MA4269 Tutorial 3 Solution
Question 1. Consider the Brownian motion Wt , t
0. Show that for any s, t
0, we have
Cov [Wt , Ws ] = mincfw_s, t.
Here Cov[X, Y ] is the covariance between X and Y .
Solution. Without loss of generality, let s < t. We need to s

MA4269 Tutorial 1 (Week 3)
Question 1. Suppose the current forward price for a 6-month forward contract is $35, and the
underlying non-dividend paying stock is currently at $30. Suppose the risk-free interest rate
is 12% per annum with continuous compound

MA4269 Tutorial 1 Solution
Question 1. Suppose the current forward price for a 6-month forward contract is $35, and the
underlying non-dividend paying stock is currently at $30. Suppose the risk-free interest rate
is 12% per annum with continuous compound

MA4269 Tutorial 3 (Week 5)
Question 1. Consider the Brownian motion Wt , t 0. Show that for any s, t 0, we have
Cov [Wt , Ws ] = mincfw_s, t.
Here Cov[X, Y ] is the covariance between X and Y .
Question 2. A companys cash position, measured in millions of

MA4269 Assignment 4 (2 Questions)
Important: due on Wednesday 2 November 2016. Submit to the boxes in
the front of LT33 before the beginning of the lecture, according to your
tutorial group number.
Please remember to write your name & matric number & tuto

MA4269 MATHEMATICAL FINANCE II
Semester 2, 2013/2014
Mid-term Test
Problem 1 [10 marks]
A non-dividend paying stock is currently $25. It is know that in 2 months it will
be either $23 or $27. The risk-free interest rate is 10% per annum with continuous
co

MA4269 Mid-Semester Test
1 OCT 2014.
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. Writ

MA4269 Mid-Semester Test
2 OCT 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. Writ

Monte Carlo Methods in
Mathema0cal Finance
Lecture 11: Sensi0vity Analysis
Introduc0on
Consider an op0on on a stock with maturity T. Let V (S,
r, ,
t)
the value of the op0on at 0me t

Mid-Semester Test Information
MA4269 Mathematical Finance II Sem 1 AY2015-16.
Date: 28 SEPTEMBER 2015 (Wed)
Time: 7:30pm 9:00pm
Venue: MPSH 1 B
1. You are allowed to use ONE A4-size two-sided handwritten help sheet during the test.
2. Definition of a help

MA4269 Tutorial 4 (Week 6)
Question 1. What is the Black-Scholes 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 ann

MA4269 Tutorial 5 (Week 7)
Question 1. Determine
c
, the rate of change of the value of the European call option relative
q
to the dividend yield.
Question 2. Does dividend yield increase or decrease the value of the put option?
Question 3. Assume the Bla

MA4269 Tutorial 4 Solution
Question 1. What is the Black-Scholes 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 ann

MA4269 Tutorial 5 Solution
Question 1. Determine
c
, the rate of change of the value of the European call option relative
q
to the dividend yield.
Solution. Recall from previous tutorial that
St N 0 (d1 ) = Ker N 0 (d2 ).
With dividend yield q, it can be

Monte Carlo Methods in
Mathema0cal Finance
Lecture 7: Importance Sampling
Introduc0on
Consider the problem of es0ma0ng the expected value
= E[h(X)].
Suppose X
has the density f (x)
.
Plain MC:

Monte Carlo Methods in
Mathema0cal Finance
Lecture 6: Variance Reduc0on Techniques
Variance Reduc0on Techniques
Recall the error in Monte Carlo es0mate:
Let
= E[H] : the true value
1
= (H1 + + Hn ) : MC estimate
n

Chapter 2
Stochastic calculus: A crash course
1
Wiener process/Brownian motion
In 1828, Robert Brown observed irregular movement of pollen suspended in water.
This motion is now known to be caused by the bueting of the pollen by water
molecules, as explai

Chapter 3
The Black-Scholes PDE
Throughout, we assume the following Black-Scholes model for our market
consisting of one stock and the money market account.
(1) The money market account (riskless asset) Mt is given by
dMt = rMt dt.
(2) The stock price fol

Chapter 5
American options
1
Introduction
While most index options are of European-style, most traded stock options are of
American-style.
The distinctive feature of American options is the early exercise privilege,
that is, the holder of an American opti

Chapter 6
Barrier options
1
Introduction
One of the most commonly traded path-dependent options is the continuous barrier
option. This is an option with the ordinary call/put payo subject to an additional
event of whether a perscribed level has been cross

Chapter 4
The Martingale Approach to Pricing
1
Introduction
The aim of this chapter is to introduce the martingale method for pricing nancial
derivatives. This modern approach to pricing has far reaching consequences, even
beyond the Black-Scholes framewo

Chapter 8
Lookback options
1
Introduction
Lookback options are path dependent options whose payos depend on the maximum or the minimum of the underlying stock price attained over a certain period of
time (called the lookback period). Throughout, we set th

Chapter 7
Asian options
1
Introduction
As usual, we assume the Black-Scholes framework with no dividends.
Asian options are path-dependent options whose payo depends on some form
of averaging of the price of the underlying asset. The average is usually ta

Chapter 9
Multi-asset options
1
Introduction
This chapter covers options which depend on more than one risky asset. We focus
on two special and important types of multi-asset options: exchange options and
cross-currency options.
2
Two-assets options
We as

MA4269
Question 1. [15 marks]
Note that under P:
2 /2)+0.2W
S1 = S0 e(0.10.2
and under Q:
2
1
= 100e0.08+0.2
S1 = S0 e(0.050.2 /2)+0.2W1 = 100e0.03+0.2
ft is a Q-Brownian motion.
where is standard normal, W
f
Thus, under P:
ln S1 = ln 100 + (0.08 + 0.02).

MA4269
NATIONAL UNIVERSITY OF SINGAPORE
MA4269 MATHEMATICAL FINANCE II
(SEMESTER 1: AY 2014-2015)
Time allowed: 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
1. Please write your matriculation/student number only. Do not write your
name.
2. This examinati