Derive the stochastic differential equation for a derivative dependent on the prices of two securities by...

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Derive the stochastic differential equation for a derivative dependent on the prices of two securities by forming a riskless portfolio consisting of the derivative and the two traded securities. Assume that the securities are non-dividend-paying. You can assume that the price, f, of the derivative depends on the prices, S and S2, of two traded securities. The prices of these securities are given by dS = Sidt+Sdz and dS2 = H2S 2dt+02S2dz2, where 1, 1, 2 and 2 are constants and dz and dz2 are Wiener processes with correlation p. Suggestions: 1) use It's lemma to write down a stochastic differential equation for f; 2) construct your portfolio II; 3) impose the risk-free condition (denote the risk-free interest rate by the letter r); this will lead to the desired stochastic differential equation. The stochastic discount factor (SDF) in a consumption asset pricing model can be written as: Mt+1 = B u' (C++1) u'(ct) Where u (c) is a utility function and u'(c) is its first derivative with respect to c. Suppose the utility function is of the form: 1-Y Ct u(c) = 1-Y If consumption today is 0.5, consumption tomorrow is 0.4, is equal to 0.98 and y is equal to 2, the value of the SDF is: A. 1.531 B. 0.627 C. 1.562 D. 1.225 generated by the process Sn 2. Let Mr be the symmetric random walk defined in the previous example. Define Io = 0 and n-1 In = EM,(Mj+-M), n=1,2,... j=0 a) Show that In = M. b) Show that {In, n 0} is a Markov process, i.e. En f(In+1)] = g(In). 3. Suppose (Mn, n 0} is a martingale, and let An, n 0 be an adapted process. Define the discrete-time stochastic integral (sometimes called a martingale transform) by Io = 0 and = n-1 In A(Mj+1M;). Show that In, n > 0} is a martingale. j=0 4. (Divident-paying stock) Consider a binomial asset model such that, after each movement in the in stock price, a dividend is paid and the stock price is reduced accordingly. Define Ju, Yn+1 (W1, ..., Wn, wn+1)= d, if wn+1 = H if wn+1 = T Derive the stochastic differential equation for a derivative dependent on the prices of two securities by forming a riskless portfolio consisting of the derivative and the two traded securities. Assume that the securities are non-dividend-paying. You can assume that the price, f, of the derivative depends on the prices, S and S2, of two traded securities. The prices of these securities are given by dS = Sidt+Sdz and dS2 = H2S 2dt+02S2dz2, where 1, 1, 2 and 2 are constants and dz and dz2 are Wiener processes with correlation p. Suggestions: 1) use It's lemma to write down a stochastic differential equation for f; 2) construct your portfolio II; 3) impose the risk-free condition (denote the risk-free interest rate by the letter r); this will lead to the desired stochastic differential equation. The stochastic discount factor (SDF) in a consumption asset pricing model can be written as: Mt+1 = B u' (C++1) u'(ct) Where u (c) is a utility function and u'(c) is its first derivative with respect to c. Suppose the utility function is of the form: 1-Y Ct u(c) = 1-Y If consumption today is 0.5, consumption tomorrow is 0.4, is equal to 0.98 and y is equal to 2, the value of the SDF is: A. 1.531 B. 0.627 C. 1.562 D. 1.225 generated by the process Sn 2. Let Mr be the symmetric random walk defined in the previous example. Define Io = 0 and n-1 In = EM,(Mj+-M), n=1,2,... j=0 a) Show that In = M. b) Show that {In, n 0} is a Markov process, i.e. En f(In+1)] = g(In). 3. Suppose (Mn, n 0} is a martingale, and let An, n 0 be an adapted process. Define the discrete-time stochastic integral (sometimes called a martingale transform) by Io = 0 and = n-1 In A(Mj+1M;). Show that In, n > 0} is a martingale. j=0 4. (Divident-paying stock) Consider a binomial asset model such that, after each movement in the in stock price, a dividend is paid and the stock price is reduced accordingly. Define Ju, Yn+1 (W1, ..., Wn, wn+1)= d, if wn+1 = H if wn+1 = T