simulation methods chapter 13 of chris brook's book · die or a roulette wheel. computers...

Introduction Dickey-Fuller Test Option Pricing Bootstrapping

Simulation MethodsChapter 13 of Chris Brook’s Book

Christopher Ting

Christopher Ting

http://www.mysmu.edu/faculty/christophert/

k: [email protected]: 6828 0364

ÿ: LKCSB 5036

April 26, 2017

Christopher Ting QF 604 Week 5 April 26, 2017 1/30

http://www.mysmu.edu/faculty/christophert/

mailto:[email protected]


Table of Contents

1 Introduction

2 Dickey-Fuller Test

3 Option Pricing

4 Bootstrapping



Learning Outcomes

Design simulation frameworks to solve a variety of problems

Explain the difference between pure simulation and bootstrapping

Describe the various techniques available for reducing MonteCarlo sampling variability



Monte Carlo Method

Monte Carlo technique is often used when the properties of aparticular estimation method are not known. Examples:

1 Quantifying the simultaneous equations bias induced by treating anendogenous variable as exogenous.

2 Determining the appropriate critical values for a Dickey-Fuller test.

3 Determining what effect heteroscedasticity has upon the size andpower of a test for autocorrelation.

4 The pricing of exotic options (no an analytical pricing formula)

5 Determining the effect on financial markets of substantial changesin the macroeconomic environment.

6 “Stress-testing” risk management models to determine whetherthey generate capital requirements sufficient to cover losses in allsituations.



Conducting Simulations Experiments

In all of the above examples, the basic way that such a studywould be conducted (with additional steps and modificationswhere necessary) is as follows.

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.

N is the number of replications, and should be as large as isfeasible.

The central idea behind Monte Carlo is that of random samplingfrom a given distribution.

Therefore, if the number of replications is set too small, the resultswill be sensitive to “odd” combinations of random number draws.



Random Number Generation

We generate numbers that are a continuous uniform(0,1)according to the following recursion:

yi+1 = (ayi + c) modulo m, i = 0, 1, . . . , T

then

Ri+1 = yi+1/m for i = 0, 1, . . . , T

for T random draws, where y0 is the seed (the initial value of y), ais a multiplier and c is an increment.An example of a discrete uniform number generator would be adie or a roulette wheel. Computers generate continuous uniformrandom number draws.These are pseudo-random numbersThe U(0,1) draws can be transformed into other standarddistributions (e.g normal, t, etc).



Variance Reduction Techniques

The sampling variation in a Monte Carlo study is measured by thestandard error estimate, denoted Sx:

Sx =

√var(x)

N

where var(x) is the variance of the estimates of the quantity ofinterest over the N replications

In order to achieve acceptable accuracy, the number ofreplications may have to be set at an infeasibly high level.

An alternative way to reduce Monte Carlo sampling error is to usea variance reduction technique

Two of the intuitively simplest and most widely used methods areantithetic variates and control variates.



Antithetic Variates

One reason that a lot of replications are typically required of aMonte Carlo study is that it may take many, repeated sets ofsampling before the entire probability space is adequatelycovered.

What is really required is for successive replications to coverdifferent parts of the probability space.

The antithetic variate technique involves taking the complement ofa set of random numbers and running a parallel simulation onthose.

For each replication conducted using random draws ut, anadditional replication with errors given by −ut is also used.



Antithetic Variates (Cont’d)

Suppose that the average value of the parameter of interestacross 2 sets of Monte Carlo replications is given by

x = (x1 + x2)/2

where x1 and x2 are the average parameter values for replicationsset 1 and 2 respectively.

The variance of will be given by:

var(x) =1

4(var(x1) + var(x2) + 2cov(x1, x2))

If no antithetic variates are used:

var(x) =1

4(var(x1) + var(x2))



Antithetic Variates (Cont’d)

However, the use of antithetic variates would lead the covarianceto be negative, and therefore the Monte Carlo sampling error to bereduced.



Control Variates

The application of control variates involves employing a variablesimilar to that used in the simulation, but whose properties areknown prior to the simulation.

Denote the variable whose properties are known by y, and thatwhose properties are under simulation by x.

The simulation is conducted on x and also on y, with the samesets of random number draws being employed in both cases.

Denoting the simulation estimates of x and y using hats, a newestimate of x can be derived from:

x∗ = y + (x− y)



Control Variates (Cont’d)

Again, it can be shown that the Monte Carlo sampling error of thisquantity, x*, will typically be lower than that of x.

In fact, it can be shown that control variates will reduce the MonteCarlo variance if

corr(x, y) >1

2

√var(y)

var(x)

For example, if the topic of interest is the pricing of an Asianoption, the control variate approach would be given by

P ∗A = (PA − ˆPBS) + PBS



Deriving the Critical Values for a DF Test

Recall, that the equation for a Dickey Fuller test applied to someseries y t is the regression

yt = φyt−1 + ut

The test is one of H0: φ = 1 against H1: φ < 1. The relevant teststatistic is given by

τ =φ− 1

SE(φ)

Under the null hypothesis of a unit root, the test statistic does notfollow a standard distribution, and therefore a simulation would berequired to obtain the relevant critical values.



Deriving the Critical Values for a DF Test (Cont’d)

The simulation would be conducted in the following steps:

Construct the data generating process under the null hypothesis –that is, obtain a series for y that follows a unit root process. Thiswould be done by

– Draw a series of length T, the required number of observations,from a normal distribution. This will be the error series, so thatut ∼ 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 y1, y2, and so on:

� y1 = y0 + u1

� y2 = y1 + u2

� y3 = y2 + u3

�, . . .

� yT = yT−1 + uT



Deriving the Critical Values for a DF Test (Cont’d)

Calculate the test statistic, τ .

Repeat steps 1 and 2 N times to obtain N replications of theexperiment. A distribution of values for τ will be obtained across thereplications.

Order the set of N values of τ from the lowest to the highest. Therelevant 5% critical value will be the 5th percentile of thisdistribution.



Price of a Financial Option

Although the example presented here is for a simple plain vanillacall option, the same basic approach would be used to value anexotic option. The steps are:

– Specify a data generating process for the underlying asset. Arandom walk with drift model is usually assumed. Specify also theassumed size of the drift component and the assumed size of thevolatility parameter. Specify also a strike price K, and a time tomaturity, T.

– Draw a series of length T, the required number of observations forthe life of the option, from a normal distribution. This will be theerror series, so that ut ∼ N(0, 1).

– Form a series of observations of length T on the underlying asset.



Price of a Financial 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 maturitydate, PT < K, the option expires worthless for this replication. If thevalue of the underlying asset on maturity date, PT > K, the optionexpires in the money, and has value on that date equal to PT −K,which should be discounted back to the present day using the riskfree rate.

– Repeat steps 1 to 4 N times, and take the average value of theoption over the N replications. This average will be the price of theoption.

The data for the underlying can be generated according to a moreempirically realistic process.



Bootstrapping

In finance, the bootstrap is often used instead of a puresimulation.

This is mainly because financial asset returns do not follow thestandard statistical distributions that are used in simulations.

Bootstrapping provides an alternative where by definition, theproperties of the artificially constructed series will be similar tothose of the actual series.

Bootstrapping is used to obtain a description of the properties ofempirical estimators by using the sample data points themselves.

Bootstrapping is similar to pure simulation but the former involvessampling from real data rather than creating new data.



Bootstrapping (Cont’d)

Suppose we have a sample of data, y = y1, y2, . . . , yT and wewant to estimate some parameter θ.

We can get an approximation to the statistical properties of θT bystudying a sample of bootstrap estimators.

We do this by taking N samples of size T with replacement from yand re-calculating θT with each new sample.

We then get a series of θ estimates.

Advantage of bootstrapping: it allows the researcher to makeinferences without making strong distributional assumptions.



Bootstrapping (Cont’d)

Instead of imposing a shape on the sampling distribution of theθ’s, bootstrapping involves empirically estimating the samplingdistribution by looking at the variation of the statistic withinsample.

We draw a set of new samples with replacement from the sampleand calculate the test statistic of interest from each of these.

Call the test statistics calculated from the new samples θ∗.

The samples are likely to be quite different from each other.

We thus get a distribution of θ∗’s.



Bootstrapping in a Regression Context

Consider a standard regression model, y = βX + u.

The regression model can be bootstrapped in two ways.1 Resample the Data

1 Take the data, and sample the entire rows corresponding toobservation i together.

The steps would then be− Generate a sample of size T from the original data by

sampling with replacement from the whole rowstaken together.

− Calculate β∗, 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.

2 Problem with this approach is that we are sampling from theregressors.



Bootstrapping in a Regression Context (Cont’d)

3 The only random influence in the regression is the errors, u, so whynot just bootstrap from those?

2 Resampling from the Residuals

The steps are1 Estimate the model on the actual data, obtain the fitted values y, and

calculate the residuals, u.

2 Take a sample of size T with replacement from these residuals (andcall these u∗), and generate a bootstrapped dependent variable:

y∗ = y + u∗

3 Then regress the original X data on this new dependent variable toget a bootstrapped coefficient vector, β∗.

4 Go back to stage 2, and repeat a total of N times.



Situations where the Bootstrap will be Ineffective

If there are extreme outliers in the data, the conclusions of thebootstrap may be affected.

Use of the bootstrap implicitly assumes that the data from whichthe sampling is done are independent. This would not hold, forexample, if the data were autocorrelated.



Capital Risk Requirements (Hsieh, 1993)

Financial MotivationAn important use of bootstrapping in a financial context is in thecalculation of value at risk. VaR gives the amount in financialterms that is expected to be lost with a given probability over agiven time interval.

E.g. 95%, 10-day VaR = $100m

How do we calculate, say, the amount of capital required to cover95% of the daily losses on a particular position?

This is a forecasting problem considered by Hsieh (1993).

Data: daily log returns on foreign currency (against the US dollar)futures series from 22 February 1985 until 9 March 1990 (1275observations) for the British pound (denoted BP), the Germanmark (GM), the Japanese yen (JY), and the Swiss franc (SF).



Capital Risk Requirements (Hsieh, 1993) (Cont’d)

The first step in the analysis is to find a model that describes themain features of the data that can be used to construct thebootstraps.

Hsieh shows that his FX returns have no linear or non-linearstructure in the mean, but they have evidence of non-linearstructure in the variance.

What models are examples of this type? The GARCH family.

Hsieh uses EGARCH and Autoregressive Volatility models.

The standardised residuals (e.g. ut/σt for the EGARCH model)are tested for non-randomness; little evidence of any structure isfound.



Capital Risk Requirements (Hsieh, 1993) (Cont’d)

We can thus sample with replacement from these standardisedresiduals.

From these, we can simulate a set of future paths for σt and thenyt by bootstrapping from vt = (ut/σt).

Hsieh simulates the future path of returns for 180 days after theend of the estimation sample.



Calculating the Capital Risk Requirements

In each case, the maximum drawdown (loss) can be calculatedover a given holding period by

Q = (P0 − P1) ×number of contracts

where P0 is the initial value of the position, and P1 is the lowestsimulated price (for a long position) or highest simulated price (fora short position) over the holding period.Hsieh repeats this procedure 2,000 times to get 2,000 maximumlosses.The maximum loss is calculated assuming holding periods of 1, 5,10, 15, 20, 25, 30, 60, 90 and 180 days.It is assumed that the futures position is opened on the final day ofthe sample used to estimate the models, 9 March 1990.Then he takes the 90th percentile of these 2000 maximum lossesto get the amount of capital required to cover losses on 90% ofdays.



Minimum Capital Risk RequirementsLong position Short position

Unconditional UnconditionalNo. of daysAR density EGARCHAR density EGARCH

BP 1 0.73 0.91 0.93 0.80 0.98 1.055 1.90 2.30 2.61 2.18 2.76 3.00

10 2.83 3.27 4.19 3.38 4.22 4.8815 3.54 3.94 5.72 4.45 5.48 6.6720 4.10 4.61 6.96 5.24 6.33 8.4325 4.59 5.15 8.25 6.20 7.36 10.4630 5.02 5.58 9.08 7.11 8.33 12.0660 7.24 7.44 14.50 11.6412.87 20.7190 8.74 8.70 17.91 15.4516.90 28.03

180 11.3810.67 24.25 25.8127.36 48.02DM 1 0.72 0.87 0.83 0.89 1.00 0.95

5 1.89 2.18 2.34 2.23 2.70 2.9110 2.77 3.14 3.93 3.40 4.12 5.0315 3.52 3.86 5.37 4.36 5.30 6.9220 4.05 4.45 6.54 5.19 6.14 8.9125 4.55 4.90 7.86 6.14 7.21 10.6930 4.93 5.37 8.75 7.02 7.88 12.3660 7.16 7.24 13.14 11.3612.38 20.8690 8.87 8.39 16.06 14.6816.16 27.75

180 11.3810.35 21.69 24.2526.25 45.68JY 1 0.56 0.74 0.72 0.68 0.87 0.86

5 1.61 1.99 2.22 1.92 2.36 2.7310 2.59 2.82 3.46 3.06 3.53 4.4115 3.30 3.46 4.37 4.11 4.60 5.7920 3.95 4.10 5.09 5.13 5.45 6.7725 4.42 4.58 5.78 5.91 6.30 7.9830 4.95 4.92 6.34 6.58 6.85 8.8160 6.99 6.84 8.72 10.5310.74 13.5890 8.43 8.00 10.51 13.6114.00 17.63

180 10.9710.27 13.99 21.8622.21 27.39SF 1 0.82 0.97 0.89 0.93 1.12 0.98

5 1.99 2.51 2.48 2.23 2.93 2.9810 2.87 3.60 4.12 3.37 4.53 5.0915 3.67 4.35 5.60 4.22 5.67 7.0320 4.24 5.10 6.82 5.09 6.69 8.8625 4.81 5.65 8.12 5.90 7.77 10.9330 5.23 6.20 9.12 6.70 8.47 12.5060 7.69 8.41 13.73 10.5513.10 21.2790 9.23 9.93 16.89 13.6017.06 27.80

180 12.1812.57 22.92 21.7227.45 45.47



Minimum Capital Risk Requirements: Discussion

The MCRRs derived from bootstrapping the price changesthemselves (the “unconditional approach”) are in most caseshigher than those generated from the other two methods,especially at short investment horizons.

As the holding period increases from 1 towards 180 days, theMCRR estimates from the ARV model converge upon those of theunconditional densities. On the other hand, those of the EGARCHmodel do not converge, even after 180 days

It can also be observed that the MCRRs for short positions arelarger than those of comparative long positions.



Problems with Simulation

1 It might be computationally expensive

2 The results might not be precise

3 The results are often hard to replicate

4 Simulation results are experiment-specific.

To conclude, simulation is an extremely useful tool that can beapplied to an enormous variety of problems. The technique hasgrown in popularity over the past decade, and continues to do so.However, like all tools, it is dangerous in the wrong hands.


simulation methods chapter 13 of chris brook's book · die or a roulette wheel. computers...

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