MATH 5510 Mathematical Models of Financial Derivatives
Topic 1 Risk neutral pricing principles under discrete securities models
1.1 Law of one price and Arrow securities
1.2 No-arbitrage theory and risk neutral probability measure Fundamental theorem of a

MATH5510 - Mathematical Models of Financial Derivatives
Homework Four
Instructor: Prof. Y.K. Kwok
1. Consider the function
(
f (S, ) =
S
B
(
)
cE
B2
,
S
)
,
where cE (S, ) is the price of a vanilla European call option, is a constant parameter. Show
that

MATH 5510 Mathematical Models of Financial Derivatives
Topic 2 Black-Scholes-Merton framework and Martingale
Pricing Theory
2.1 Stochastic processes and Ito calculus
2.2 Change of measure: Girsanovs theorem
2.3 Riskless hedging principle and dynamic repli

MATH 5510 Mathematical Models of Financial Derivatives
Topic 4 - Financial derivatives with embedded barrier features
4.1 Product nature of barrier options
Accumulators
4.2 Partial dierential equation approach and method of images
4.3 Probabilistic appro

MATH 5510 Mathematical Models of Financial Derivatives
Topic 6 - Derivatives with averaging style payos
6.1
Pricing models of Asian options
Partial dierential approach for continuous models
Closed form pricing formulas for discretely monitored models
6.

MATH 5510
Mathematical Models of Financial Derivatives
Homework One
Course Instructor: Prof. Y.K. Kwok
1. Show that a dominant trading strategy exists if and only if there exists a trading strategy satisfying V0 < 0
and V1 () 0 for all .
Hint: Consider th

MATH 5510
Mathematical Models of Derivatives
Homework Three
Course Instructor: Prof. Y.K. Kwok
1 Let Q be the martingale measure with the money market account as the numeraire and
Q denote the equivalent martingale measure where the asset price St is used

MATH 5510
Mathematical Models of Derivatives
Homework Two
Course Instructor: Prof. Y.K. Kwok
1. Consider a portfolio containing units of asset and M dollars of riskless asset in the form of money market account. The portfolio is dynamically adjusted so as

MATH5510 - Mathematical Models of Financial Derivatives
Homework Six
Instructor: Prof. Y.K. Kwok
1. Apply the exchange option price formula to price the oating strike Asian call option based on
the knowledge of the price formula of the xed strike Asian ca

MATH5510 - Mathematical Models of Financial Derivatives
Homework Five
Instructor: Prof. Y.K. Kwok
1. As an alternative approach to derive the value of a European oating strike lookback call, we
consider
cf (S, m, ) = er EQ [ST min(m, mT )]
t
= S er EQ [mi