The Monte Carlo Method for pricing financial derivatives Part II – Optimizing and working with exotic options

1. The Monte Carlo Method for pricing financial derivativesPart II – Optimizing and working with exotic options John Sonchack

2. Overview • Recap from last time • Questions / Clarifications of first presentation • Time optimization • Lowering variance / increasing accuracy • Actual uses • Examples

3. Basically, a Monte Carlo method is a way of solving complex problems through approximation using many random numbers. They are very versatile and can help us solve difficult problems, but are often slower and less accurate than other available methods.

4. Recap from last time: A general approach to investigating problems with a Monte Carlo Method 1)Draw a random number 2)Process this random number in some way, i.e. plug it into an equation or use it in some way 3)Repeat steps 1 and 2 a large number of times 4)Analyze the cumulative results to find an estimation for a non random value

5. How we apply this idea to financial derivatives • 1) generate a random walk of stock based on the Brownian assumption • 2) Get an expected payoff for the given derivative based on this walk (Put, Call, lookback put, lookback call, asian call, etc) • 3) Repeat steps 1 and 2 a large number of times • 4) Average the cumulative results and discount the value to the present to get an estimate for the value of the option

6. A simple example Option value based on run 1 Option value based on run 2 Option value based on run 3 Estimate of option value

7. Questions / Clarifications • The Monte Carlo method was first suggested as a way to price options in 1977 by Phelim Boyle in his paper: “Options: A Monte Carlo Approach” • Pricing European style options is a bit less complicated than pricing American (i.e. early exercise) options. Thus, for these examples, we will only be pricing options for which there is no early exercise.

8. In order to price path independent options accurately, we only need to use a small number of steps in each trial run, like 20. We cannot go too low, however, because this will skew the distribution. How many steps are needed?

9. How many trials are needed? • How many trials we should run depends on many factors such as the kind of derivative we are pricing, how optimized our code is, and - most importantly - how much time we have to spare. • Generally though, somewhere between 1000 (takes about 10 seconds with 20 steps) and 100,000 (takes about 1.5 minutes with 20 steps) trials are sufficient

10. Optimization • One of the main criticisms of the Monte Carlo method is how time consuming it is. This can often be fixed with clever optimization. • There are two main ways in which a Monte Carlo program can be made more efficient. • Time optimization – making sure the code is as efficient as possible, choosing the best procedures to use, etc • Variance reduction – making a set of a certain number of trials more accurate.

11. Time optimization • Without any optimization, the original European Call pricing method from last time takes about 21.5 seconds to run a simulation of step size 20 and trial size 1000. • There are a number of changes that we can make to the code itself that will make it more efficient to run.

12. Time optimization • Instead of multiplying the payoff by e-rt for every trial, we can assign e-rt to a variable ahead of time and use that value in the trials. (Allows us to do less *e^rt calculations) • Instead of calling another procedure to generate the next step in a random walk, we can put the code right into the main method • these optimizations combined make the program about 5% faster, bringing the time down to ~20 seconds.

13. Time optimization • The largest benefit to this program can be achieved by changing the way we work with random numbers. • Previously, the program would create a method to generate a random number and then generate a random number for each step. If we generate the random numbers as a set outside of the loop, we can lower the time significantly. • If we change the procedure that maple uses to generate random numbers, we can cut the run time of the procedure even more.

14. Chart about time optimization

15. Variance reduction • Variance reduction is a statistical term that basically means trying to lower the standard deviation without increasing the sample size. • Normally, in order to half the standard deviation, we must quadruple the sample size.

16. Variance reduction: Antithetic Variate • Because the random variable used in the procedure is normally distributed with a mean of 0 and is symmetric, the probability that a random path is drawn is exactly the same as the probability that the negative of that path is drawn.

17. In the probability distribution curve of a normal variable, the chances of drawing say, a -2 are exactly the same as drawing a 2. In this case, -2 would be the antithetic variate of 2 and vice versa. The Antithetic Variate

18. The Antithetic Variate Since the chances of drawing a random number for the random part of the brownian motion are exactly the same as drawing the negative of that number, and since the “drift” is the same for all trials, we can effectively generate two random walks from one set of random numbers.

19. The Antithetic Variate • This lets us effectively double our sample size without doing much extra work. • A 50,000 run trial using Antithetic Variates will have the accuracy of a 100,000 run trial and so on. • This helps more for lower trial sizes.

20. Another method is the Control Variate Method. This method uses the value of a option that has a high correlation to the one which we are trying to predict a value for which can be priced using formulas to lower the Variance (error) of our trials. Let x be the simulated option value, y be the value of a highly correlated option obtained with a formula, and z be a simulated option value for y. Instead of just generating lots of values of x, we would generate lots of values of: x+(y-z) Essentially, this strategy evaluates (x-z)+y. Var(x-z)=Var(x)+Var(z)-2*cov(x,z) If Var(z)<2*cov(x,z), then Var(x-z)<Var(x) Other Variance Reduction strategies

21. Control Variate continued • The Control Variate technique is more useful if you can find an option that is both highly correlated to the one you’re trying to estimate and has a definite value. • Usefulness varies based on what option you’re trying to price – its particularly useful for Asian options. • Not currently implemented in this method. • There are many other ways to lower variance ( increase accuracy ) that depend on the option.

22. Actual Uses • Any option that has a payoff formula can be priced with a Monte Carlo method. • Very versatile - Multiple underlying, changing volatility, a different distribution assumption for the random walk, etc. • For some options, like the Asian option, variance reduction strategies can make the Monte Carlo method very accurate at low trial sizes – meaning that it is both accurate and quick. • Besides being used for these particular options, this method is also often used to double check other implementations.

23. Three examples of pricing exotic options using the Monte Carlo method • We will use an implementation of the Monte Carlo method to price: an Asian Call, a Lookback Call, and a Barrier option; and compare it to results from both formulas and the binomial pricing example that Joe and James did.

24. Asian option payoffs are calculated using the average over the entire run rather than the value of the underlying at time t. Asian options

25. Fixed Lookback option payoffs are calculated using the optimal value (maximum for a call, minimum for a put) Lookback options

26. Barrier options either knock in or knock out above or below a certain barrier. Barrier options

27. Closing thoughts • The Monte Carlo method is primarily used for pricing of several kinds of exotic options for which there is no formula or the formula is difficult. • It is also good as a way to double check any implementation of option pricing. • Unfortunately, there aren’t really any programs that simply do Monte Carlo approximations of options  you basically have to write an implementation yourself, or copy code from someone.

28. Works cited • Options pricing by simulation -http://finance-old.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html • http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN96-03.pdf • http://en.wikipedia.org/wiki/Monte_Carlo_option_model • Pricing exotic options using the binomial tree method presentation