Question Matlab project Q a) Programing Matlab, please, price out natural gas basis swap option contracts in two ways: i) bivariate binomial tree; ii) simple Monte-Carlo simulation. For the bivariate binomial tree, set each subinterval to be 1 calendar day ( 1 365 year). For the simple Monte-Carlo simulation, set the subinterval to be one 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Maturity (year) Correlation between two futures contracts 4 calendar day ( 1 365 year), and control the absolute error within plus/minus 0.25 cent. When programming the bivariate binomial tree and Monte-Carlo simulation, assume that the correlation at the beginning of each sub-interval does not change. Price out the natural gas basis swap option contracts maturing from Jan 1, 2016 to Dec 31, 2016; in other words, you need to price 366 basis swap options. Compare the calculation speeds of these two methods. (Hint: The bivariate binomial tree will be faster than the MC simulation. You still need to answer how fast.) b) Use various variance reduction techniques to improve the speed of the MC simulation in a) and make a recommendation of the variance reduction technique of your choice. Then, price the natural gas basis swap option contracts maturing from Jan 1, 2016 to Dec 31, 2016 using the MC simulation with the variance reduction technique you recommend. Report how much calculation time improved by using the variance reduction technique. (Hint: There is not a single right answer for this task. Use your creativity. I will measure calculation time when I grade this task, though.) c) As I stated earlier, in the real world, a futures contract with shorter maturity is more volatile than a futures contract with longer maturity – sometimes such a phenomenon is called the "Samuelson's effect" (Samuelson, 1965). Now you want to model such time-varying volatility; in other words, you want to relax the simplifying assumption of 𝜎1 = 𝜎1 = 0.40. Propose a model which can price out the natural gas basis swap options under such assumptions, and price out the natural gas basis swap option contracts maturing from Jan 1, 2016 to Dec 31, 2016. Then, compare the results with those in a) and b). (Hint: Unfortunately, you cannot use a bivariate binomial tree, because a tree will not be recombining any more. Hence, you should use a MC simulation. There is not a single way of modeling the Samuelson Effect. Use your creativity or search literature, and propose a reasonable way of modeling it. You also need to make a reasonable assumption for the degree of Samuelson effect.) d) For Stuart Energy Company, propose another business opportunity directly or indirectly using the bivariate binomial tree and/or MC simulation you programmed in a) – c). Try to relate your proposal to an academic paper, a magazine/ newspaper article or interview with a real world professional. Write one white paper to address a) to d) in the above. The white paper should contain the following sections: 1) Executive Summary 2) Introduction 3) Methodology: Binomial Tree and Monte-Carlo Simulation 4) Numerical Results and Discussion 5) Recommendation for improving the calculation speed of MC simulation 6) Extension to time-varying volatility 7) Proposal of another Business Opportunity 8) Conclusion 9) Appendix 10) Tables and Figures 11) Reference Sections 3) and 4) in your white paper correspond to task a); section 5) corresponds to tasks b); section 6) corresponds to task c); section 7) corresponds to task d). All the tables and figures should be in section 10) – the results of task a) should be in a table reporting all 366 days' results