Chapter 2 Questions for Chapter 2
Warm-up
Exercise 2.1 Recall Example 1.1: A stock has current price 100 and its price will be either 200 or 50 at time 1. Now suppose the risk free interest rate is \(r \neq 0\) per period. In other words, suppose that \(B_0 = 1\), and \(B_1 = 1+r\).
- Find the risk neutral probabilities and deduce from them the no-arbitrage
prices of
- a call option with strike price 150 and expiry time 1.
- a put option with strike price 150 and expiry time 1.
- Check that the put and call price satisfy the put-call parity formula.
Hint: Your answer to part (a.i) should match the price we found in Example 1.1 on setting \(r=0\).
Exercise 2.2 Consider a one-period financial market \(\mathcal{M}=(B_t, S_t)\). Assume that the current stock price is \(£ 28\), and after three months the stock price may either rise to \(£ 32\) or decline to \(£ 26\). Assume that the three-months interest rate for deposits and loans is \(r=3\%\). Find the no arbitrage prices of a call option and a put option with the same strike price \(K=£ 28\). Also find the replicating portfolios of both of these call and put options.
Main problems
Exercise 2.3 Consider a financial market with one risk-free asset with price process \(B_0=1, B_T=1.1\) and one risky-asset with price process \(S_0=100\) and \[ S_T=\left \{ \begin{array}{ll} 120& \mbox{with probability $0.5$},\\ 60& \mbox{with probability $0.5$}. \end{array} \right. \]
- Show that this financial market is arbitrage-free.
- Find the replicating portfolio of a call option with strike price \(K=90\). What is the premium that you would be willing to pay to purchase this call option at time \(t=0\)?
- Find the price of a put option with the same strike price \(K=90\).
- Find a risk-neutral measure for this financial market. Is this financial market complete? Justify if it is.
- Find the replicating portfolio for the contingent claim \(X=(S_T)^2\). What is the price of this contingent claim at time \(t=0\)?
Exercise 2.4 Let \(\mathcal{M}=(B_t, S_t^1, S_t^2)\) be a one-period financial market with three assets. In this market a portfolio is specified by a vector \(h=(x, y, z)\in \R^3\), and the value process of \(h\) is given by \(V_t^h=xB_t+yS_t^1+zS_t^2\). Suppose that the asset price \(B_t\) is risk-free with dynamics \(B_0=1, B_T=1.1\); the asset prices \(S_t^1\) and \(S_t^2\) are risky and have the following joint dynamics: \((S_0^1, S_0^2)=(10,100)\) \[ (S^1_T,S^2_T)=\left \{ \begin{array}{ll} (12,130)& \mbox{with probability $0.3$},\\ (7, 60)&\mbox{with probability $0.7$}, \end{array} \right. \] so that the prices of the two risky assets either both increase, or both decrease. As before, arbitrage means a portfolio \(h\) with \(V_0^h = 0\), \(\P(V_T^h \geq 0) = 1\) and \(\P(V_T^h > 0) > 0\). Is this financial market arbitrage-free? If it is not, find an arbitrage portfolio for this financial market.
Exercise 2.5 Suppose that in Example 1.1 / Exercise 2.1 the stock price after one period is one of 50, 100 or 200 (so the price can remain unchanged). Show there is no longer a unique no-arbitrage price for a \((150, 1)\) call option and identify a range of values for its price \(C\) at time 0 which ensure no arbitrage is possible.
Exercise 2.6 Consider a 1-period market \(\mathcal{M} = (B_t,S^1_t, S^2_t)\) with prices satisfying: \(B_0 =1\) and \(B_1 = 1.1\); \((S^1_0, S^2_0) = (10,20)\) and \((S^1_1,S^2_1)\) has the joint distribution: \[\begin{equation*} (S^1_1, S^2_1) = \begin{cases} (15,21) & \text{with probability 0.5}, \\ (10,22.5) & \text{with probability 0.25}, \\ (10,22.2) & \text{with probability 0.25}. \end{cases} \end{equation*}\]
- Prove this market contains no arbitrage.
- Consider the contingent claim \(X = \sqrt{S^1_1}+\sqrt{S^2_1}\). Show that there is a portfolio \(h^\star\) that replicates \(X\). (You need not find \(h^\star\) explicitly but you must justify why such an \(h^\star\) exists.)
- Prove that the arbitrage-free price of \(X\) is the value of the portfolio \(h^\star\) at time 0.
Exercise 2.7 A share price (with initial value \(S_0\)) will be one of the values \(s_1\), \(s_2\), \(\ldots\) \(s_n\) after one period. The risk-free interest rate per period is \(r\). What condition on \(r\), \(S_0\) and the \(s_i\) ensures no arbitrage for this market? And what is the no-arbitrage price of an option to buy a share at time 1 for strike price \(K < \min_i s_i\)?