EXAMINATION GUIDE 2021
MATH3975 Financial Mathematics (Advanced)
• You should know fundamental results, understand the general principles of arbitrage-free
pricing of financial derivatives and be ready to solve problems regarding general discrete-time
market models, the CRR binomial model, and the continuous-time Black-Scholes model.
• No aids other than standard non-programmable calculators are permitted. You should under-
stand formulae, concepts and computational techniques needed to answer typical questions,
as outlined below. The best preparation is to go through lecture notes and tutorial questions
for MATH3975.
1. Single-period model (Exercise 1 from tutorial 6)
(a) verify the arbitrage-free property and completeness of a model,
(b) describe the class of all attainable contingent claims,
(c) find the set of all risk-neutral probability measures,
(d) compute the replicating strategy and arbitrage price for a contingent claim,
(e) analyse the properties of a general single-period model.
2. Multi-period model: European claims (Exercise 4 from tutorial 7, Exercise 2 from tutorial 8,
Exercises 1 and 2 from tutorial 9)
(a) find the class of all risk-neutral probability measures (i.e., martingale measures),
(b) describe the replicating strategy for a given European contingent claim,
(c) compute the arbitrage price of a given European contingent claim,
(d) verify the martingale property of the discounted price process,
(e) identify the range of prices for a non-attainable contingent claim.
3. The CRR model: American claims (Exercises 1 and 2 from tutorial 10)
(a) find the martingale measures for the CRR model,
(b) compute the arbitrage price of a given American contingent claim,
(c) find the rational exercise time for the holder,
(d) compute the replicating strategy for the issuer,
(e) analyse the martingale property of the discounted price of an American claim.
4. The Black-Scholes model: European claims (Exercises 1, 2 and 3 from tutorial 11)
(a) apply the Black-Scholes formula to price European call and put options,
(b) compute the price in the Black-Scholes model of a given path-independent European
claim with a piecewise linear payoff through a decomposition into a combination of put
and call options,
(c) identify the replicating strategy for a European claim,
(d) analyse the asymptotic properties of the arbitrage price of a European claim with a piece-
wise linear payoff.