Please adhere to what the questions is looking for. I need the work well done
For simulation exercises, you may use any software or programming language of your choice, but Matlab or R should be easier to use. C(S1(T ), S2(T )) = max(aS1(T )− bS2(T ), 0) Payoff analysis: 1. By choosing a suitable numeraire, show that the option payoff is merely the payoff of a call option on a new underlying with a given strike. Give the expression for the new underlying in function of S1 and S2 as well as the strike value. 1 2. Similarly, by choosing a suitable numeraire, show that the option payoff is merely the payoff of a put option on a new underlying with a given strike. Give also the expression for the new underlying in function of S1 and S2 as well as the strike value. Pricing: Consider the case of the put option derived above. We assume a Black-Scholes world where S1 and S2 have geometric Brownian motion dynamics as follow: dS1t S1t = µ1dt+ σ1dz1t dS2t S = µ2dt+ σ2dz2t where µ1 and µ2 are respectively the expected return of the S&P500 and the FTSE indices, σ1 and σ2 their respective volatilities, and z1t and z2t are correlated Brownian motions in the sense that dz1 · dz2 = ρdt. Moreover, you may assume that there is a money market account denotes Bt and paying a risk-free rate r: dBt Bt = rdt 1. Using risk-neutral pricing, derive an analytical formula for the price of the option. 2. What do you observe in this formula regarding the risk-free rate ? Can you explain your obser- vation ? Monte-Carlo implementation and convergence analysis: Assume that: S10 = 2725, S20 = 2050, a = 10%, b = 10%, µ1 = 12%, µ2 = 9%, r = 3%, σ1 = 20%, σ2 = 10%, ρ = 0.4 and T = 1 (1year). 1. Implement a Monte-Carlo pricing engine to price the option. 2. Implement also an analytical pricing engine for the option. 3. Study the convergence of the Monte-Carlo pricing to the true price (the analytical price). 4. How could you improve this convergence (give briefly an overview of some techniques that could be used for doing so) ? dS = rSdt+ √ V Sdz1 dV = µV dt+ ξV dz2, where z1 and z2 are two Wiener processes with instantaneous correlation ρ. Let fi, (i = 1, 2) be the value the value of the down-and-in (put) and the up-and-out (call) barrier options on S with barrier level Hi, strike price Xi and maturity Ti, (i = 1, 2). 1. What is the payoff of each option at their respective maturity dates. 2 2. By numerical simulation, give the price of each option when Ti = 0.5 (6 months), H1 = 95 and H2 = 105. Consider the current stock price S0 = 100 and the parameters values: V0 = 0.15 2, µ = 0, the risk-free rate r = 3%, ξ = 1 and ρ = 0.2. (Give the value of B and n simulated.) 3. Obtain the prices when S follows a geometric Brownian motion with volatility σ = 0.15 4. Compare the prices obtained at 2. and 3. Question 2: Assume that the risk-neutral dynamics of the price S of an underlying asset is given by: