7CCMFM07 King’s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. FOLLOW the instructions you have been given on how to upload your solutions MSc Examination 7CCMFM07 Interest Rate and Credit Risk (Mock Exam) Summer 2021 Time Allowed: Two Hours All questions carry equal marks. Full marks will be awarded for complete answers to FOUR questions. If more than four questions are attempted, then only the best FOUR will count. You may consult lecture notes and use a calculator. 2021 King’s College London 7CCMFM07 1. Assume that the zero coupon bond term structure is given by P (0, T ) = exp(kT 2), k > 0. a. Compute the LIBOR spot rate L(0, T ) as a function of T . [15%] b. Compute the LIBOR forward rate F (0, T, T+1) as a function of T . [15%] c. Compute the swap rate S(0, 0, 2) = S0,2(0) for a swap with annual tenor in both legs. [30%] d. Find an explicit expression for the continuously compounded spot rate R(0, T ) at time t for maturity T defined as R(t, T ) = 1 T t lnP (t, T ) when the bond has the structure given above. [20%] e. Using part d. find the short rate r0 at time 0. [20%] 2. Consider the following curve of zero coupon bonds for the maturities T0 = 1y, T1 = 2y, . . . , T4 = 5y: P (0, T0) = 0.961538462, P (0, T1) = 0.924556213, P (0, T2) = 0.888996359, P (0, T3) = 0.854804191, P (0, T4) = 0.821927107. a. Compute the forward swap rates S1,4(0) and S2,4(0). [40%] b. Compute the forward libor rates F1(0), F2(0), ... and verify that the swap rate S2,4(0) is a weighted average of the forward LIBOR rates (in partic- ular, compute the weights). [60%] - 2 - See Next Page 7CCMFM07 3. Consider the Black-Derman-Toy short rate model: rt = r0e µ(t)+(t)Wt whereW is a Brownian motion underQ. Assume one unit of currency is invested in the bank account at time 0. a. Show that the expectation of the investment at time t > 0 can be approximated as: E(Bt) ⇡ E(et r0+rt 2 ). [35%] b. Prove that the bank account explodes: E(Bt) = 1. (This is true for arbitrarily small investment time and any lognormal short rate model.) Discuss the result. [35%] c. Give another example of a short rate model where the bank account would exhibit this behaviour. [30%] (You may use that all moments of the lognormal distribution exist, but the moment generating function does not, i.e. if X = exp(Y ) is lognormal with Y ⇠ N(µ, 2), then E(Xn) exists 8n, but E(etX) =1.) - 3 - See Next Page 7CCMFM07 4. Consider the short rate process rt = xt + yt + '(t), where dxt = axtdt+ dW 1t , x(0) = 0, dyt = bytdt+ ⌘dW 2t , y(0) = 0, dW 1t dW 2 t = ⇢dt with r0, a, b, , ⌘ positive constants, ' deterministic function and 1  ⇢  1. a. What is the name of this model and what is its main advantage? [30%] b. Using the fact thatZ T t xudu = 1 ea(Tt) a xt + a Z T t (1 ea(Tu))dW 1u prove that the random variable I(t, T ) := R T t (xu + yu)du conditional on Ft is normally distributed with mean M(t, T ) := 1 ea(Tt) a xt + 1 eb(Tt) b yt [35%] and variance V (t, T ) : = 2 a2 (T t+ 2 a ea(Tt) 1 2a e2a(Tt) 3 2a ) + ⌘2 b2 (T t+ 2 b eb(Tt) 1 2b e2b(Tt) 3 2b ) +2⇢ ⌘ ab ⇣ T t + e a(Tt) 1 a + eb(Tt) 1 b e (a+b)(Tt) 1 a+ b ⌘ . [35%] - 4 - See Next Page 7CCMFM07 5. a. Explain how to obtain the default time ⌧ in an intensity model where we are given a value ⇠0 of an exponential random variable with mean 1 and independent of any default intensity t. [20%] b. Compute the price at time 0 of a defaultable zero coupon bond with zero recovery in terms of the default-free short rate interest rate (rt)t and the default intensity (t)t. [30%] c. Explain how a Credit Default Swap (CDS) works. [20%] d. Assume that calibration to CDS gives the following piecewise constant hazard rate: (t) = 8<: 0.02, 0  t < 1y 0.04, 1y  t < 2y 0.02, 2y  t < 3y Compute the probability of survival in three years and use it to compute the price of the protection leg of a CDS that sells protection from today to three years assuming zero interest rate and a recovery of 0.5. [30%] - 5 - Final Page


















































































































































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