Problem 1
What view about the market is reflected in each of the following strategies? (a) Bullish vertical spread: Buy one European call and sell a second one with the same expiry date, but a larger strike price. (b) Bearish vertical spread: Buy one European call and sell a second one with the same expiry date but a smaller strike price. (c) Strip: Buy one European call and two European puts with the same exercise date and strike price. (d) Strap: Buy two European calls and one European put with the same exercise date and strike price. (e) Strangle. Buy a European call and a European put with the same expiry date but different strike prices (consider all possible cases).
Problem 2
A butterfly spread represents the complementary bet to the straddle. It has the following payoff at expiry: Find a portfolio consisting of European calls and puts, all with the same expiry date, that has this payoff.
Problem 3
Suppose that the price of a certain asset has the lognormal distribution. That is \(\log \left(S_{T} / S_{0}\right)\) is normally distributed with mean \(v\) and variance \(\sigma^{2} .\) Calculate \(\mathbb{E}\left[S_{T}\right] .\)
Problem 5
Suppose that at current exchange rates, \(£ 100\) is worth \(£ 160\). A speculator believes that by the end of the year there is a probability of \(1 / 2\) that the pound will have fallen to \(\in 1.40\), and a \(1 / 2\) chance that it will have gained to be worth \(\epsilon 2.00\). He therefore buys a European put option that will give him the right (but not the obligation) to sell \(£ 100\) for \(\in 1.80\) at the end of the year. He pays \(\in 20\) for this option. Assume that the risk-free interest rate is zero across the Euro-zone. Using a single period binary model, either construct a strategy whereby one party is certain to make a profit or prove that this is the fair price.
Problem 8
Put-call parity: Denote by \(C_{t}\) and \(P_{t}\) respectively the prices at time \(t\) of a European call and a European put option, each with maturity \(T\) and strike \(K\). Assume that the risk-free rate of interest is constant, \(r\), and that there is no arbitrage in the market. Show that for each \(t \leq T\), $$ C_{t}-P_{t}=S_{t}-K e^{-r(T-t)} $$.
Problem 12
Suppose that the value of a certain stock at time \(T\) is a random variable with distribution \(\mathbb{P}\). Note we are not assuming a binary model. An option written on this stock has payoff \(C\) at time \(T\). Consider a portfolio consisting of \(\phi\) units of the underlying and \(\psi\) units of bond, held until time \(T\), and write \(V_{0}\) for its value at time zero. Assuming that interest rates are zero, show that the extra cash required by the holder of this portfolio to meet the claim \(C\) at time \(T\) is $$ \Psi \triangleq C-V_{0}-\phi\left(S_{T}-S_{0}\right) $$ Find expressions for the values of \(V_{0}\) and \(\phi\) (in terms of \(\mathbb{E}\left[S_{T}\right], \mathbb{E}[C], \operatorname{var}\left[S_{T}\right]\) and \(\left.\operatorname{cov}\left(S_{T}, C\right)\right)\) that minimise $$ \mathbb{E}\left[\Psi^{2}\right] $$ and check that for these values \(\mathbb{E}[\Psi]=0\) Prove that for a binary model, any claim \(C\) depends linearly on \(S_{T}-S_{0} .\) Deduce that in this case we can find \(V_{0}\) and \(\phi\) such that \(\Psi=0\). When the model is not complete, the parameters that minimise \(\mathbb{E}\left[\Psi^{2}\right]\) correspond to finding the best linear approximation to \(C\) (based on \(S_{T}-S_{0}\) ). The corresponding value of the expectation is a measure of the intrinsic risk in the option.
