Problem 1. Black-Scholes comparison We put ourselves in the setting of Prob. 3 of HW#3. There, we worked out (using a computer) the price of a European call and a European put options for large N. Use the Black-Scholes formula to find the price of the European call and put-call parity (or the Black-Scholes formula for a put) to find the price of the European put of Prob. 3 of HW#3. Compare the results with the prices obtained in HW#3 and comment on the quality of the approximation. Hint: The Black-Scholes formula for a put is given by Po= Ke¯(-d_) — SöÞ(−d+), where d are the same as for the call. Problem 3. (N-period model with N large) Note: The purpose of this problem below is to use computational techniques (Excel spreadsheet, Matlab, R, Python, etc.) and code the dynamic programming ideas seen in class. Please provide the numerical answer to the questions as well as a sample of your work (spreadsheet, code file, etc.). = 60, the We consider an N-period binomial model with the following properties: N current stock price is So = 1000; on each period, the stock price increases by 0.5% when it moves up and decreases by 0.3% when it moves down. The annual interest rate on the money market is 5%. (Notice that this model is a CRR model, which means that the binomial tree is recombining.) (a) Find the price at time to = 0 of a (European) call option with strike price K and maturity T = 1 year. = = 1040 (b) Find the price at time to = 0 of a (European) put option with strike price K = 1040 and maturity T = 1 year. (c) We consider now, that you are at time t5 (i.e. after 5 periods, which represents 1 month later). Assume that the stock went up for the first three periods and then down for the next two. Determine the price of the stock at that position, the price of each of the put and call and determine how many shares of stock you should be holding in the hedging portfolio for each of the call and the put. (d) Repeat questions (a)-(c) in the case where r = 0. Comment on the results. Hint: take advantage of the fact that the model is recombining: moving up, then down takes you to the same value as moving first down, and then up. It reduces the amount of calculations necessary and will speed up your code.

Problem 1. Black-Scholes comparison We put ourselves in the setting of Prob. 3 of HW#3. There, we worked out (using a computer) the price of a European call and a European put options for large N. Use the Black-Scholes formula to find the price of the European call and put-call parity (or the Black-Scholes formula for a put) to find the price of the European put of Prob. 3 of HW#3. Compare the results with the prices obtained in HW#3 and comment on the quality of the approximation. Hint: The Black-Scholes formula for a put is given by Po= Ke¯(-d_) — SöÞ(−d+), where d are the same as for the call. Problem 3. (N-period model with N large) Note: The purpose of this problem below is to use computational techniques (Excel spreadsheet, Matlab, R, Python, etc.) and code the dynamic programming ideas seen in class. Please provide the numerical answer to the questions as well as a sample of your work (spreadsheet, code file, etc.). = 60, the We consider an N-period binomial model with the following properties: N current stock price is So = 1000; on each period, the stock price increases by 0.5% when it moves up and decreases by 0.3% when it moves down. The annual interest rate on the money market is 5%. (Notice that this model is a CRR model, which means that the binomial tree is recombining.) (a) Find the price at time to = 0 of a (European) call option with strike price K and maturity T = 1 year. = = 1040 (b) Find the price at time to = 0 of a (European) put option with strike price K = 1040 and maturity T = 1 year. (c) We consider now, that you are at time t5 (i.e. after 5 periods, which represents 1 month later). Assume that the stock went up for the first three periods and then down for the next two. Determine the price of the stock at that position, the price of each of the put and call and determine how many shares of stock you should be holding in the hedging portfolio for each of the call and the put. (d) Repeat questions (a)-(c) in the case where r = 0. Comment on the results. Hint: take advantage of the fact that the model is recombining: moving up, then down takes you to the same value as moving first down, and then up. It reduces the amount of calculations necessary and will speed up your code.

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter12: Algebra Of Matrices

Section12.4: Systems Of Linear Inequalities Linear Programming

Problem 44PS

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