” EE 512编程 写作、 辅导c/c++程序设计EE 512 – Fall 2020 Class ProjectDue Friday, Dec. 4th.Solve one of the two simulation problems below in any computational language of yourchoice. Submit a report (strongly recommend either a Jupyter notebook or an R Markdowndocument) with your results, including all relevant captioned figures.1. [Options Pricing via Monte Carlo Estimation]: Generate T = 500-day samplepaths of the geometric Brownian motion (GBM) for different values of (, ) using theEuler-Maruyama Simulation scheme. Recall that the GBM is the stochastic processthat satisfies the following SDE:dSt = (t, St)dt + (t, St)dBt = Stdt + (t, St) StdBt.(a) Assume a GBM model for an underlying assets spot price St. Suppose we have aEuropean call option for the asset St given a strike price of K. Write and test anevaluation function that takes a GBM sample path and K as input. The outputis the payoff for the call option. Recall that European call options may only beexercised at the expiration time T = 500 and has the value ZZ = g(ST ) = max{ST K, 0} .(b) Specify 5 sets of GBM parameters (, ) for 5 different assets {Skt }k. Use MonteCarlo to estimate the expected value of European call options on each of theseunderlying assets. And provide confidence intervals for your estimates. You maymake any simplifying assumptions (e.g. constant asset volatilities) as long as youstate your assumptions.(c) Suppose the risk-free rate of return is r = 0.05. Modify your value estimationprocess to estimate the discounted expected payoff for the option.(d) Replace the European call option with a Asian call option monitored every 50days. This is a path-dependent option also exercised at the expiration time T butvalued at ZAsZAs = gAs(ST ) = max{S K, 0}where S = Avg ({S50, S100, , ST })Use Monte Carlo to estimate the value of Asian options on the same 5 sets ofGBM-modeled assets you chose above. Compare the value of the European andAsian call options based on the results of your experiments.(e) Repeat the European option value estimation process using Jump-Diffusion stochasticprocesses for the underlying asset prices instead of just a diffusion GBM process.2. [Optimal City Paths]: The famous Traveling Salesman Problem (TSP) is an NP-hardrouting problem. The time to find optimal solutions to TSPs grows exponentially withthe size of the problem (number of cities). A statement of the TSP goes thus:A salesman Needs to visit each of N cities exactly once and in any order.Each city is connected to other cities via an air transportation network. Finda minimum length path on the network that goes through all N cities exactlyonce (an optimal Hamiltonian cycle).A TSP solution is just an ordered list of the N cities with minimum path length. Wewill be exploring MCMC solutions to small and larger scale versions of the problem.(a) Pick N = 10 2-D points in the [0, 1000] [0, 1000] rectangle. These 2-D sampleswill represent the locations of N = 10 cities.i. Write a function to capture the objective function of the TSP problem:ii. Start with a Random path through all N cities ~c (a random permutation of thecities), an initial high Temperature T0, and a cooling schedule Tk = f(T0, k).iii. Randomly pick any Two cities in your current path. Swap them. Use thedifference between the new and old path length toiv. calculate a Gibbs acceptance probability. Update the path accordingly.v. Update your annealing temperature and repeat the previous city swap step.Run the Simulated annealing procedure to convergence.vi. Plot the values of your objective function from each step. Plot your final TSPcity tour.(b) Run the Simulated Annealing TSP solver you just developed for N = {40, 400, 1000}cities. Explore the speed and convergence properties at these different problemsizes. You Might want to play with the cooling schedules.如有需要，请加QQ：99515681 或邮箱：99515681@qq.com
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