UNIVERSITY COLLEGE LONDON EXAMI:NATION· FOR ··1 NTERNAL STU DENTS MODULECODE COMPG012 ASSESSMENT COMPG012A PATTERN MODULE NAME Financial Engineering DATE Wednesday 9 May 2018 TIME 14:30 TIMEALLOWED 2 hrs 30 'mins This paper is suitable ,for candidates who attended classes for this module in the following academic year(s): Year 2016/17 and 2017/18 .' EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE .2016/17-COMPG012A-001-EXAM-Computer Science 117 © 2016 University College London TURN OVER L Department of Computer Science University College London COMPG012 Financial Engineering MSc Examination 2018 Time allowed: 2.5 hours Throughout this examination Wt or W (t) is a standard Brownian motion. You may assume (where appropriate) the following: P (8, t) = _8e-D (T-t)N (-d1) +Ee-r(T-t)N (-d2) d _ log (8/E) + (r - D ± !o-2) (T - t) 1,2- .~ , o-y.L - t N (x) is the standard Normal Cumulative Distribution Function N (x) _ _ l_jX e-s2 / 2ds .j2; -00 1 -x2/2N ' (x) --e .j2; SDE refers to Stochastic Differential Equation. Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR questions will count towards the total mark. Each question is worth 25 marks. CALCULATORS ARE PERMITTED. COMPG012 PLEASE TURN OVER 1 1. a. [10 Marks] Suppose that X is a normally distributed random variable, where X rv N (j..L, 0-2) . Show that where f is a suitable function and eE 1R is a scalar. Hint: \iVrite X = j..L + o-¢; ¢ rv N (0,1) and calculate the resulting integral. b. [15 Marks] The ordinary differential equation for the function u (8) is to be solved with boundary conditions j..L and 0- are constants. Show that the solution is given by Hint: \iVhen solving for the particular integral, assume a solution of the form Clog 8, where C is a constant. COMPG0l2 CONTINUED 2 2. a. [8 Marks] A spot rate rt, evolves according to the popular form (2.1) where v and /3 are constants. Suppose such a model has a steady state transition probability density functionpoo (r) that sat isfies the forward Fokker Planck Equation. Show that this implies the drift structure of (2.1) is given by u (rt) 2 2(3-1= v /3rt . 1 2 2(3 d+ -v rt - (logpoo) . 2 dr b. [12 Marks] The Ornstein-Uhlenbeck process satisfies the spot rate SDE and initial condition given by where J)" e and CT are constants. Solve this SDE by setting yt = eK.trt and using Ito's lemma to show that c. [5 Marks] Calculate the mean of rt given by (2.2). You may use the result in your calculation where it is a time-dependent function. COMPG012 PLEASE TURN OVER 3 3. a. [9 Marks] An option V (8, t) is to be written on a dividend paying stock 8t , that satisfies the SDE d8t8 = p,dt + adWt . t p, and a are constants. Assume that' the asset receives a con tinuous and constant dividend yield, D, across each time-step. By constructing a hedged portfolio derive the partial differential equation (3.1) for the fair price of an option with r the risk-free interest rate. b. [8 Marks] Separable solutions of (3.1) of the form V (8, t) = A (t) B (8) are sought. By substitution show that (3.1) leads to a first order differential equation in A (t) and a second order differ ential equation in B (8) . You are not required to solve this pair of differential equations. c. [8 Marks] An At-The-Money-Forward (ATMF) option is struck when its' strike price is E = 8e(r-D) (T-t) . What is the Black Scholes pricing formula for an ATfVIF put option? COMPG012 CONTINUED 4 4. a. [11 Marks] Consider the pricing equation for the value of a deriv ative security V (S, t), 8V 1 2 282V 8V fit+2,O"S 8S2 +(r-D)S8S- rV =O, (4.1) where S ~ 0 is the spot price of the underlying equity, 0 < t :S T is the time, r ~ 0 the constant rate of interest, and 0" is the constant volatility of S. The variables (t, S) can be written as t = m8t 0 :S m :S M; S = n5S o:S n:S N, where (8t,5S) are fixed step sizes in turn. V (S, t) is written dis cretely as vnm. An Explicit Finite Difference Method is to be developed to solve (4.1) using a backward marching scheme. 1. Derive a difference equation in the form where an, bn, en should be defined; you may use the following as a starting point, 11. Obtain expressions for the final and boundary conditions in finite difference form for a European Put Option. b. [8 Marks] A binary put option is to be priced. Outline a Monte Carlo method to do this. c. [6 Marks] Give four advantages of the Monte Carlo methods for pricing derivatives COMPG012 PLEASE TURN OVER 5 5. a.. [9 Marks] Suppose the spot interest rate r, which is a function of time t, satisfies the stochastic differential equation dr = u (r, t) dt + w (r, t) dWt . Derive the bond pricing equation (by hedging with a bond of dif ferent maturity) for a security Z = Z (r, t; T) oZ 1 202Z oZ m+2w or2 +(u(r,t)->.(r,t)w(r,t)) or -rZ=O, where T is the maturity of the bond. b. [16 Marks] The Vasicek model has risk-neutral drift and diffusion in turn, given by u (r, t) - >. (r, t) w (r, t) = a - br, w (r, t) = c1/ 2 , where a, b and c are constants. Deduce that the value of a zero coupon bond, (which has Z (r, T; T) = 1) in the Vasicek model is given by Z (r t" T) = eA(t;T)-rB(t;T), , , where B(t; T) _ ~(1 - e-b(T-t)), 1 1 cB(t;T)2A(t; T) b2(B(t; T) - T + t)(ab - 2c) - 4b You are required to show all steps in solving the Bond Pricing Equation. COMPG012 END OF PAPER 6
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