© LSE ST2021/FM441 Page 1 of 5 Summer Assessment 2021 Assessment paper and instructions to candidates: FM441 – Derivatives Suitable for all candidates Instructions to candidates This paper contains four questions. Answer all four questions. All questions will be given equal weight (25 marks). If at any point in this examination you feel that anything is unclear, please make any additional assumptions that you feel are necessary, state them clearly, and proceed with your answers. Precision, clarity, and legibility will all be valued and rewarded in grading. Specify the question numbers that you answered in the box provided on the coversheet for submission. Instructions on how to scan and upload your answers can be found on the course Moodle page. It is your responsibility to ensure that your answers are legible, including the scanned versions of your answers. Time Allowed: You have a 24-hour window to complete and submit your assignment. The exam was written with the intention that it should take 2 hours for students to complete. We recommend that you spend 15 minutes reading the exam paper before you begin writing your answers. Permitted materials: This is an open-book, open-notes exam. Calculators: Calculators are permitted in this assessment. 1. (a) Consider a two-date economy with three states of the world and two assets. The assets are traded at date 0 and pay off at date T . The payoff matrix is D =   1 −2 −1 3 1 0   . The (i, j)’th entry of D is the payoff of the j’th asset in the i’th state. The price of asset 1 is 0 and the price of asset 2 is 1. i. [5 marks] Is there an arbitrage? If there is no arbitrage, explain why. If there is an arbitrage, show how it can be exploited. ii. [4 marks] In addition to the information about the two assets given above, suppose the riskfree interest rate is zero. Is there an arbitrage? If there is no arbitrage, explain why. If there is an arbitrage, show how it can be exploited. (b) Suppose there are two trading dates, t = 0 and t = T . The date 0 price of a stock is S0. Between dates 0 and T , the stock price can go up by 10%, down by 10%, or stay the same, with equal probability. There is also an asset-or-nothing call option on the stock expiring at T , with strike price 0.95S0. The date 0 price of this option is V0. The riskfree interest rate is zero. i. [8 marks] Calculate V0 as a function of S0. If you cannot price the option exactly (as a function of S0), explain why, and calculate the range of prices that are consistent with no arbitrage. ii. [8 marks] Now suppose you know that V0 = 0.6S0. Consider an at-the-money European call option on the stock expiring at date T . Let C0 be the date 0 price of this option. Calculate C0 as a function of S0. If you cannot price the option exactly (as a function of S0), explain why, and calculate the range of prices that are consistent with no arbitrage. c© LSE ST 2021/FM441 Page 2 of 5 2. (a) [8 marks] Show that the Black-Scholes formula for the price of a call option, C(S, t), satisfies the following condition: C(S, t) ≥ [S − e−r(T−t)K]+ . In order to answer this question, use the Black-Scholes formula. Do not use put-call parity, or a general argument (not specific to the Black-Scholes setting) that establishes no-arbitrage bounds for option prices. (b) [7 marks] Under Black-Scholes assumptions, the price of an asset-or-nothing digital call option with strike price K and maturity date T is given by CA(S, t) = SΦ(d1), where Φ(·) is the standard normal cdf, and d1 = log(S/K) + (r + σ2/2)(T − t) σ √ T − t . Calculate the delta of this option, δA. What is the limit of at-the-money δA as t→ T ? (c) [10 marks] Suppose the instantaneous riskfree interest rate follows a stochastic process. Consider a European call and put on a non-dividend paying underlying with the same maturity date T and strike price K. For t ≤ T , let Ct and Pt be the time t prices of these options, F (t, T ) the time t forward price for delivery at T , and B(t, T ) the time t price of a zero-coupon bond with face value 1 maturing at T . Derive a no-arbitrage relationship between i. Ct, Pt, St and B(t, T ); ii. Ct, Pt, F (t, T ) and B(t, T ). c© LSE ST 2021/FM441 Page 3 of 5 3. Consider two European call options on the same underlying, with the same strike price K, and expiration dates T1 and T2 (T1 < T2). A calendar spread strategy involves taking a short position in the short-dated option and a long position in the long-dated option. Suppose a calendar spread is entered into at date 0 and S0 = K. (a) [10 marks] Represent graphically the value of the spread at date T1 as a function of ST1 . What is the limit of the spread value at date T1 as ST1 goes to infinity? (b) [5 marks] How does the value of the calendar spread depend on realized volatility between dates 0 and T1? How does it depend on implied volatility at date T1? (c) [5 marks] For this question, suppose that Black-Scholes assumptions are satisfied. Under Black- Scholes, the absolute value of the theta of an at-the-money call increases with time-to- expiration. Analyze how the value of the spread, between dates 0 and T1, is affected by the passage of time, assuming that the options remain at the money. (d) [5 marks] For this question, suppose that Black-Scholes assumptions are satisfied. Under Black- Scholes, vega is increasing in time-to-expiration. Analyze how the value of the spread, between dates 0 and T1, is affected by a change in the level of the volatility parameter σ. c© LSE ST 2021/FM441 Page 4 of 5 4. (a) In this question all interest rates are quoted as annual rates with continuous compound- ing, and all bonds have a face value of £100. The current one-year rate is 7% and the current price of a three-year zero-coupon bond is £78. One year from now, the one-year interest rate is either 10% with probability 1/4, or 4% with probability 3/4, and remains fixed thereafter. Do all your calculations to at least two decimal places. i. [3 marks] Draw a tree representing the evolution of the one-year rate over the next two years and the evolution of the price of the three-year zero-coupon bond. ii. [7 marks] Price a 3-year callable zero-coupon bond with face value £100, that gives the issuer the option to buy back the bond in one year for £85. Without calculating a repli- cating strategy, determine the no-arbitrage price of this bond. How does this price compare to the price of the three-year zero? Interpret. iii. [6 marks] Now price the callable bond in part (ii) above via replication. (b) Consider a swap, a cap and a floor with the same characteristics (i.e. the same floating interest rate, strike (or fixed) interest rate, notional, and payment dates). The fixed interest rate is exogenously given; it is not necessarily equal to the fair swap rate. i. [4 marks] Derive a no-arbitrage relationship between the values of these three financial instru- ments. ii. [1 mark] Comment on the case where the fixed interest rate is equal to the fair swap rate. iii. [4 marks] How do the relative values of the swap, cap and floor change if the yield curve shifts upwards? Explain your answer. c© LSE ST 2021/FM441 Page 5 of 5



















































































































































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