7242019 MATH_3075_3975_t3

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MATH30753975

Financial Mathematics

School of Mathematics and StatisticsUniversity of Sydney

Semester 2 2012

Tutorial sheet Week 3

Exercise 1 What is the price at time 0 of a contingent claim representedby the payoff h(S 1) = S 1 Give at least two explanations

Exercise 2 Give a proof of the put-call parity relationship

Exercise 3 Compute the hedging strategies for the European call and theEuropean put in Examples 211 and 212

Exercise 4 Consider the elementary market model with the parameters r =1

4 S 0 = 1 u = 3 d = 1

3 p = 4

5 Compute the price of a digital call option

with strike price K and the payoff function given by

h(S 1) =

983163 1 if S 1 ge K

0 otherwise

Exercise 5 Prove that the condition d lt 1 + r lt u implies that there is noarbitrage in the elementary single-period market model

Exercise 6 (MATH3965) Under the assumptions of Section 21 show thatthere exists a random variable Z such that the price x of a claim h(S 1) canbe computed using the formula

x = EP(Zh(S 1))

where the expectation is taken under the original probability measure P Arandom variable Z is then called a pricing kernel (note that it does notdepend on h)

Hint You may use the fact that the probability measures P and P are equi-valent

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