Introduction to the Mathematics of Finance. HOMEWORK 3. Due March 31, 2021, 11.59pm Please write a pledge that homework solutions represent your own work and that you did not copy solutions from the...

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Introduction to the Mathematics of Finance. HOMEWORK 3. Due March 31, 2021, 11.59pm Please write a pledge that homework solutions represent your own work and that you did not copy solutions from the work of other students. 1. Suppose that the price Xt of Euro in terms of USD follows dXt = 0.01 Xtdt + 0.09 XtdWt Write the equation for Yt the price of USD in terms of Euro. 2. Suppose that Xt follows the process dXt = 0.05 Xt dt + 0.4 Xt dWt. Using Ito’s Lemma find the equation for the process for Yt = ln Xt. 3. A stock price is $100 now. In 1 month it can go to $110 or $90. The annual interest rate is 10% with continuous compounding. Using risk-free portfolios, determine the value of the one-month European put with strike price 100. 4. Use risk-neutral valuation to calculate the probabilities that will give you the correct put prices in problem 3. 5. Construct trading strategies in stock only that replicate each of the two puts of problem 3. That means construct a) synthetic long put strategy with strike price 100. What is the cost of each synthetic trading strategy. 6. A stock price is $40 now. In one month it can go 10% up or down. In the second month it can go 10% up or down. The annual interest rate is 11% with continuous compounding. Use risk-free portfolios to determine the value of the a two-month European call with strike price 40. 7.Use risk-neutral valuation to calculate the probabilities in the model that will give you the correct call prices in the previous problem. 8. Create a spreadsheet modeling trajectories of geometric Brownian motion starting at 50 with growth rate 1 percent (it is also risk free rate) and volatility 22 percent. Make a spreadsheet that calculates European calls maturing in 1 year with strikes 50 and 51 on non-dividend paying stock using Monte- Carlo method and using 10,000 trajectories with 250 steps in each trajectory. Compare Monte-Carlo price with 20,000 trajectories to theoretical model price. Calculate with 40,000 trajectories. Compare Monte-Carlo price with 40,000 trajectories to a Black-Scholes theoretical model price. 9.Create Matlab code doing the same thing as problem 8. Compare Matlab and spreadsheet results.