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MMF1941H: Stochastic Analysis - Assignment # 1

1 Instructions

1. Please have your final report typeset using LATEX and submit your report individually. Provide code separate in a Python script file that you attach in your submission.

2. You may discuss these questions with your fellow students, however the write-up must be yours and yours alone, sharing of the write-up before the deadline is not allowed.

2 Problem: Bachelier Call Option Pricing

Let X be a standard normal random variable and let  and variance  parameters. We are looking at the value of a call option in the Bachelier Model ie

for a given strike K.

1. (10 pts) Show that for 

holds where  is the pdf of the standard normal distribution and  the respective cdf. (Hint: you can exploit that  holds).

2. (5 pts) Use the previous result to show that analytically

holds.

3. (10 pts) For a parameter  we can define a measure  via the definition

For  calculate the  and conclude that under  X again follows a normal distribution and determine its parameters.

4. (5 pts) Write Python code to simulate the option value 1000 times under the measure P with a sample size of 5000 simulations each for  and K = 8. Share the code and provide a histogram of the results. Also calculate the exact value analytically per the above formula.

5. (5 pts) Denoting vj a single MC estimate (based on 5000 simulations) for j = 1, . . . , M with M = 1000 we can define the sample variance as

Calculate the sample variance for your previous experiment.

6. (10 pts) Note that for any a the option value can be re-written as

which mathematically will yield the same answer for any a. Define

which can be evaluated through MC simulation given the fact the distribution of X under the measure  is known. Write Python code plot the function g(a) for the above selection of parameters and  Note that equivalently,

holds which you could use alternatively for implementation purposes.

7. (5 pts) Determine the minimum of the function g numerically (and approximately) from the prior plot and repeat the experiment of simulating the option value but now calculated through

– where g attains its minimum at  times with a sample size of 5000 simulations each and again determine the sample variance. Plot the histogram of this experiment again.