Chapter 5 Questions for Chapter 5
For Exercises 5.1–5.5 you should use the 3-period binomial model with \(S_0 = 10\), \(u=2,d=1/2\) and \(r=0.01\). Remember that for these exotic options you need to calculate the payoff for all possible paths through the tree (i.e., all 8 states of \(\Omega_3\)).
Warm-up
Exercise 5.1 On the market defined above, what are the fair prices at times 0, 1, 2, 3 of:
- a lookback call option, with payoff \(X=S_3 - \min\{S_0,S_1,S_2,S_3\}\),
- a lookback put option, with payoff \(X=\max\{S_0,S_1,S_2,S_3\} - S_3\)?
Exercise 5.2 On the market defined above, find the fair price at time 0 of an up-and-out put option, with strike price \(K=30\), barrier \(L=30\) and expiry date \(T=3\).
Exercise 5.3 On the market defined above, find the fair price of the Asian put option whose payoff is \[ P_T^{\rm Asian} = \Big( 15 - \frac{S_1 + S_3}{2} \Big)^+ \]
Main problems
Exercise 5.4 On the market defined above, calculate the no-arbitrage price at time 0 of:
- a down-and-out call option, with strike price \(K=10\), barrier \(L=8\) and expiry date \(T=3\).
- a down-and-in call option, with the same strike price, barrier and expiry date.
What is the price of a standard European call option with strike price \(K=10\) and expiry date \(T=3\)?
Exercise 5.5 On the market defined above, find the price at time 0 of the Asian call option whose payoff \(C_T^{\rm Asian}\) is \[ C_T^{\rm Asian} = \Big( \frac{S_0+S_1+S_2+S_3}{4} - 10\Big)^+. \]
Exercise 5.6 A stock price evolves on a binomial tree with prices \(S_0 = 100\), \(S_1 \in \{120, 80\}\), \(S_2 \in \{140, 100, 60\}\) and \(S_3 \in \{160, 120, 80, 40\}\). The tree is recombining, so that both states \(S_2 = 140\) and \(S_2 = 100\) have branches to state \(S_3 = 120\) and so on. Draw the tree and compute risk-neutral probabilities for it when the interest rate is \(r = 0.1\) per period. (Be aware that these probabilities can be different at different nodes.)
Calculate the no arbitrage price at time 0 of an American put on this stock with strike price \(K = 90\) that expires at \(T = 3\). Identify the states where it is optimal to exercise the option before expiry. Compare this price with that of the European put with the same parameters \((K, T)\).
Use put-call parity to find the price of the European call with the same \((K, T)\) on this stock.
Calculate the no arbitrage price at time 0 of an American call with the same \((K,T)\) and confirm that (i) there are no states where it is better to exercise before expiry than wait, and (ii) the price is the same as the price of the European call with the same \((K,T)\).