Solution
MA3245 Financial Mathematics 1
1.
(a) The probability distributions are
N (2 + 0.1, 0.16 1) = N (2.1, 0.16)
N (2 + 0.6, 0.16 6) = N (2.6, 0.96)
N (2 + 1.2, 0.16 12) = N (3.2, 1.92)
(b) The chance of a random variable from N (2.6, 0.96)

Tutorial 8 Solution
MA3245 Financial Mathematics 1
1. We have
St = 69, K = 70, r = 0.05, = 0.32, = 0.5.
Therefore, the put price is
p = Ker N (d ) St N (d+ ) = 5.828.
2. Note that
Y
2Y
Y
= et Xt ,
= et ,
= 0.
t
X
X 2
By Itos Lemma
Y
1 2 2Y
Y
Y
Xt
+
dWt

Tutorial 3 Solution
MA3245 Financial Mathematics 1
1.
(a) We first prove that
p(K2 ) p(K1 ) .
Suppose that p(K1 ) >
and long one unit of the
p(K2 ) . Form the portfolio: short one unit of the K1 -strike European put
K2 -strike European put.
Obviously, the

Tutorial 5 Solution
MA3245 Financial Mathematics 1
1. At the end of two months the value of the derivative will be either 529 (if the stock price is
and 729 (if the stock price is 27 ). Consider the portfolio consisting of + shares and
derivative.
23 )
1

Tutorial 4 Solution
MA3245 Financial Mathematics 1
1. Consider the following strategy:
sell the 55-strike American put
buy the 55-strike American call
short-sell the stock
invest cash worth of $56 in money market
The initial cost of this strategy is
1

Tutorial 1 Solution
MA3245 Financial Mathematics 1
1. Consider two portfolios A and
B
A
T = T in all states of nature.
B which have the same value at time T with certainty, i.e.
B
Suppose A
0 > 0 . Consider the portfolio
opposite position for all position

CHAPTER 10
Greeks and Hedging, Futures Options
10.1
The Greeks for options
Option Greeks are formula that express the change in the option price when an input to the formula
changes, taking all other inputs as fixed. In other words, the Greeks are partial

CHAPTER 9
Risk-Neutral Valuation
In the binomial model, we learned that we can price derivatives using the risk-neutral probability
measure Q . Recall that this measure Q is determined by the risk-neutral probability p that was
obtained from solving a sys

CHAPTER 1
Forward contracts and options
1.1
Overview
The aim of this chapter is to introduce some basic financial derivatives which have become increasingly important in finance. Derivatives (or rather their misuse) are often blamed for almost destroying

CHAPTER 8
The Black-Scholes PDE
The Black-Scholes partial differential equation (PDE) is an equation that must be satisfied by the
price of any derivative whose underlying stock follows a Geometric Brownian motion.
The argument used to derive this equatio