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Mathematical Finance Mathematical Finance 1. What is the effective annual interest rate when the nominal interest rate of 10% is compounded (a) twice yearly (b) quarterly (c) continuously? 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 (a) double when interest is compounded continuously and r =0.1; (b) 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 αD when the interest is compounded annually. 3. An investment offers returns of 500 at the end of year one and A at the end of year two for an initial payment of 1,000 at the start of year 1. Find the rate of return when the value of A is: (a)300 (b)500 (c)700. 4. You plan to invest amount a at the start of each of the next 60 months. The an- nual 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. 5. The start-of-year cash flows of an investment are −1, 000 −1, 200 800 900 800 Suppose you can borrow or invest cash at an interest rate of 6% compounded annually. Is the investment worthwhile? 6. 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 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%. 7. You have decided to borrow 120, 000 from a bank. There is a fee of 2, 400 (which is deducted from the loan) and interest is calculated monthly at rate 0.5%. You will pay the interest monthly for 36 months and then repay the initial sum of 120, 000. What is the monthly rate of return on this loan? 8. For any non-trivial cash flow stream ci, i = 0, 1, ... m + n with ci = −ai ≤ 0 for i = 0, ... m and cm+i = bi ≥ 0 for i = 1, ... n show there is a unique rate of return r∗ ∈ (−1, ∞). 9. Suppose that interest is compounded annually, the borrowing rate is 0.08 but the investment rate is 0.05. You start with no capital. An investment has cash flows −1000, 900, 800, −1200, 700 at the start of the next five years. Should you invest? Michælmas 2010 Mathematical Finance: Questions 3 10. Suppose the risk-free interest r(t) varies in time. Letr ¯(t) denote average of the spot rate up to time t and P (t)= e−tr¯(t). What is the financial meaning of P (t)? Show that if r(·) is non-decreasing in t then so isr ¯(·). Additionally show thatr ¯(·) is non-decreasing in t if and only if P (αt) ≥ P (t)α for all α ∈ [0, 1], t ≥ 0. 11. Suppose in the following portfolios all options are based on the same stock, have expiry date T and strike price K (unless otherwise stated). In each case find the portfolio value at time T in terms of S(T ), K: (i) one call option and one put option; (ii) two call options and one share sold (i.e. short); (iii) one share, short one call option; (iv) one (K1, T ) call option, short one (K2, T ) put option. 12. A stock has current price 100 and its price will be either 200 or 50 at time 1. The risk free interest rate is r. Find the risk neutral probabilities and deduce from them the no-arbitrage price of a put option with strike price 150 at time 1. Check that the put and call price satisfy the put-call parity formula. 13. A share price will be one of the values s1, s2, ... sn after one period. The risk-free interest rate per period is r. What is the no-arbitrage price of an option to buy a share at time 1 for K < mini si? 14. 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 when both options have price 2.5. 15. 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 )+ ≤ C ≤ S. Find and prove similar bounds for the price P (K, T ) of the European put option for the same stock. Show that for K1 >K2, P (K1, T ) − P (K2, T ) ≤ K1 − K2 . 16. A double call option can be exercised either at time T1 with strike price K1 or at time T2 > T1 with strike price K2. The risk free interest rate is r. Show that it is not −rT1 −rT2 optimal to exercise at time T1 if K1e >K2e . 17. Consider three European (Ki, T ) call options with K3 − K2 = K2 − K1 > 0. Denote their prices by Ci, i = 1, 2, 3. Show that C2 ≤ (C1 + C3)/2 directly i.e. not using Theorem 1.3(a). What is the corresponding inequality for European put options? 18. 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? (a) P ≤ S (b) P ≤ K. 19. Show that P (t), the price of an American put option with expiry date t, is non- decreasing in t..
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