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Module Code MAT00015H BA, BSc and MMath Examinations 2019/20 Department: Mathematics Title of Exam: Mathematical Finance I Time Allowed: 2 hours Allocation of Marks: Question 1 carries 30 marks. Question 2 carries 35 marks. The marking scheme shown on each question is indicative only. Instructions for Candidates: Answer all questions. Please write your answers in ink; pencil is acceptable for graphs and diagrams. Do not use red ink. Materials Supplied: Green booklet Calculator Do not write on this booklet before the exam begins. Do not turn over this page until instructed to do so by an invigilator. Page 1 (of 4) MAT00015H 1 (of 3). (a) Consider a market consisting of two risky securities with expected returns and covariances between the returns given respectively by the matrices m = ( μ1 μ2 ) = ( 2 3 ) and C = ( 2 1 1 4 ) . (i) Consider a portfolio V with weights w = (w1, w2) = (0.4, 0.6). Compute the expected return of V and the standard deviation of return of V . [6] (ii) Find the minimum variance portfolio. [6] (iii) Let Vˉ be a portfolio on the efficient frontier. If wˉ denotes the weight vector of Vˉ , and C and m are as above, find γ such that γ wˉC = m μu with μ = 0.5 and u = (1, 1). [4] (b) Suppose a market consists of n risky securities S1, . . . , Sn with expected returns μ1, . . . , μn and standard deviations of returns σ1, . . . , σn, respectively. Let C be the covariance matrix of returns. Assume also that the risk-free return is R > 0. (i) Explain what is meant by the market portfolio in the Capital Asset Pricing Model. [3] (ii) Write down the equation for the Capital Market Line (CML) under the as- sumptions of the Capital Asset Pricing Model. [3] (iii) Suppose that portfolios V1 and V2 have expected returns μV1 = 8 and μV2 = 5, and standard deviations of returns σV1 = 2 and σV2 = 1, respectively. If V1 and V2 are both on the CML, find the risk-free return R. [8] Page 2 (of 4) MAT00015H 2 (of 3). (a) Let the market consist of three securities S1, S2, S3. Suppose the correspond- ing vector of expected returns is m = (2, 1, 1). Let C be the covariance matrix of returns. Suppose C and C 1 are respectively given by C = 1 20 15 5 5 5 7 3 5 3 7 and C 1 = 2 1 11 4 1 1 1 4 . Assume also that the risk-free return is R = 0.4. (i) Compute the weights of the market portfolio. [10] (ii) Compute the expected return and the standard deviation of the return on the market portfolio. [8] (iii) Compute the beta factor and the risk premium for a portfolio with weights w1 = 0.7, w2 = 0.1, w3 = 0.4. [5] (b) Consider a market which consists of a stock with price S(t), for t = 0, 1, 2, . . . , T , and a risk-free asset A(t) with constant one-step return R > 0. (i) State the no-arbitrage principle. [4] (ii) Suppose we have a forward contract written on a stock S with initial value S(0) and terminal value S(T ). Suppose also that we have a risk- free asset with initial value A(0) and terminal value A(T ). Let F (0, T ) be the forward price. Show that if F (0, T ) < A(T ) A(0) S(0) then there is an arbitrage opportunity. [8] Page 3 (of 4) Turn over MAT00015H 3 (of 3). (a) Explain what is meant by a European call option with the strike price X and expiry time T written on a stock S. What is the payoff of a European call option [5] (b) If CE(0) denote the initial price of a call option written on stock with strike price X and expiry time T . Assume also that the risk-free rate is R > 0. Show that max { 0, S(0) X (1 +R)T } ≤ CE(0) . [10] (c) Let CE(0) and PE(0) be the initial prices of a European call and put option, respectively, written on a stock S with strike price X and expiry time T = 1. Assume also that the risk-free rate is R > 0. Show that there is an arbitrage opportunity if CE(0) PE(0) > S(0) X (1 +R) . [20] Page 4 (of 4) End of examination. SOLUTIONS: MAT00015H 1. (a) (i) The expected return of V is given by μV = wm T = ( 0.4 0.6 )(2 3 ) = 0.4× 2 + 0.6× 3 = 2.6 . The variance of return of V is given by σ2V = wCw T = ( 0.4 0.6 )(2 1 1 4 )( 0.4 0.6 ) = ( 0.4 0.6 )(1.4 2.8 ) = 2.24 . So the standard deviation is σV = √ 2.24 ≈ 1.497. 6 Marks (ii) The weights of the minimum variance portfolio is given by w = uC 1 uC 1uT , where u = (1, 1). Note that if C = ( 2 1 1 4 ) then C 1 = 1 det(C) ( 4 1 1 2 ) = 1 7 ( 4 1 1 2 ) . Therefore the minimum variance portfolio has weights w = 1 7 ( 1 1 )( 4 1 1 2 ) 1 7 ( 1 1 )( 4 1 1 2 )( 1 1 ) = ( 3 1 ) ( 3 1 )(1 1 ) = (3/4 1/4) = (0.75 0.25) . 6 Marks (iii) Multiplying the equation γ wˉC = m μu on the right by C 1 gives γ wˉ = (m μu)C 1 . Now multiplying on the right by uT and noting that wˉuT = 1 we obtain γ = (m μu)C 1uT . 5 SOLUTIONS: MAT00015H Hence γ = 1 7 (( 2 3 ) 0.5 (1 1))( 4 1 1 2 )( 1 1 ) = 1 7 ( 1.5 2.5 )( 4 1 1 2 )( 1 1 ) = 1 7 ( 1.5 2.5 )(3 1 ) = 7 7 = 1 . 4 Marks (b) (i) The Market Portfolio is a portfolio which consists of all risky securities with weights equal to their relative share in the whole market. Alternative definition: The Market Portfolio is the tangency point of the efficient frontier and the capital market line. (A diagram showing the tangency point in the (σ, μ)-plane is also an acceptable answer.) 3 Marks (ii) The equation for the Capital Market Line is μV = R + μM R σM σV , where μV is the expected return of the portfolio V , σV is the standard deviation of return on the portfolio V , μM is the expected return of the market portfolio, and σM is the standard deviation of return on the market portfolio. 3 Marks (iii) We have μV1 = R + μM R σM σV1 , and μV2 = R + μM R σM σV2 . From the above we obtain μV1 R σV1 = μM R σM , and μV2 R σV2 = μM R σM . Therefore μV1 R σV1 = μV2 R σV2 , which gives R = μV2σV1 μV1σV2 σV1 σV2 = 5× 2 8× 1 2 1 = 2 . 6 SOLUTIONS: MAT00015H 8 Marks Total: 30 Marks 2. (a) (i) We have m = (2, 1, 1), R = 0.4 and let u = (1, 1, 1). The weights of the market portfolio are given by wM = (m Ru)C 1 (m Ru)C 1uT = (( 2 1 1) (0.4 0.4 0.4)) 2 1 11 4 1 1 1 4 (( 2 1 1) (0.4 0.4 0.4)) 2 1 11 4 1 1 1 4 11 1 = ( 1.6 0.6 1.4) 2 1 11 4 1 1 1 4 ( 1.6 0.6 1.4) 2 1 11 4 1 1 1 4 11 1 = ( 5.2 2.6 6.6) ( 5.2 2.6 6.6) 11 1 = 1 1.2 ( 5.2 2.6 6.6) = ( 13/3 13/6 11/2) ≈ (4.333 2.167 5.5) . 10 Marks (a) (ii) The expected return of the market portfolio which has weights wM = (4.333, 2.167, 5.5) is given by μM = wM m T = ( 4.333 2.167 5.5) 21 1 = 49/3 ≈ 16.333 . The variance of the return on the market portfolio is given by σ2M = wMCw T M = 1 20 ( 4.333 2.167 5.5) 15 5 5 5 7 3 5 3 7 4.3332.167 5.5 = 1 20 ( 26.66 10.004 23.336) 4.3332.167 5.5 ≈ 265.5444 20 ≈ 13.277 . 7 SOLUTIONS: MAT00015H So the standard deviation of return on the market portfolio is σM = √ 13.277 ≈ 3.6438. 8 Marks (a) (iii) The expected return for the portfolio V with weightswV = (0.7, 0.1, 0.4) is given by μV = wV m T = ( 0.7 0.1 0.4) 21 1 = 0.9 . The Capital Asset Pricing Model states that μV = R + β(μM R) and this gives the beta factor β = μV R μM R = 0.9 0.4 16.333 0.4 = 0.5 15.933 ≈ 0.0314 . The risk premium is given by μV R = 0.9 0.4 = 0.5. 5 Marks (b) (i) There is no admissible strategy such that the value of the portfolio at time t = 0 is V (0) = 0 and the future value V (t) satisfies: V (t) ≥ 0 with probability 1 and V (t) > 0 with positive probability for some t > 0. 4 Marks (b) (ii) We consider the following strategy ( 1, S(0) A(0) , 1) i.e., borrow one risky security and with it to buy S(0)/A(0) risk free securities, and take one long forward contract (to buy a risky security). Then the wealth at time t = 0 is S(0) + S(0) A(0) A(0) = 0 and the wealth at time t = T is W (T ) = S(T ) + S(0) A(0) A(T ) + (S(T ) F (0, T )) = S(0) A(0) A(T ) F (0, T ) > 0 so we have an arbitrage opportunity. 8 Marks Total: 35 Marks 8 SOLUTIONS: MAT00015H 3. (a) A European call option is a contract which gives the holder the right to buy an asset for a price X fixed in advance (the exercise or strike price) at a specified time T (called the exercise or expiry time). The payoff for a European call option is CE(T ) = (S(T ) X)+ , where S(T ) is the price of stock at time T and x+ = max{0, x}. 5 Marks (b) Note that the contract is worth something so CE ≥ 0. Hence we need to show that CE ≥ S(0) X (1 +R)T . Recall from the put-call parity that CE PE = S(0) X (1 +R)T , which implies that CE ≥ CE PE = S(0) X (1 +R)T and then (since PE(0) ≥ 0) CE ≥ max{0, S(0) X (1 +R)T } . This proves the desired inequality. 10 Marks (c) Consider the following strategy. At time t = 0, we: buy 1 share, sell a European call option, buy one European put option. This costs us PE(0) CE(0)+S(0) so we do this by borrowing PE(0) CE(0)+ S(0) or equivalently investing CE(0) S(0) PE(0). At time T = 1 we have (CE(0) S(0) PE(0))(1 +R) + S(1) and our contracts. (1) If S(1) < X we do not exercise the EC but the EP option will be exercised giving X S(1) and wealth (CE(0) S(0) PE(0))(1 +R) + S(1) +X S(1) = (CE(0) S(0) PE(0))(1 +R) +X . 9 SOLUTIONS: MAT00015H (2) If S(1) > X then the EC is exercised to give (S(1) X). The put option will not be exercised. Again our wealth is (CE(0) S(0) PE(0))(1 +R) + S(1) (S(1) X) = (CE(0) S(0) PE(0))(1 +R) +X . By the assumption we have CE(0) PE(0) > S(0) X (1 +R) , which is equivalent to (CE(0) S(0) PE(0))(1 +R) +X > 0 . So (1) and (2) above show that this strategy gives an arbitrage opportunity. 20 Marks Total: 35 Marks 10