MSc Financial Mathematics – SMM306 1Question 1Let L be composed by a Brownian motion with drift and a rate compound Poisson processwith jump size X N (X; 2X), under the real probability measure. Assume the marketparameters are as in the following table.St = 100; r = 5% p.a.; = 13:12% p.a.; X = ??0:0537 X = 0:07 = 0:68:Consider a European call option on S, with strike price K = 100 and maturity in 3 months.a) The price of the European call under the Merton measure isCt =X1n=0e??(T??t) ( (T ?? t))nn!f (n)dn =ln StK +rn + v2n2(T ?? t)vnpT ?? t; d0n = dn ?? vnpT ?? trn = r ?? ( ?? 1) +nT ?? tln ; = eX+2X2 ; v2n= 2 +nT ?? t2X:Consider the problem of implementing this formula in Excel (Note: f() is the Black-Scholes formula).i) The pricing equation requires the computation of the innite series determining thePoisson distribution. Using Excel, show that for the given parameters set, the rstthree or four terms of the series are enough to guarantee a good approximationof the contract price.ii) Using Excel, calculate the price of the option for the given set of parameters.iii) Perform a sensitivity analysis for S = 80; 85; 90; 105; 110; 115; 120, and plot yourresults.b) Now consider the traditional Black-Scholes framework under whichdSt = Stdt + StdWtunder the real probability measure.i) Show that if = 15% p.a., the overall instantaneous volatility of the log-returns isthe same under the Lévy based model and the Black-Scholes model (Hint: usethe results from Unit 3 – Question 1).ii) Calculate the price of the European call option using the Black-Scholes formula forthe same range of values of S.iii) Calculate the ratio between the prices you obtained under the two framework andexplain the results you obtain.1. An individual sets up a pension fund. Let X(t) represent the value of the fund at time t.Assum X(0) = 0.The fund is invested in a mixture of a risk-free asset, paying a return of r, and a risky asset,whose value evolves according to a geometric Brownian motiondS(t)S(t)= µ dt + dW(t),where W is a standard Brownian motion. The control variable, u(t), is the amount of moneyinvested in the risky asset at time t.The individual pays money into the fund at a deterministic rate p(t) dt for 0 t T/2. Attime T/2 the individual retires. After retirement the individual intends to consume the fundat determinstic rate c(t) dt until time T.The success of the individual’s investment strategy will be measured by the objective function-1(X(T)+K), where K > 0, 0 < < 1. In other words, the optimal value function will beV (t, x) = sup{E[-1(X(T) + K)]},where the supremum is taken over all possible strategies.(i) Show thatdX(t) = rX(t) dt + (µ – r)u(t) dt + u(t) dW(t) + y(t) dt,wherey(t) =p(t) if t < T/2-c(t) if T/2 < t < T(ii) Write down the HJB equation and show thatVt + (rx + y(t))Vx -µ222V 2xVxx= 0.(iii) Verify that there is a solution of the formV (t, x) = A(t)(x + B(t)),where A and B are functions which you should find.(iv) Write down the optimal amount u(t, x) to invest in the risky asset.(v) Show that, subject to optimal control, X(t) + B(t) is a geometric Brownian motion.(vi) Calculate P[X(T) > 0] in the case where the net present value at time 0 of thepayment stream is 0, i.e., whereZ T/20e-rtp(t) dt =Z TT/2e-rtc(t) dt.