P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com
Chris Brooks, Chapter 13. Simulation Methods DESCRIBE THE BASIC STEPS TO CONDUCT A MONTE CARLO SIMULATION... 4 DESCRIBE WAYS TO REDUCE MONTE CARLO SAMPLING ERROR... 5 EXPLAIN HOW TO USE ANTITHETIC VARIATE TECHNIQUE TO REDUCE MONTE CARLO SAMPLING ERROR.... 5 Please note: The previous syllabus assignment for (P1.T2.) Simulations was Jorion s Chapter 12 called Monte Carlos Methods. Strictly speaking, because the Jorion chapter has been replaced by the Brooks chapter, it is optional for you to review. However, we have included this reading because it remains conceptually relevant. In particular, we do recommend that you review the following learning objectives: Describe how to simulate various distributions using the inverse transform method. Explain how to simulate correlated (normal) random variables 2
Chris Brooks, Chapter 13. Simulation Methods Describe the basic steps to conduct a Monte Carlo simulation. Describe ways to reduce Monte Carlo sampling error. Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective. Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them. Describe the bootstrapping method and its advantage over Monte Carlo simulation. Describe the pseudo-random number generation method and how a good simulation design alleviates the effects the choice of the seed has on the properties of the generated series. Describe situations where the bootstrapping method is ineffective. Describe disadvantages of the simulation approach to financial problem solving. In many situations, Monte Carlo simulation is the best approach. Often, analytical solutions are tractable but too simplistic to capture realistic and complex dynamics. Historical simulation, on the other hand, does offer the advantage of plentiful data, but may not give us the means to effectively imagine possible future outcomes. Consider interest rate paths. With relatively few assumptions, simulation allows us to randomize possible future interest rate paths. The two charts below can be found in the Learning Spreadsheet associated with this reading. Each chart simulates ten (10) different interest rate paths over a ten year (120 month) horizon. The models employed are Cox-Ingersoll-Ross (CIR) and the Vasicek model. These interest rate paths become inputs into a credit exposure (in the XLS) or can feed into a value at risk (VaR) model. Such is the flexibility of simulations, which allow us to manufacture data! 3
Describe the basic steps to conduct a Monte Carlo simulation Simulations studies are used to investigate the properties and behavior of various statistics of interest. The technique is used in econometrics when the properties of a particular estimation method are not known. The following are the basic steps to conduct a Monte Carlo simulation (Brooks Box 13.1): 1. Generate the data according to the desired data generating process (DGP), with the errors being drawn from some given distribution 2. Do the regression and calculate the test statistic 3. Save the test statistic or whatever parameter is of interest 4. Go back to stage 1 and repeat N times. These four steps contain two key stages : The first stage is the specification of the (data generating) model. This model may be either a pure time series model or a structural model. o o Pure time series models are simpler to implement. A full structural model is harder because it requires (in addition) the specification of the DGP for the explanatory variables also. Once the time series model is selected, the next choice refers to the probability distribution specified for the errors. Usually, standard normal draws are used, although any other empirically plausible distribution (e.g., Student s t) could also be used. The second stage involves estimation of the parameter of interest in the study. The parameter of interest might be, for example, the value of a coefficient in a regression, or the value of an option at its expiry date. It could instead be the value of a portfolio under a particular set of scenarios governing the way that the prices of the component assets move over time. The quantity N is known as the number of replications, and this should be as large as is feasible. The central idea behind Monte Carlo is that of random sampling from a given distribution. If the number of replications is set too small, the results will be sensitive to odd combinations of random number draws. It is also worth noting that asymptotic arguments apply in Monte Carlo studies as well as in other areas of econometrics. The results of a simulation study will be equal to their analytical counterparts (assuming that the latter exist) asymptotically. 4
Describe ways to reduce Monte Carlo sampling error There are two basic ways to reduce sampling error a) Increase the sample size, b) Employ a variance reduction technique; aka, acceleration method Sampling variation scales with the square root of the sample size Suppose that the value of the parameter of interest for replication i is denoted xi. If the average value of this parameter is calculated for a set of, say, N=1,000 replications, and another researcher conducts an otherwise identical study with different sets of random draws, a different average value of x is almost guaranteed. This situation is akin to the problem of selecting only a sample of observations from a given population in standard regression analysis. The sampling variation in a Monte Carlo study is measured by the standard error estimate, Sx: = var( ) where var(x) is the variance of the estimates of the quantity of interest over the N replications. It can be seen from this equation that to reduce the Monte Carlo standard error by a factor of 10, the number of replications must be increased by a factor of 100. As a result, to achieve acceptable accuracy, the number of replications may have to be set at an infeasibly high level. An alternative way to reduce Monte Carlo sampling error is to use a variance reduction technique. Two of the intuitively simplest and most widely used variance reduction methods are: Antithetic variate technique: This involves taking the complement of a set of random numbers and running a parallel simulation on those so that the covariance is negative, and therefore the Monte Carlo sampling error is reduced. Control variate technique: This involves using employing a variable similar to that used in the simulation, but whose properties are known prior to the simulation. The effects of sampling error for the problem under study and the known problem will be similar, and hence can be reduced by calibrating the Monte Carlo results using the analytic ones. Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. The antithetic variate technique involves taking the complement of a set of random numbers and running a parallel simulation on those. For example, if the driving stochastic force is a set of T N (0,1) draws, denoted ut, for each replication, an additional replication with errors given by ut is also used. It can be shown that the Monte Carlo standard error is reduced when antithetic variates are used. This is illustrated here: 5
Suppose that the average value of the parameter of interest across two sets of Monte Carlo replications is given by = ( + )/2 where x 1 and x 2 are the average parameter values for replications sets 1 and 2, respectively. The variance of will be given by: ( ) = 1 4 ( ( ) + ( ) + 2 (, )) If no antithetic variates are used, the two sets of Monte Carlo replications will be independent, so that their covariance will be zero, i.e. var( ) = 1 4 (var( ) + var( )) However, the use of antithetic variates would lead the covariance to be negative, and therefore the Monte Carlo sampling error to be reduced. It may appear that the reduction in Monte Carlo sampling variation from using antithetic variates will be huge since, by definition, (, ) = (, ) = 1. It is however important to remember that the relevant covariance is between the simulated quantity of interest for the standard replications and those using the antithetic variates. The perfect negative covariance is between the random draws (i.e. the error terms) and their antithetic variates. For eg. in the context of option pricing, the production of a price for the underlying security constitutes a non-linear transformation of ut. Therefore, the covariances between the terminal prices of the underlying assets based on the draws and based on the antithetic variates will be negative, but not 1. Several other variance reduction techniques that operate using similar principles are available, including stratified sampling, moment-matching and low discrepancy sequencing. Low discrepancy sequencing is also known as quasi-random sequences of draws. These involve the selection of a specific sequence of representative samples from a given probability distribution. Successive samples are selected so that the unselected gaps left in the probability distribution are filled by subsequent replications. The result is a set of random draws that are appropriately distributed across all of the outcomes of interest. The use of low-discrepancy sequences leads the Monte Carlo standard errors to be reduced in direct proportion to the number of replications rather than in proportion to the square root of the number of replications. o For example, to reduce the Monte Carlo standard error by a factor of 10, the number of replications would have to be increased by a factor of 100 for standard Monte Carlo random sampling, but only 10 for low-discrepancy sequencing. 6