Chapter 1 Questions for Chapter 1
Warm-up
Exercise 1.1 What is the effective annual interest rate when the nominal interest rate of 10% is compounded: (a) twice yearly, (b) quarterly, (c) continuously?
Exercise 1.2 You deposit amount \(D\) in a bank which pays interest at nominal rate \(r\). How many years does it take for your money to
- double when interest is compounded continuously and \(r = 0.1\);
- quadruple when interest is compounded annually and \(r = 0.05\)?
Find an expression for the number of years required for your money to increase to \(\alpha D\) when the interest is compounded annually.
Exercise 1.3 Consider European \((K, T)\) call and put options on the same stock with \(K = 10\), \(T = 1/4\) (i.e. they expire in three months time). The current stock price is 11 and the risk free interest rate is 6% (compounded continuously). Identify an arbitrage portfolio when both options have price \(2.5\).
Main problems
Exercise 1.4 You plan to invest amount \(a\) at the start of each of the next 60 months. The annual interest rate is 6% compounded monthly. How big should \(a\) be to ensure your investment has value 100,000 at the end of 60 months?
Exercise 1.5 A five-year bond with a 10% coupon rate costs 10,000 initially, pays its holder 500 at the end of each six month period for five years and then repays the initial 10,000. Find its present value when interest is compounded twice annually at rate: (a) 6%, (b) 10%, (c) 12%.
Repeat the calculation for a similar bond costing 1,000 with coupon rate 6% (so the investor receives 30 every six months plus the initial 1,000 at the end of five years) but now suppose that interest is continuously compounded at rate 5%.
Exercise 1.6 Let \(S\) denote the initial price of a stock and \(C\) the cost of the option to buy it for \(K\) at time \(T\). Suppose that interest is compounded continuously at rate \(r\). Prove, by considering appropriate portfolios, that \[ (S - Ke^{-rT})^+ \leq C \leq S. \]
Find and prove similar bounds for the price \(P(K, T)\) of the European put option for the same stock. Show that for \(K_1 > K_2\), \[ P(K_1, T) - P(K_2, T) \leq (K_1 - K_2)e^{-rT}\ . \]
Exercise 1.7 A double call option can be exercised either at time \(T_1\) with strike price \(K_1\) or at time \(T_2 > T_1\) with strike price \(K_2\). The risk free interest rate is \(r\). Show that it is not optimal to exercise at time \(T_1\) if \(K_1e^{-rT_1} > K_2e^{-rT_2}\).
Exercise 1.8 Let \(P\) be the price of a European \((K, T)\) put option on a stock with initial price \(S\). Which of the following are true?
- \(P \leq S\);
- \(P \leq K\).
Exercise 1.9 Show that \(P(t)\), the price of an American put option with expiry date \(t\), is non-decreasing in \(t\).
Exercise 1.10 Suppose interest is compounded discretely at rate \(r\) per period, and suppose that the market contains a bond (risk-free asset) with price \(B_t\) satisfying \(B_t = (1+r)^t B_0\) for \(t=1,2,\dots,T\).
For any portfolio \(P\) generating a cash flow \((x,t)\) find a “bond-only” portfolio \(P_B\) that generates the same cash flow \((x,t)\).
Using part a. show that for any portfolio \(P\), the portfolio \(\widetilde{P} := P - P_B\) is self-financing.
If \(P\) is also an arbitrage portfolio, in the sense that \(\alpha^T V_T-V_0>0\) whatever happens to the share prices, show further that the value \(\widetilde{V}_t\) of \(\widetilde{P}\) satisfies: (i) \(\widetilde{V}_0 = 0\); and (ii) \(\widetilde{V}_T > 0\) whatever happens to the share prices.