Week 7 Quantitative Analysis of Financial Markets Simulation Methods

Transcription

1 Week 7 Quantitative Analysis of Financial Markets Simulation Methods Christopher Ting Christopher Ting : christopherting@smu.edu.sg : : LKCSB 5036 November 21, 2017 Christopher Ting QF 603 November 21, /30

2 Lesson Plan 1 Introduction 2 Techniques 3 Examples 4 Bootstrapping 5 Capital Risk 6 Takeaways Christopher Ting QF 603 November 21, /30

3 Introduction A The Monte Carlo Method technique is often used in econometrics and finance when the properties of a particular estimation method are not known. 1 Quantifying the simultaneous equations bias induced by treating an endogenous variable as exogenous. 2 Determining the appropriate critical values for a Dickey-Fuller test. 3 Determining what effect heteroscedasticity has upon the size and power of a test for autocorrelation. 4 Pricing of exotic options 5 Determining the effect on financial markets of substantial changes in the macroeconomic environment. 6 Stress-testing risk management models o determine whether they generate capital requirements sufficient to cover losses in all situations. Christopher Ting QF 603 November 21, /30

4 A Procedures Conducting Simulations Experiments 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. A N is the number of replications, and should be as large as is feasible. A The central idea behind Monte Carlo is that of random sampling from a given distribution. A Therefore, if the number of replications is set too small, the results will be sensitive to odd combinations of random number draws. Christopher Ting QF 603 November 21, /30

5 QA-16 Simulation Methods Chapter 13. Chris Brooks, Introductory Econometrics for Finance, 3rd Edition (Cambridge, UK: Cambridge University Press, 2014). A Describe the basic steps to conduct a Monte Carlo simulation. A Describe ways to reduce Monte Carlo sampling error. A Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. A Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective. Christopher Ting QF 603 November 21, /30

6 QA-16 Simulation Methods (cont d) A Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them. A Describe the bootstrapping method and its advantage over Monte Carlo simulation. A 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. A Describe situations where the bootstrapping method is ineffective. A Describe disadvantages of the simulation approach to financial problem solving. Christopher Ting QF 603 November 21, /30

7 Random Number Generation B We generate numbers that are a continuous uniform(0,1) according to the following recursion: then y i+1 = (ay i + c) modulo m, i = 0, 1,..., T R i+1 = y i+1 /m for i = 0, 1,..., T for T random draws, where y 0 is the seed (the initial value of y), a is a multiplier and c is an increment. B An example of a discrete uniform number generator would be a die or a roulette wheel. Computers generate continuous uniform random number draws. B These are pseudo-random numbers B The U(0, 1) draws can be transformed into other standard distributions (e.g normal, t, etc). Christopher Ting QF 603 November 21, /30

8 Variance Reduction Techniques B The sampling variation in a Monte Carlo study is measured by the standard error estimate, denoted S x : var(x) S x = N where var(x) is the variance of the estimates of the quantity of interest over the N replications B In order to achieve acceptable accuracy, the number of replications may have to be set at an infeasibly high level. B An alternative way to reduce Monte Carlo sampling error is to use a variance reduction technique B Two of the intuitively simplest and most widely used methods are antithetic variates and control variates. Christopher Ting QF 603 November 21, /30

9 Antithetic Variates B One reason that a lot of replications are typically required of a Monte Carlo study is that it may take many, repeated sets of sampling before the entire probability space is adequately covered. B What is really required is for successive replications to cover different parts of the probability space. B The antithetic variate technique involves taking the complement of a set of random numbers and running a parallel simulation on those. B For each replication conducted using random draws u t, an additional replication with errors given by u t is also used. Christopher Ting QF 603 November 21, /30

10 Antithetic Variates (cont d) B Suppose that the average value of the parameter of interest across 2 sets of Monte Carlo replications is given by x = (x 1 + x 2 )/2, where x 1 and x 2 are the average parameter values for replications set 1 and 2 respectively. B The variance of will be given by: var( x) = 1 4 ( var(x1 ) + var(x 2 ) + 2cov(x 1, x 2 ) ) B If no antithetic variates are used: var( x) = 1 4 (var(x 1) + var(x 2 )) Christopher Ting QF 603 November 21, /30

11 Control Variates B The application of control variates involves employing a variable similar to that used in the simulation, but whose properties are known prior to the simulation. B Denote the variable whose properties are known by y, and that whose properties are under simulation by x. B The simulation is conducted on x and also on y, with the same sets of random number draws being employed in both cases. B Denoting the simulation estimates of x and y using hats, a new estimate of x can be derived from: x = y + (ˆx ŷ) Christopher Ting QF 603 November 21, /30

12 Control Variates (cont d) B Again, it can be shown that the Monte Carlo sampling error of this quantity, x, will typically be lower than that of x. B In fact, it can be shown that control variates will reduce the Monte Carlo variance if corr(ˆx, ŷ) > 1 var(ŷ) 2 var(ˆx) B For example, if the topic of interest is the pricing of an Asian option, the control variate approach would be given by PA = ( P ˆ A P ˆ BS ) + P BS Christopher Ting QF 603 November 21, /30

13 Dickey-Fuller Test d The Dickey Fuller test applied to some series y t is the regression y t = φy t 1 + u t d The test is one of H 0 : φ = 1 against H 1 : φ < 1. The relevant test statistic is given by τ = ˆφ 1 SE( ˆφ) d Under the null hypothesis of a unit root, the test statistic does not follow a standard distribution, and therefore a simulation would be required to obtain the relevant critical values. Christopher Ting QF 603 November 21, /30

14 Simulation d Construct the data generating process under the null hypothesis. that is, obtain a series for y that follows a unit root process. Draw a series of length T, the required number of observations, from a normal distribution. This will be the error series, so that u t N(0, 1). Assume a first value for y, i.e. a value for y at time t = 0. Construct the series for y recursively, starting with y 1, y 2, and so on: y 1 = y 0 + u 1 y 2 = y 1 + u 2 y 3 = y 2 + u 3... y T = y T 1 + u T Christopher Ting QF 603 November 21, /30

15 Calculate the test statistic, τ. Simulation (cont d) Repeat steps 1 and 2 N times to obtain N replications of the experiment. A distribution of values for τ will be obtained across the replications. Order the set of N values of τ from the lowest to the highest. The relevant 5% critical value will be the 5 th percentile of this distribution. Christopher Ting QF 603 November 21, /30

16 Simulate the Price of an Option d Although the example presented here is for a simple plain vanilla call option, the same basic approach would be used to value an exotic option. The steps are: Specify a data generating process for the underlying asset. A random walk with drift model is usually assumed. Specify also the assumed size of the drift component and the assumed size of the volatility parameter. Specify also a strike price K, and a time to maturity, T. Draw a series of length T, the required number of observations for the life of the option, from a normal distribution. This will be the error series, so that u t N(0, 1). Form a series of observations of length T on the underlying asset. Christopher Ting QF 603 November 21, /30

17 Simulate the Price of an Option (cont d) Observe the price of the underlying asset at maturity observation T. For a call option, if the value of the underlying asset on maturity date, P T < K, the option expires worthless for this replication. If the value of the underlying asset on maturity date, P T > K, the option expires in the money, and has value on that date equal to P T K, which should be discounted back to the present day using the risk free rate. Repeat steps 1 to 4 N times, and take the average value of the option over the N replications. This average will be the price of the option. d The data for the underlying can be generated according to a more empirically realistic process. Christopher Ting QF 603 November 21, /30

18 Introduction to Bootstrapping H In finance, the bootstrap is often used instead of a pure simulation. H This is mainly because financial asset returns do not follow the standard statistical distributions that are used in simulations. H Bootstrapping provides an alternative where by definition, the properties of the artificially constructed series will be similar to those of the actual series. H Bootstrapping is used to obtain a description of the properties of empirical estimators by using the sample data points themselves. H Bootstrapping is similar to pure simulation but the former involves sampling from real data rather than creating new data. Christopher Ting QF 603 November 21, /30

19 Taking Sample with Replacement H Suppose we have a sample of data, y = y 1, y 2,..., y T and we want to estimate some parameter θ. H We can get an approximation to the statistical properties of ˆθ T by studying a sample of bootstrap estimators. H We do this by taking N samples of size T with replacement from y and re-calculating ˆθ T with each new sample. H We then get a series of ˆθ estimates. H Advantage of bootstrapping: it allows the researcher to make inferences without making strong distributional assumptions. Christopher Ting QF 603 November 21, /30

20 Taking Sample with Replacement (cont d) H Instead of imposing a shape on the sampling distribution of the ˆθ s, bootstrapping involves empirically estimating the sampling distribution by looking at the variation of the statistic within sample. H We draw a set of new samples with replacement from the sample and calculate the test statistic of interest from each of these. H Call the test statistics calculated from the new samples ˆθ. H The samples are likely to be quite different from each other. H We thus get a distribution of ˆθ s. Christopher Ting QF 603 November 21, /30

21 Bootstrapping in a Regression Context H Consider a standard regression model, y = βx + u. H The regression model can be bootstrapped in two ways. Resample the Data 1 Take the data, and sample the entire rows corresponding to observation i together. The steps would then be Generate a sample of size T from the original data by sampling with replacement from the whole rows taken together. Calculate ˆβ, the coefficient matrix for this bootstrap sample. Go back to stage 1 and generate another sample of size T. Repeat this a total of N times. Christopher Ting QF 603 November 21, /30

22 Bootstrapping in a Regression Context (cont d) 2 Problem with this approach is that we are sampling from the regressors. 3 The only random influence in the regression is the errors, u, so why not just bootstrap from those? Resampling from the Residuals The steps are 1 Estimate the model on the actual data, obtain the fitted values ŷ, and calculate the residuals, û. 2 Take a sample of size T with replacement from these residuals (and call these û ), and generate a bootstrapped dependent variable: y = ŷ + û 3 Then regress the original X data on this new dependent variable to get a bootstrapped coefficient vector, ˆβ. 4 Go back to stage 2, and repeat a total of N times. Christopher Ting QF 603 November 21, /30

23 Situations where the Bootstrap will be Ineffective H If there are extreme outliers in the data, the conclusions of the bootstrap may be affected. H Use of the bootstrap implicitly assumes that the data from which the sampling is done are independent. This would not hold, for example, if the data were autocorrelated. Christopher Ting QF 603 November 21, /30

24 Financial Motivation o An important use of bootstrapping in a financial context is in the calculation of value at risk. VaR gives the amount in financial terms that is expected to be lost with a given probability over a given time interval. o E.g. 95%, 10-day VaR = $100m o How do we calculate, say, the amount of capital required to cover 95% of the daily losses on a particular position? o This is a forecasting problem considered by Hsieh (1993). o Data: daily log returns on foreign currency (against the US dollar) futures series from 22 February 1985 until 9 March 1990 (1275 observations) for the British pound (denoted BP), the German mark (GM), the Japanese yen (JY), and the Swiss franc (SF). Christopher Ting QF 603 November 21, /30

25 Which Model to Use? o The first step in the analysis is to find a model that describes the main features of the data that can be used to construct the bootstraps. o Hsieh shows that his FX returns have no linear or non-linear structure in the mean, but they have evidence of non-linear structure in the variance. o What models are examples of this type? The GARCH family. o Hsieh (1993) uses EGARCH and Autoregressive Volatility models. Christopher Ting QF 603 November 21, /30

26 Simulating by Bootstrapping o The standardised residuals (e.g. û t /ˆσ t for the EGARCH model) are tested for non-randomness; little evidence of any structure is found. o We can thus sample with replacement from these standardised residuals. o From these, we can simulate a set of future paths for σ t and then y t by bootstrapping from v t = (û t /ˆσ t ). o Hsieh simulates the future path of returns for 180 days after the end of the estimation sample. Christopher Ting QF 603 November 21, /30

27 Calculating the Capital Risk Requirements o In each case, the maximum drawdown (loss) can be calculated over a given holding period by Q = (P 0 P 1 ) number of contracts where P 0 is the initial value of the position, and P 1 is the lowest simulated price (for a long position) or highest simulated price (for a short position) over the holding period. o Repeat this procedure 2,000 times to get 2,000 maximum losses. o The maximum loss is calculated assuming holding periods of 1, 5, 10, 15, 20, 25, 30, 60, 90 and 180 days. o It is assumed that the futures position is opened on the final day of the sample used to estimate the models, 9 March o Then he takes the 90th percentile of these 2000 maximum losses to get the amount of capital required to cover losses on 90% of days. Christopher Ting QF 603 November 21, /30

28 Minimum Capital Risk Requirements Long position Short position Unconditional Unconditional No. of days AR density EGARCH AR density EGARCH BP JY Christopher Ting QF 603 November 21, /30

29 Minimum Capital Risk Requirements: Discussion o The MCRRs derived from bootstrapping the price changes themselves (the unconditional approach ) are in most cases higher than those generated from the other two methods, especially at short investment horizons. o As the holding period increases from 1 towards 180 days, the MCRR estimates from the ARV model converge upon those of the unconditional densities. On the other hand, those of the EGARCH model do not converge, even after 180 days o It can also be observed that the MCRRs for short positions are larger than those of comparative long positions. Christopher Ting QF 603 November 21, /30

30 Takeaways k It might be computationally expensive. k The results might not be precise. k The results are often hard to replicate. k Simulation results are experiment-specific. k To conclude, simulation is an extremely useful tool that can be applied to an enormous variety of problems. k The technique has grown in popularity over the past decade, and continues to do so. Christopher Ting QF 603 November 21, /30