A LEVEL 4 MODULE, SPRING SEMESTER 2020-2021 ADVANCED FINANCIAL MATHEMATICS Time to complete THREE Hours plus THIRTY Minutes upload time Paper set: 16/06/2021 - 10:00 am BST Paper due: 16/06/2021 - 13:30 BST 

Answer ALL questions Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or typeset. Do not write close to the margins. Your solutions should include complete explanations and all intermediate derivations. Your solutions should be based on the material covered in the module and its prerequisites only. Any notation used should be consistent with that in the Lecture Notes. Submit your answers as a single PDF with each page in the correct orientation, to the appropriate dropbox on the modules Moodle page. Use the standard naming convention for your document: [StudentID]_[ModuleCode].pdf. A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable and does not require magnification. Text which is not in focus or is not legible for any other reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced in size (e.g. scan in black & white, use a lower dpi but not so low that readability is compromised). Staff are not permitted to answer assessment or teaching queries during the assessment period. If you spot what you think may be an error on the exam paper, note this in your submission but answer the question as written. Where necessary, minor clarifications or general guidance may be posted on Moodle for all students to access. If you submit your paper after the deadline, you will receive a mark of zero, unless you have an EC accepted. If you submit your paper less than thirty minutes after the deadline, please submit a fast track ECF. If you submit your paper more than thirty minutes after the deadline, please submit a normal ECF. MATH4061-E1 MATH4061-E1 Turn over 1. (a) Assume that our market has two riskless assets: USD money market account $ with short rate $ and euro money market account with short rate , where their dynamics and /$ exchange rate are described by the following differential equations $ = $$, = , = + 1(), (1) = ( ) + (1() + 1 22()) . Here () is a stochastic volatility, $, , 0, > 0, > 0, > 0, [1,1] are constant parameters so that 2 2, and 1() and 2() are independent standard Wiener processes on a filtered probability space (, F, {F}, ) with being the real probability measure and F being the natural filtration for (1(),2()). Note that USD is the domestic currency in this market. i) Find an EMM in this market and re-write the equation for () under this . ii) Is this market complete? Provide an explanation. iii) For a European call option /$ to buy one EUR for $ at time , write the PDE problem. [18 marks] Let (), 0 , be a one-dimensional standard Wiener process on the filtered probability space (, F, {F}, ), where {F} is the natural filtration for (). Assume that the stock price () (in GBP) is modelled as = + (), where the mean rate of return and the volatility > 0 are constant. Further, assume that the model for the GBP money market account (the current account) is the following differential equation = , where is a constant short rate. Find an analytical formula for the arbitrage price of a European option with maturity which pays 1 if the stock price at the maturity exceeds a given strike > 0, otherwise it pays nothing. [7 marks] 1 MATH4061-E1 (b) MATH4061-E1 =[]+ +(), (0)=0, 2 MATH4061-E1 2. (a)Let()beacompoundPoissonprocesswithintensity>0andjumpsizedistribution . Assume that jumps are so that [12] < . In the following questions you may use any results from the Lecture Notes and Homework 3 without reproving them but appropriately citing any results you use. i) Show that [ 2 ()] = 2 + 2 ()2 , where = [1 ] and 2 = [12 ] . 2 22 iii) Prove that () is a martingale. (b) Consider the jump-diffusion short rate model ii) Compute [ ()] , where () is the compensated compound Poisson process. where , and ar positive constants and () is a one-dimensional standard Wiener process and () is a compensated compound Poisson process under an EMM . i) Compute (). ii) Give reasons (three in total) why to use or why not to use this model in practice. [10 marks] MATH4061-E1 Turn Over [15 marks] 3 MATH4061-E1 3. In this question assume that there exists a market for zero-coupon (non-defaultable) bonds (, ) for every maturity . Also, assume that there is an EMM under which all discounted prices are martingales. (a) Consider the following jump-diffusion model for a LIBOR rate (, ) written under the +1-forward measure +1 +1 (,)=(,)[ ()+()], 0, (2) where +1 () is a standard one-dimensional Wiener process, > 0 is a constant, and () is a compensated compound Poisson process with intensity > 0 and the jumps 1, 2, ... are so that +1= =exp(), and are i.i.d. Gaussian random variables with mean and variance 2. i) To modify the diffusion LIBOR model from Section 6.4 of the Lecture Notes to obtain (2), we added the compensated compound Poisson process. Explain why this is a sensible choice while adding the usual (i.e. non-compensated) compound Poisson process would result in a wrong model. ii) You may use appropriate results from the Lecture Notes and homeworks 3 and 4 to answer the following questions. A) Write down an explicit expression for the solution of (2). B) Using part (a)ii.A or otherwise, find an analytical expression for the price of a caplet (; , +1) assuming the model (2), where the caplet price notation as in the Lecture Notes. [15 marks] (b) In Example 47 on pp. 93-95 of the Lecture Notes, we derived how a caplet can be hedged, when the short rate is following the Vasicek model. Modify the argument of this example for hedging a floorlet (it is sufficient to write down the changes required without re-writing the whole example). You may use without a proof that the floorlet price (; , + ) is equal to (;, +) = (1+)(, +) ( ln (,+)[1+] (,) ( (,) , (,) , ) ,, 2 ln (,+)[1+] ) + , 2 MATH4061-E1 where the notation is as in Section 6.3. of the Lecture Notes. [10 marks] 4 MATH4061-E1 4. (a)Youworkatabankwheretherisk-freeborrowingrateiscurrently0.01%pa.Thebank lends money at the following rates: 3.8% pa for mortgages, 8% pa for secured loans, 15% pa for unsecured loans. Compute credit spreads for these three products. In credit risk terms, discuss which products are riskier and why. [6 marks] (b) Let (, F, {F}, ) be a filtered probability space on which two correlated Wiener processes 1() and 2() with correlation coefficient [1, 1] are defined. Assume that the short rate () and the conditional intensity of default () are described under an EMM by the model: = + 1, (0) = 0, (3) = + 2, (0) = 0, where > 0, > 0, > 0, and > 0 are constants. It is known that the price of defaultable bond is given by (you do not need to derive this formula): (0,)={>0}(exp( [()+()])), 0, 0 where = () is a default time for this bond. i) Solve the system (3) of two SDEs to find () and (). ii) Compute the expectation of the Gaussian random variable () + (). iii) Show that covariance Cov (() + (), () + ()) = (2 + 2 + 2) , where , 0. [Hint: covariance of a standard Wiener process 1() is equal to Cov (1(), 1()) = .] iv) Using parts (ii) and (iii) or otherwise, compute the expectation and variance of the Gaussian random variable [() + ()] . 0 v) Using the result of part (iv), compute (0, ) in terms of the parameters of the model. vi) Give reasons (three in total) why to use or why not to use the model (3) in practice. [19 marks] MATH4061-E1 END