1. Hedge with an Edge An Introduction to the Mathematics of Finance Riaz Ahmed & Adnan Khan Lahore Uviersity of Management Sciences Monte Carlo Methods

2. Topics Simulating Bernoulli Random Variable Generating Random Variables – Inverse Transform Method – Box Muller Method – Rejection Method Simulate a 1-D random Walk – Calculate the mean – Calculate the Variance Simulating Brownian Motion Geometric Brownian Motion Arithmetic Brownian Motion Variance Reduction Techniques

3. Simulating a Binomially Distributed Random Variable Note sum of Bernoulli trials is a binomial Let X i be a Bernoulli trial with probability ‘p’ of success is binomial ‘n’, ‘p’

4. Some Properties Distribution of successes in trials Expected Value Variance

5. Simulation of Binomial Generating Bernoulli Binomial as the sum of Bernoulli Monte Carlo Simulation Numerical vs. Exact Mean and Variance

6. Simulation of Binomial

7. Continuous Random Variables Inverse Transform Method – Suppose a random variable has cdf ‘F(x)’ – Then Y=F -1 (U) also had the same cdf Generating the exponential Generate the exponential, compare with exact cdf Generate a r.v. with cdf

8. Simulating the Exponential

9. Simulating Normal using Inverse Transform Cannot get a closed form in terms of elementary functions Excel has built in command normsinv() Use normsinv(rand())

10. Simulation of Normal

11. Rejection Method Simulate & To Simulate look @ If accept, else reject To Simulate N(0,1) let If set

12. Box Muller Method Recall the cdf for the standard normal is We saw one way was to invert this Another technique is to generate Then and where

13. Simulation

14. Weiner Process W(t) CT-CS process is a Weiner Process if W(t) depends continuously on t and the following hold a) b) are independent c)

15. Simulating Brownian Motion Initialize at 0 as W(0)=0 Simulate Weiner Increments according to The Weiner Process then follows

16. Simulation

18. Stock Price Model Modeled by Geometric Brownian Motion Note To simulate use the ‘Euler Scheme’

19. Simulating GBM

21. Mean Reverting Process Arithmetic Brownian Motion is mean reverting Interest rate models The numerical scheme is

22. Simulating ABM

24. Option Pricing using Monte Carlo Generate several risk-neutral random walks for the asset starting at the asset price today and going on till expiry. For each path generated calculate the payoff. Calculate average the average of all the payoffs Take the present value of this average to get the option value today.

25. Pricing of European Call

26. Challenge Problem Simulate using Monte Carlo techniques the price of a European call option where the underlying with volatility 0.5 interest rate 3% exercise price 100 and currently underlying at 90