In this problem, you are asked to price an Asian option using the simulation method we...

  

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In this problem, you are asked to price an Asian option using the simulation method we talked about in class. Suppose that the stock price follows Geometric Brownian Motion d.StuStdt + o SidBt. = The initial stock price So is $10. The drift = 15% and the volatility = 40%. The continuously-compounded riskfree rate r is 5%. Consider an Asian put option on the stock with maturity T = 1 and strike price K = $10. The average stock price is calculated using the closing stock price of each trading day. That is, letting N = 252 and At = T/N, the average stock price is defined as Savg = Sat + S2At +...+ SNAT N Act To ensure the reproducibility of your results, you can set random seed in Python using np.random.seed (0). Got This problem asks you to numerically calculate Delta and Gamma for the Asian put option in Problem 1. All parametric settings are the same as that of Problem 1, so you will need to build on the program you write for Problem 1. Choose step size € = = 0.1. Instead of fixing the initial stock price So at $10 as in Problem 1, we now consider the initial stock price So on grid points €, 2€, 3€, . . .‚ 20 – €, 20. (a) (8 points) For each possible initial stock price So on the grid points, calculate the current price of the Asian put option P(So). Plot the option price P(S) against the initial stock price So. Make sure that you see a downward sloping line. Hint: • As we discussed in class, each time your iteration changes to a different starting So, make sure you reset to the same random seed in Python using np.random.seed(0). Can you think of a smarter programming way to implement the same thing? • Your codes could take up to 10 minutes to run depending on your computer and how efficient your codes are. To save time, first do the calculation with step size € = 1. Once you see the results and are sure your codes are correct, change back to € = = 0.1 to produce the required figure. (b) (5 points) The option Delta can be approximated by the following finite-difference equation, P(So + €) P(So) P'(So)~ € Use the option price P(So) in part (a) to calculate the Delta P'(So) of the Asian put option. Plot the option Delta P'(So) against the initial stock price So. Is this curve upward or downward slopping? Why? (c) (5 points) The option Gamma can be approximated by the following finite-difference equation, P(So + €) 2P (So) + P(So - €) €² - P" (So)~ Use the option price P(So) in part (a) to calculate the Gamma P"(So) of the Asian put option. Plot the option Gamma P"(So) against the initial stock price So. What is the shape of this curve? Why? In this problem, you are asked to price an Asian option using the simulation method we talked about in class. Suppose that the stock price follows Geometric Brownian Motion d.StuStdt + o SidBt. = The initial stock price So is $10. The drift = 15% and the volatility = 40%. The continuously-compounded riskfree rate r is 5%. Consider an Asian put option on the stock with maturity T = 1 and strike price K = $10. The average stock price is calculated using the closing stock price of each trading day. That is, letting N = 252 and At = T/N, the average stock price is defined as Savg = Sat + S2At +...+ SNAT N Act To ensure the reproducibility of your results, you can set random seed in Python using np.random.seed (0). Got This problem asks you to numerically calculate Delta and Gamma for the Asian put option in Problem 1. All parametric settings are the same as that of Problem 1, so you will need to build on the program you write for Problem 1. Choose step size € = = 0.1. Instead of fixing the initial stock price So at $10 as in Problem 1, we now consider the initial stock price So on grid points €, 2€, 3€, . . .‚ 20 – €, 20. (a) (8 points) For each possible initial stock price So on the grid points, calculate the current price of the Asian put option P(So). Plot the option price P(S) against the initial stock price So. Make sure that you see a downward sloping line. Hint: • As we discussed in class, each time your iteration changes to a different starting So, make sure you reset to the same random seed in Python using np.random.seed(0). Can you think of a smarter programming way to implement the same thing? • Your codes could take up to 10 minutes to run depending on your computer and how efficient your codes are. To save time, first do the calculation with step size € = 1. Once you see the results and are sure your codes are correct, change back to € = = 0.1 to produce the required figure. (b) (5 points) The option Delta can be approximated by the following finite-difference equation, P(So + €) P(So) P'(So)~ € Use the option price P(So) in part (a) to calculate the Delta P'(So) of the Asian put option. Plot the option Delta P'(So) against the initial stock price So. Is this curve upward or downward slopping? Why? (c) (5 points) The option Gamma can be approximated by the following finite-difference equation, P(So + €) 2P (So) + P(So - €) €² - P" (So)~ Use the option price P(So) in part (a) to calculate the Gamma P"(So) of the Asian put option. Plot the option Gamma P"(So) against the initial stock price So. What is the shape of this curve? Why?

Answer rating: 100% (QA)

To price an Asian put option using the simulation method youll need to perform the following steps Generate Stock Price Paths Use the Geometric Browni... View the full answer