Consider a one-step market model with two time instants \(t=0\) and \(t=1\) and two assets:

- a riskless asset \(\pi\) with price \(\pi_{0}\) at time \(t=0\) and value \(\pi_{1}=\pi_{0}(1+r)\) at time \(t=1\),

- a risky asset \(S\) with price \(S_{0}\) at time \(t=0\) and random value \(S_{1}\) at time \(t=1\).

We assume that \(S_{1}\) can take only the values \(S_{0}(1+a)\) and \(S_{0}(1+b)\), where \(-1

a) What are the possible values of \(R\) ?

b) Show that under the probability measure \(\mathbb{P}^{*}\) defined by

the expected return \(\mathbb{E}^{*}[R]\) of \(S\) is equal to the return \(r\) of the riskless asset.

c) Does there exist arbitrage opportunities in this model? Explain why.

d) Is this market model complete? Explain why.

e) Consider a contingent claim with payoff \(C\) given by

Show that the portfolio allocation \((\eta, \xi)\) defined* by

hedges the contingent claim with payoff \(C\), i.e. show that at time \(t=1\), we have

Hint: Distinguish two cases \(R=a\) and \(R=b\).

f) Compute the arbitrage-free price \(\pi_{0}(C)\) of the contingent claim payoff \(C\) using \(\eta, \pi_{0}, \xi\), and \(S_{0}\).
g) Compute \(\mathbb{E}^{*}[C]\) in terms of \(a,b, r, \xi, \eta\).
h) Show that the arbitrage-free price \(\pi_{0}(C)\) of the contingent claim with payoff \(C\) satisfies

i) What is the interpretation of Relation (1.23) above?
j) Let \(C\) denote the payoff at time \(t=1\) of a put option with strike price \(\mathrm{K}=\$ 11\) on the risky asset. Give the expression of \(C\) as a function of \(S_{1}\) and \(K\).
k) Letting \(\pi_{0}=S_{0}=1, r=5 \%\) and \(a=8, b=11, \xi=2, \eta=0\), compute the portfolio allocation \((\xi, \eta)\) hedging the contingent claim with payoff \(C\).
l) Compute the arbitrage-free price \(\pi_{0}(C)\) of the claim payoff \(C\).

Transcribed Image Text:

R= S1 - So So