Find an expression for the stochastic process Xt given that it follows the stochastic differential equation (SDE),

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dXt = (1+t)2 1+tXt dt+ (1+t)2dWt.

5 Marks

2. The current stock price of of ABC inc. is $50. You have signed an agreement such that you owe 100 if the stock price ends $45 four months later and 0 otherwise. You intend to replicate that liability with a portfolio made of shares of ABC inc. and the risk-free asset. Assume r = 0.03 and = 0.18.

(a) At time 0, what is the delta-hedging portfolio for this agreement? 5 Marks

(b) Inthescenariothat,amonthlater,thestockpriceisupto$52,determinethedeltahedging

portfolio such that the portfolio remains self-financing. 5 Marks

3. Consider a mean-variance portfolio model with two securities, 1 & 2, where the expected return and the variance of return for 2 are twice the corresponding values for 1. Suppose the correlation between the returns on the two securities is .

(a) Determine the values of which allow the possibility of constructing a zero-risk port-

folio, by calculating the variance of the return on a portfolio with weights w1 and w2 is contructed from the two assets. 2 Marks

(b) Calculate the portfolio weights that lead to the most efficient zero-risk portfolio. 2 Marks

(c) Calculate the expected return on the portfolio in part (i)(b) in terms of the expected return

on security 1, R1.

1. Find an expression for the stochastic process Xgiven that it follows the stochastic differential equation (SDE). 1 dX: -x+ (1 + ) 1)2 5 Marks = (a +42 14cxi) dt + (+ jadw 2. The current stock price of of ABC inc. is $50. You have signed an agreement such that you owe 100 if the stock price ends $45 four months later and otherwise. You intend to replicate that liability with a portfolio made of shares of ABC inc. and the risk-free asset. Assume r = 0.03 and a = 0.18 (a) At time 0, what is the delta-hedging portfolio for this agreement? 5 Marks (b) In the scenario that, a month later, the stock price is up to $52, determine the delta hedging portfolio such that the portfolio remains self-financing 5 Marks 3. Consider a mean-variance portfolio model with two securities, 1 & 2, where the expected return and the variance of return for 2 are twice the corresponding values for 1. Suppose the correlation between the returns on the two securities is p. (a) Determine the values of p which allow the possibility of constructing a zero-risk port- folio, by calculating the variance of the return on a portfolio with weights w, and w, is contructed from the two assets. 2 Marks (b) Calculate the portfolio weights that lead to the most efficient zero-risk portfolio. 2 Marks (c) Calculate the expected return on the portfolio in part (1)(b) in terms of the expected return on security 1. R 1 Mark Total: 20 Marks 1 1. Find an expression for the stochastic process Xgiven that it follows the stochastic differential equation (SDE). 1 dX: -x+ (1 + ) 1)2 5 Marks = (a +42 14cxi) dt + (+ jadw 2. The current stock price of of ABC inc. is $50. You have signed an agreement such that you owe 100 if the stock price ends $45 four months later and otherwise. You intend to replicate that liability with a portfolio made of shares of ABC inc. and the risk-free asset. Assume r = 0.03 and a = 0.18 (a) At time 0, what is the delta-hedging portfolio for this agreement? 5 Marks (b) In the scenario that, a month later, the stock price is up to $52, determine the delta hedging portfolio such that the portfolio remains self-financing 5 Marks 3. Consider a mean-variance portfolio model with two securities, 1 & 2, where the expected return and the variance of return for 2 are twice the corresponding values for 1. Suppose the correlation between the returns on the two securities is p. (a) Determine the values of p which allow the possibility of constructing a zero-risk port- folio, by calculating the variance of the return on a portfolio with weights w, and w, is contructed from the two assets. 2 Marks (b) Calculate the portfolio weights that lead to the most efficient zero-risk portfolio. 2 Marks (c) Calculate the expected return on the portfolio in part (1)(b) in terms of the expected return on security 1. R 1 Mark Total: 20 Marks 1