last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part one)

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5.7 Exercises on Chapter 5

All questions use the notation \(u,d,p_u,p_d,s\) and \(r\), which has been used throughout this chapter. In all questions we assume that the models are arbitrage free and complete: \(d<1+r<u\).

On the one-period model
  • 5.1 Suppose that we hold the portfolio \((1,3)\) at time \(0\). What is the value of this portfolio at time \(1\)?

  • 5.2 Find portfolios that replicate the following contingent claims.

    • (a) \(\Phi (S_1)=1\)

    • (b) \(\Phi (S_1)=\begin {cases} 3 & \text { if }S_1=su,\\ 1 & \text { if }S_1=sd. \end {cases}\).

    Hence, write down the values of these contingent claims at time \(0\).

  • 5.3 Find the contingent claims \(\Phi (S_1)\) for the following derivatives.

    • (a) A contract in which the holder promises to buy two units of stock at time \(t=1\), each for strike price \(K\).

    • (b) A European put option with strike price \(K\in (sd,su)\) (see Section 5.3).

    • (c) A contract in which we promise that, if \(S_1=su\), we will sell one unit of stock at time \(t=1\) for strike price \(K\in (sd,su)\) (and otherwise, if \(S_1=sd\) we do nothing).

    • (d) Holding both the contracts in (b) and (c) at once.

  • 5.4 Let \(\Pi ^{call}_t\) and \(\Pi ^{put}_t\) be the price of European call and put options, both with the same strike price \(K\in (sd,su)\), at times \(t=0,1\).

    • (a) Write down formulae for \(\Pi ^{call}_0\) and \(\Pi ^{put}_0\).

    • (b) Show that \(\Pi ^{call}_0-\Pi ^{put}_0=s-\frac {K}{1+r}\).

On the binomial model
  • 5.5 Write down the contingent claim of a European call option (that matures at time \(T\)).

  • 5.6 Let \(T=2\) and let the initial value of a single unit of stock be \(S_0=100\). Suppose that \(p_u=0.25\) and \(p_d=0.75\), that \(u=2.0\) and \(d=0.5\), and that \(r=0.25\). Draw out, in a tree-like diagram, the possible values of the stock price at times \(t=0,1,2\). Find the price, at time \(0\), of a European put option with strike price \(K=100\).

    Suppose instead that \(p_u=0.1\) and \(p_d=0.9\). Does this change the value of our put option?

  • 5.7 Let \(T=2\) and let the initial value of a single unit of stock be \(S_0=120\). Suppose that \(p_u=0.5\) and \(p_d=0.5\), that \(u=1.5\) and \(d=0.5\), and that \(r=0.0\). Draw out, in a tree-like diagram, the possible values of the stock price at times \(t=0,1,2\). Annotate your tree to show a hedging strategy for a European call option with strike price \(K=60\). Hence, write down the value of this option at time \(0\).

  • 5.8 Let \(T=2\) and let the initial value of a single unit of stock be \(S_0=480\). Suppose that \(p_u=0.5\) and \(p_d=0.5\), that \(u=1.5\) and \(d=0.75\), and that \(r=0\). Draw out, in a tree-like diagram, the possible values of the stock price at times \(t=0,1,2\). Annotate your tree to show a hedging strategy for a European call option with strike price \(K=60\). Hence, write down the value of this option at time \(0\).

    Comment on the values obtained for the hedging portfolios.

  • 5.9 Recall that \((S_t)_{t=1}^T\) is the price of a single unit of stock.

    • (a) Find a condition on \(p_u,p_d,u,d\) that is equivalent to saying that \(S_t\) is a martingale under \(\P \).

    • (b) When is \(M_t=\log S_t\) is a martingale under \(\P \)?

Challenge questions
  • 5.10 Write a computer program (in a language of your choice) that carries out the pricing algorithm for the binomial model, for a general number \(n\) of time-steps.