Stochastic Processes and Financial Mathematics
(part one)
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
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5.1 Suppose that we hold the portfolio \((1,3)\) at time \(0\). What is the value of this portfolio at time \(1\)?
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5.2 Find portfolios that replicate the following contingent claims.
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(a) \(\Phi (S_1)=1\)
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(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 \(0\).
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5.3 Find the contingent claims \(\Phi (S_1)\) for the following derivatives.
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(a) A contract in which the holder promises to buy two units of stock at time \(t=1\), each for strike price \(K\).
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(b) A European put option with strike price \(K\in (sd,su)\) (see Section 5.3).
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(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).
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(d) Holding both the contracts in (b) and (c) at once.
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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 \(t=0,1\).
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(a) Write down formulae for \(\Pi ^{call}_0\) and \(\Pi ^{put}_0\).
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(b) Show that \(\Pi ^{call}_0-\Pi ^{put}_0=s-\frac {K}{1+r}\).
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On the binomial model
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5.5 Write down the contingent claim of a European call option (that matures at time \(T\)).
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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?
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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 \(0\).
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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.
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5.9 Recall that \((S_t)_{t=1}^T\) is the price of a single unit of stock.
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(a) Find a condition on \(p_u,p_d,u,d\) that is equivalent to saying that \(S_t\) is a martingale under \(\P \).
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(b) When is \(M_t=\log S_t\) is a martingale under \(\P \)?
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