Assignment 1

Exercise 1.11 Suppose in the following portfolios all options are based on the same stock, have expiry date \(T\) and strike price \(K\) (unless otherwise stated). In each case find an expression for the payoff of the portfolio at time \(T\) in terms of \(S_T\), \(K\), and draw the graph of the payoff as a function of \(S_T\):

  1. one call option and one put option;
  2. two call options and one share sold (i.e. short);
  3. one share, short one call option;
  4. one \((K_1, T)\) call option, short one \((K_2, T)\) put option.

Exercise 1.12 Suppose that interest is compounded continuously at rate \(r\). Show that if the market contains a risk-free asset with price \(B_t\) for \(t \in [0,T]\), then this price must satisfy \(B_t = B_0 \e^{rt}\) for all \(t \in [0,T]\) or else there is arbitrage.

Exercise 1.13 Let \(K_1 < K_2\). A bull spread can either be created by

  • buying a \((K_1, T)\) call option and selling a \((K_2,T)\) call option, or
  • buying a \((K_1, T)\) put option and selling a \((K_2,T)\) put option.
  1. Which of these strategies involves a positive initial investment?
  2. Use put-call parity to relate the initial investment for a bull spread created using call options to the initial investment for a bull spread created using put options.

Exercise 1.14 Over the next few weeks, we will be working with \(\sigma\)-algebras in the context of the binomial model. Go to the prerequisite notes and copy out one of the criteria for a set \(\mathcal{F}\) to be a \(\sigma\)-algebra.