Turun kauppakorkeakoulu • Turku School of Economics

REVIEW OF MONTE CARLO METHODS IN QUANTITATIVE FINANCE

Practical comparison of Monte Carlo and quasi-Monte Carlo simulations in option valuation

Bachelor’s Thesis Accounting and Finance

Author: Joonas Häkkinen

Supervisor: M.Sc. Matti Niinikoski

4.4.2013 Turku

TABLE OF CONTENTS

1   INTRODUCTION ...................................................................................................... 5  

1.1   Preface ............................................................................................................ 5  1.2   Objective, scope and exclusions ..................................................................... 5  1.3   Thesis walkthrough ........................................................................................ 6  

2   BASICS OF OPTION VALUATION.....................................................................7  

2.1   Introduction to options ................................................................................... 7  2.2   The Black-Scholes-Merton model .................................................................. 7  

3   MONTE CARLO METHODS ................................................................................. 10  

3.1   Introduction to Monte Carlo methods .......................................................... 10  3.2   General properties of Monte Carlo models .................................................. 11  

3.2.1   Upsides ............................................................................................. 11  3.2.2   Downsides ........................................................................................ 12  

3.3   Improving the Monte Carlo method ............................................................. 12  3.3.1   Pseudorandom methods ................................................................... 12  3.3.2   Low-discrepancy sequences ............................................................ 13  3.3.3   Beyond quasi-Monte Carlo .............................................................. 14  

4   OPTION PRICING USING MONTE CARLO SIMULATIONS ............................ 16  

4.1   Model implementation ................................................................................. 16  4.2   Results .......................................................................................................... 18  

4.2.1   Simulation results and discussion .................................................... 18  4.2.2   Execution times and technical notes ................................................ 19  

5   SUMMARY .............................................................................................................. 21  

REFERENCES ............................................ ERROR! BOOKMARK NOT DEFINED.  

TABLE OF FIGURES

Figure 1   Fundamental process flow of the Monte Carlo method ............................ 10  

Figure 2   Difference in distribution between pseudorandom and low-discrepancy numbers ............................................................................................... 14  

Figure 3   European call option pricing using Monte Carlo simulations of up to 10 000 iterations. ................................................................................. 18  

Figure 4   European call option pricing using Monte Carlo simulations of 10 000 to 100 000 iterations. ............................................................................... 19  

Figure 5   Processing times of Monte Carlo simulations of 2 000 to 100 000 iterations. ............................................................................................. 20  

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1 INTRODUCTION

1.1 Preface

Monte Carlo methods are mathematical algorithms widely used in physics and mathe-matics. The first such methods were developed in late 40’s, with David B. Hertz pub-lishing the first financial application in 1964. Later, Boyle (1977) followed with an ap-plication to option pricing. Monte Carlo methods are especially useful in problems with no analytic solution. The problem at hand is first described to a computer mathematics processor and then reiteratively solved numerous times with the baseline inputs generat-ed randomly for each round of iteration. Thus Monte Carlo methods can also be seen as a departure from estimation based on historical data to a simulation that creates a distri-bution consisting of random realistic outcomes. The properties of this distribution are then used to whatever end one seeks to achieve, e.g., to price an option.

Financial applications of Monte Carlo methods are numerous. Examples include common problems such as risk management, portfolio optimisation and derivatives pricing. However, they are useful in almost any problems that include uncertainty, e.g., real options and project present value calculations. Monte Carlo algorithms are typically complex, highly customised and slow to develop and compute. To balance these down-sides, they offer more reliable results as degrees of freedom and the amount of uncer-tainty grow. In practice, the methods are often configured to use tens or hundreds of variables.

Technological evolution has transformed the once expensive, clumsy and rare math-ematical tool – the computer – into an everyday item. Even personal workstations now-adays have the power and software required for remarkably sophisticated mathematical tasks. McLeish (2005) likens this development to the mathematical equivalent of the invention of the printing press; advanced mathematical sciences, such as finance, are no longer limited by the availability of tools.

1.2 Objective, scope and exclusions

The main objective of this work is to highlight the differences between Monte Carlo and quasi-Monte Carlo methods in financial applications. I aim to amplify theoretical differ-ences with a demonstrative study and examples in option pricing. While I use the Black-Scholes-Merton model as a benchmark, it is not an objective of this thesis to derive or otherwise study the model. Also the mathematical procedure to generate the Faure se-quences is out of scope here.

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1.3 Thesis walkthrough

I begin by covering the basics of options and their valuation using the Black-Scholes-Merton model in section 2. This model is later used as a benchmark to assess the accu-racy of simulated option values. Section 3 introduces the reader to the variety of the Monte Carlo methods. I provide a review about the positive and negative properties of the methods and discuss techniques to improve them along with a brief view of what the future might hold. Throughout the work I make the attempt to link theoretical pieces to finance and option pricing where applicable.

Section 4 consists of developing a model to simulate option prices and a demonstra-tion of the results of this model applied to three different flavours of the Monte Carlo methods. I present the results graphically and provide analysis. In addition, I give a view of the processing times and some technical notes.

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2 BASICS OF OPTION VALUATION

2.1 Introduction to options

Derivatives are financial contracts of obligation or option. Various types of derivatives exist with vastly different properties, yet they all are similar in that they do not have an intrinsic value. Rather, their value is derived from the underlying asset, which is the object of the contract. Options are perhaps the most widely known types of derivatives, not least because of their use as bonus compensation and the resultant negative publici-ty. Other types of derivatives worth mentioning include futures, forwards and swaps. Underlying assets are most commonly stocks, bonds, commodities, interest rates or cur-rencies. (Hull 2012, 1–18.)

Options are contracts for the right to buy or sell the underlying asset at a specific price. The time at which the option can be executed is either at the end of maturity (Eu-ropean option) or at any time before expiry (American option). An option to buy a se-curity is termed a call option and one to sell is a put option. An important aspect is that options do not obligate the buyer to anything after settling the purchase price. This ef-fectively limits the buyer’s maximum loss. For a seller this would turn the other way around, with the contract price being their maximum profit and the option being actual-ly an obligation. (Hull 2012, 7–9.) For most part, this paper is written from the buyer’s point of view.

Options and other derivatives are traded both on public exchanges and in the over-the-counter market (OTC market). The former are similar to stock exchanges and the latter consist of individual parties dealing directly with each other, which allows for greater flexibility in terms of the financial instruments used. OTC market dwarfs the volume of exchange-traded derivatives many times over (in the magnitude of $600 bil-lion to some $100 billion, respectively). (Hull 2012, 2–4.)

Derivatives markets are populated mainly by three types of traders. Hedgers use the-se instruments to limit the amount of potential losses on other investments. Speculators aim to make a profit directly from derivatives trading and arbitrageurs look for pricing errors that enable them to make riskless profit. Hedge funds are widely known to use derivatives in all of these ways and in large scale. (Hull 2012, 9–16.)

2.2 The Black-Scholes-Merton model

The Black-Scholes-Merton model of option pricing is widely used in both research and the financial industry today. As an indication of its impact, Scholes and Merton were

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awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1997. Across the literature I have reviewed for this work, the BSM model appears as a theoretical benchmark for subsequent option valuation methods. Thus I have opted to use it as such in this work. Following is a short introduction to the fundamentals of this model.

Researchers Black, Scholes and Merton came up with their renowned option pricing formula in 1973. Black and Scholes derived the original equation using the assumptions made in the capital asset pricing model (CAPM). Merton further finessed the work and ridded it of the CAPM assumptions.1 (Hull 2012, 299, 307–308.) The Black-Scholes-Merton model’s assumptions after Merton (1973) and Hull (2012, 309) are as follows:

• Options are not dominated by nor dominate any securities, i.e., riskless arbi-trage opportunities do not exist.

• Markets are frictionless; there are no transaction costs or differential taxes. Trading is continuous. Borrowing and short selling are unrestricted.

• Stock prices follow geometric Brownian motion with constant expected return and constant, or at most a known function of time, volatility.

• Underlying securities pay no dividends. 2 • The risk-free rate is constant or a know function of time and the same for all

maturities. Since options offer only naught or positive return to the holder at maturity, we can

write the discounted return on a European call option as

c = Ε[max(ST − K , 0)]e−rT (2.1)

and the discounted return on a European put option as

p = Ε[max(K − ST , 0)]e−rT , (2.2)

where ST is the stock price at time T (which is the time to maturity), K is the strike

price at which the option can be executed and r is the risk-free rate of return. (Hull 2012, 315.)3

1 See the original papers: Black, Fischer – Scholes, Myron (1973) The Pricing of Options and Corporate

Liabilities. Journal of Political Economy, Vol. 81, 637–659, and Merton, Robert (1973) Theory of Ra-

tional Option Pricing. Bell Journal of Economics and Management Science, Vol. 4, 141-183. 2 Merton (1973) extended the model to include also dividends and to account for volatility and risk-free

rate that vary as a known function of time. 3 To accomodate for later use in Monte Carlo simulations, I have used the symbol ST for the stock price

at maturity rather than the S0erT in Hull (2012).

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Equations (2.1) and (2.2) are theoretical option prices. Since ST is a stochastic varia-

ble, these equations can be expressed as integrals over the stock price at maturity. These are the integrals that Monte Carlo simulations for option pricing attempt to estimate. To give an example, I have adapted the integral function of a European call option price from Hull (2012, 329) for use in this work:

c = e−rT (ST − K ) f (ST )dSTK

∞

∫ (2.3)

In essence, this function takes every possible payoff from the option (ST − K ) and weights those payoffs with their respective probabilities f (ST ) .

An option and the underlying stock can be combined to form a portfolio that is essen-tially riskless for a short period of time. The return on the portfolio must equal the risk-free rate of return lest there exist a riskless arbitrage opportunity in the market. From this basis Robert Merton was able to derive the price of an option. (Hull 2012, 299.)

Finally, after Hull (2012, 313), the actual Black-Scholes-Merton equations for the prices of European call and put options, respectively, are:

c = S0N(d1)− Ke−rT N(d2 ) (2.4)

and

p = Ke−rT N(−d2 )− S0N(−d1) , (2.5)

where

d1 =ln(S0 /K )+ (r +σ

2 / 2)Tσ T

(2.6)

and

d1 =ln(S0 /K )+ (r −σ

2 / 2)Tσ T

, (2.7)

where S0 is the current stock price, σ is the stock’s volatility and the function N(x) is the

cumulative probability distribution function for a standardised normal distribution. There exist a variety of option pricing models that are more sophisticated versions of

the Black-Scholes-Merton model or make use of other approaches to the problem. The-se include binomial models, stochastic volatility models and models based on alterna-tive (and often more realistic) non-normal distributions, along with others. Also, numer-ical methods can be distinguished as their own group, which Monte Carlo methods are a part of.

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3 MONTE CARLO METHODS

3.1 Introduction to Monte Carlo methods

In the inceptive4 article of the Monte Carlo method5 Nicholas Metropolis and Stanislaw Ulam (1949) describe it as ”essentially, a statistical approach to the study of differen-tial equations”. This is a fair description and pays homage to the fact that statistical sampling had already been around for decades. Whereas this was earlier a laborious and impractical task, involving crowds of people with mechanical calculators, the gist of Ulam’s invention was to use the early emerging computers to take care of the rou-tines. (Metropolis 1987.)

The classic or crude Monte Carlo method is a numerical procedure based on random numbers. Drawing numbers at random from a certain statistical distribution and using them as inputs to the function under scrutiny produces a sample of the actual outcome distribution. The properties of this resultant distribution can then be studied with stand-ard statistical methods. (Metropolis & Ulam 1949.)

Figure 1 Fundamental process flow of the Monte Carlo method

In the scope of this work, Figure 1 can be seen to represent option pricing using the Monte Carlo method. First, a random number from the normal distribution is selected. This distribution is assumed to match the distribution of stock returns. Then the payoff in this particular scenario is calculated. Finally, after numerous reiterations of this pro-cess, a sample of the option’s payoff distribution emerges. From this distribution we can, e.g., compute the current price of the option. (Hull 2012, 446–448.)

Since this particular example is fairly simple in nature and can be solved analytically (e.g. with the Black-Scholes-Merton formula), it needs to be stressed that the Monte

4 Physicist Enrico Fermi had develop essentially the same method already in the 1930s, but at the time did

not publish anything on it. 5 The name Monte Carlo method was coined by Metropolis, knowing that Ulam had an uncle who would

often borrow money from relatives and hit the casinos of Monte Carlo.

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Carlo method is very flexible and frequently used when it comes to more complicated problems. It has a number of attractive properties, which I discuss in the next chapter.

Historically speaking, the Monte Carlo method has gained popularity and has been adapted to more uses as the computing power available to researchers has increased. It has played an important role in a wealth of sciences, including mathematics, physics, chemistry and finance. Indeed, Metropolis (1987) ties together the development of the electronic computer and the Monte Carlo method as the inception of experimental mathematics. Moreover, for the usefulness of a method that gains in speed and accuracy as we develop ever more computational power, it is hard to see even a theoretical upper limit.

Nowadays, multiple flavours of the original Monte Carlo method exist. These are of-ten divided into two classes: crude Monte Carlo methods and quasi-Monte Carlo meth-ods. The former are based on Ulam’s idea of random number sampling, while the latter are based on deterministic, low-discrepancy number sequences as input. I describe both more thoroughly in section 3.3.

3.2 General properties of Monte Carlo models

3.2.1 Upsides

Monte Carlo models easily accept different statistical distributions of input variables. In financial applications, this can be useful when dealing with security returns, for exam-ple. Asset returns appear to exhibit fat-tailed and asymmetric distributions in reality, as opposed to the general theoretical assumption of normal distributions. Monte Carlo models are useful in highlighting the tail-risk and other statistical features associated with this empirical finding. (Fabozzi, Stoyanov & Rachev 2013.)

A major advantage of Monte Carlo methods is that they are very flexible. Complex models can be implemented with relative ease. In the case of option pricing, this allows, e.g., for the pricing of path-dependent options, such as Asian and lookback options. (Hull 2012, 448.) Part of this flexibility extends to the sampling process, where replac-ing the pseudorandom sampling can be replaced with low-discrepancy number sequenc-es offering faster convergence and better reliability.

In option pricing, using Monte Carlo methods allows for some of the Black-Scholes-Merton assumptions to be relaxed. According to Hull (2012, 448) the model can ac-commodate any stochastic process, and thus we can let the risk-free interest rate to fluc-tuate or allow for dividends.

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Often a problem will not consist of one but several random variables. This in itself can lead to the need for a numerical method as an analytic solution may not exist or the analytic method could be too burdensome. Hull (2012, 452) points out that as the di-mensions of a problem increase, the processing time taken by a Monte Carlo simulation will increase in a linear fashion. This is in contrast to other numerical models that will typically exhibit exponential increase in running time as dimensions increase.

3.2.2 Downsides

Perhaps the biggest downside of all Monte Carlo methods is the need for a large amount of iterations (see e.g. Boyle 1977; Metropolis 1987). Reiterative techniques are often referred to as brute force techniques: every additional numerical operation to be reiter-ated increases the computing time in a linear fashion. Lowering the error bound has even more extreme effect: for a 50% reduction in the error of a sample average (which, e.g., an option price is), quadrupling the number of iterations is required (Chance 2011).

The ability to assess the reliability of results is clearly crucial in any model based on statistical sampling. Unfortunately the crude Monte Carlo models are confined to prob-abilistic error bounds as they are based on random numbers. However, quasi-Monte Carlo models enjoy deterministic error bounds as they make use of deterministic, rather than random sequences. (Joy, Boyle & Ken 1996.)

In situations where an analytic expression of the problem under scrutiny is not avail-able, the error bound for a crude Monte Carlo method is probabilistic, rather than de-terministic. According to Fabozzi et al (2013), the Monte Carlo variability is indicative of the approximation error of the simulation. Naturally, the approximation error is re-duced by increasing the number of iterations. However, it is important to note that this does not apply to a possible error in the model being approximated. This can be seen as a general weakness and is certainly an aspect that practitioners of the Monte Carlo methods need to take into account.

3.3 Improving the Monte Carlo method

3.3.1 Pseudorandom methods

Complexity theory stipulates that for any sequence of numbers to be random, it must not be distinguishable from a uniform distribution by any efficient procedure. This is effec-tively an objective view of the randomness of a sequence. Computer random number

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generators make use of this by generating deterministic sequences of numbers with a random seed value at the beginning of the procedure. These sequences are called pseu-dorandom to differentiate from truly random, unrepeatable sequences. (Goldreich 2008, 284–285.)

As discussed earlier, the Monte Carlo methods suffer from a quadrupling of the sam-ple when a halving of the error needs to be achieved. It would thus be convenient to be able to decrease the error and improve the rate of convergence through other methods. A discussion of two such methods follows.

First, I consider the antithetic variate method. Suppose we draw a random number. In the case of a symmetric distribution, we are equally likely to draw the number on the opposite side of the distribution mean at an equal distance from it. When drawing from a normal distribution with a mean of zero, this means that we are equally likely to draw either ε or -ε. This is clearly a very efficient way to double the random sample size. (Chance 2011.) I demonstrate the antithetic method in chapter 4.

Another way to reduce the variance of the simulation results is the control variate method. This method requires the availability of a deterministic solution to a model that is similar to the one being simulated. The first model is also simulated and the error to the correct value computed. This error is then used to correct the values from the simu-lated model. In option pricing the control variate model would require the use of two options with a high covariance. (Chance 2011.)

3.3.2 Low-discrepancy sequences

Pseudorandom numbers are likely to exhibit non-uniform distribution, especially when the sample size is small. However, for purposes of Monte Carlo models it would be use-ful to have a uniform distribution already at small samples; this would dramatically im-prove convergence. This can be solved through low-discrepancy sequences. They are deterministic number sequences that are designed to distribute uniformly. In essence, the next number being generated is conveniently placed in the largest gap in the se-quence.

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Figure 2 Difference in distribution between pseudorandom and low-discrepancy numbers

Figure 2 illustrates the distributional difference between pseudorandom and Faure sequence numbers. Notice how the pseudorandom sequence leaves random empty areas in the unit square even at a sample size of 10 000. Fox (1986) discusses also Sobol’ and Halton sequences in addition to Faure sequences. The latter appear to be best suited for high-dimensional problems, which would be beneficial in many financial applications. Generally, low-discrepancy sequences degrade as dimensions increase and start to ex-hibit clustering. Fox (1986) suggests that the Faure sequence is useful at least to the 25th dimension.

An intuitive feature of the Faure sequences is brought up by Joy et al (1996): They highlight the fact that stress tests can be run using these low-discrepancy sequences in Monte Carlo simulations. The uniform distribution guarantees that even extreme scenar-ios are accounted for. One could hypothesize that this is also beneficial in simulating the tail-risk of assets.

3.3.3 Beyond quasi-Monte Carlo

The quasi-Monte Carlo methods can be further improved. Variance reduction tech-niques enhance the performance of quasi-Monte Carlo models through similar means as with the crude Monte Carlo models. For example, the control variate method discussed earlier gains from the more accurate simulation approximations and in turn the com-bined model as a whole performs better. (Joy et al 1996.)

L’Ecuyer (2009) describes an improvement called randomised quasi-Monte Carlo. This method essentially uses the quasi-Monte Carlo as a variance reduction technique.

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The low-discrepancy sequences from therein are mixed with random numbers and then used as an input sample. This method results in an increased processing time but on the other hand in better error estimates.

Numerical methods are of great importance not only in finance but in many other sciences as well. They are subject to active on-going research. This doesn’t apply only to the Monte Carlo methods: If one was to speculate on this, many kinds of emerging technologies play part. The concept of machine learning implies that possible one day we will develop a system that watches the stock market and autonomously determines the causalities at play. Neural networking attempts to create similar predictive and adap-tive systems. Indeed, the world of finance is connected to cutting-edge science and technology in a fascinating way!

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4 OPTION PRICING USING MONTE CARLO SIMULATIONS

4.1 Model implementation

I demonstrate three different Monte Carlo models to price an arbitrary European call option and compare the results to the price attained from the Black-Scholes-Merton model. The models under scrutiny differ from each other in sampling techniques: I im-plement the classic or crude Monte Carlo model based on pseudorandom numbers, an antithetic model and a low-discrepancy model based on Faure sequences.

As results, I present the errors compared to the analytic solution as a function of the number of iterations. Errors are relative absolute averages of 10 simulation runs for each number of iterations.6 I begin with the same amounts of simulations as in Joy et al (1996), from 0 up to 10 000, with steps of 200. Then I extend my analysis to cover runs of 10 000 through 100 000 simulations, with steps of 2 000. Throughout this chapter I use graphs to illustrate the otherwise very unintuitive amount of data generated.

For the European call option, I use the following properties: • Current price S0 = 100

• Strike price K = 100 • Risk-free rate r = 0.1 • Time to maturity T = 1 • Volatility σ = 0.3 The Black-Scholes-Merton formula (2.4) yields a price benchmark of 16.734 as the

result for such an option.7 Recall that the Black-Scholes Merton model includes the assumption that stocks fol-

low a geometric Brownian process with constant drift and variance. In mathematical terms ST GBM (µ,σ ) , where µ and σ are constants. The most common model for

stock prices is (Hull 2012, 287):

dSS

= µdt +σ dz GBM (µ,σ ) (4.1)

6 I do this in order to smooth out the resultant graphs to focus the reader’s attention to the effect of the

number of simulations rather than the random volatility of the curve. The value 10 is but a heuristic result

based on a few trials. The quasi-Monte Carlo trials are not reiterated in this way since their seed values

are deterministic and thus do not exhibit randomness. 7 For soundness, these values are taken from Joy et al (1996).

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Hull (2012, 448) argues that better accuracy is achieved by simulating ln S instead of S. Thus, using Itō calculus and substituting expected return µ̂ for drift µ , this is:

d lnS = µ̂ − σ 2

2⎛⎝⎜

⎞⎠⎟dt +σdz (4.2)

Now the Brownian factors dt and dz can be unwound through discretization:

ln S(t + Δt)− ln S(t) = µ̂ − σ 2

2⎛⎝⎜

⎞⎠⎟Δt +σε Δt , (4.3)

where ε is a random sample from a normal distribution with a mean of zero and a standard deviation of one. Then the equation is exponentiated, while writing Δt = T since µ̂ and σ are assumed constants, thus arriving at the simulative model of stock

prices:

ST = S0 exp µ̂ − σ 2

2⎛⎝⎜

⎞⎠⎟T +σε T

⎡

⎣⎢

⎤

⎦⎥ . (4.4)

Finally, recall equation (2.1) for discounted option return. Substituting for ST the final

form to be used in Monte Carlo simulations here emerges:

c = Ε max S0 exp µ̂ − σ 2

2⎛⎝⎜

⎞⎠⎟T +σε T

⎡

⎣⎢

⎤

⎦⎥ − K , 0

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥e−rT (4.5)

or

c = 1NerT

max S0 exp µ̂ − σ 2

2⎛⎝⎜

⎞⎠⎟T +σεi T

⎡

⎣⎢

⎤

⎦⎥ − K , 0

⎛

⎝⎜⎞

⎠⎟i=1

N

∑ , (4.6)

where ε i N(0, 1) and N is the number of simulations. This model can be referred to as

the discounted expectation over the terminal price distribution. For the quasi-Monte Carlo simulations I use Faure sequences. Fox (1986) demon-

strates the superiority of Faure sequences through to the 25th dimension compared to the Sobol’ and Halton sequences. I implement Faure sequences as described by Joy et al (1996). However, as the mathematical derivation of these sequences is out of scope for this work, I refer the reader to their work for a thorough demonstration.

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4.2 Results

4.2.1 Simulation results and discussion

I present the results for the three types of simulations discussed in the previous section. Simulation runs of up to 100 000 iterations are covered. The results are consistent with those of Joy et al (1996) with the exception of the antithetic variate method, which is not covered in their work. However, these results appear to match the theoretical expec-tations.

Figure 3 European call option pricing using Monte Carlo simulations of up to 10 000 iterations.

From Figure 3 it is evident that the quasi-Monte Carlo method outperforms other methods in terms of convergence toward the correct price. It attains a level of error that is multitudes less than that of the crude method and less than half of that of the antithet-ic method. Observe also that using the Faure sequences we gain faster convergence with small number of simulations and visibly less variation in the amount of error.

Being relatively simple compared to the quasi-Monte Carlo model, it is interesting to see, that the antithetic variate model performs remarkably close to the quasi-model. I move on to demonstrate the models’ performance with a larger number of simulations.

0 2000 4000 6000 8000 10000

6

4

2

0

Number of simulations (N)

Averageabsoluterelativeerror(%)

Crude

Antithetic

Quasi

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Figure 4 European call option pricing using Monte Carlo simulations of 10 000 to 100 000 iterations.

Figure 4 gives a view of the invisible tails of the previous figure, with number of simulations ranging from 10 000 to 100 000. Note that the scale runs only through to one per cent, opposed to the six per cent maximum in Figure 3.

The results here are quite similar to those of fewer simulations. The quasi-Monte Carlo model continues to pull ahead of the other two models, this time outrunning also the antithetic model nearly tenfold. Again, the antithetic method appears favourable to the crude method in light of simplicity and the approximate halving of the error. Never-theless, with N >10 000 all models constantly exhibit an error less than one per cent. It seems likely that in practical situations other factors will introduce more error than the Monte Carlo method.

4.2.2 Execution times and technical notes

The Monte Carlo methods are numerical techniques that rely their usefulness on brute force of the executing computer system. In light of this, it is important to consider the amount of time spent running the simulations; in today’s fast-paced market environment this maybe a crucial factor. In the following, I have run single simulations with number of iterations ranging from 2 000 to 100 000 for each model discussed and plotted the execution times.

10000 50000 100000

1

0.8

0.6

0.4

0.2

Number of simulations (N)

Averageabsoluterelativeerror(%)

Crude

Antithetic

Quasi

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Figure 5 Processing times of Monte Carlo simulations of 2 000 to 100 000 itera-tions.

We see right away from Figure 5 that the relationship between number of simulations and processing time is linear as expected. Even with N = 100 000 all simulations com-plete in a matter of few seconds. The antithetic model is expectedly approximately two times slower, as it includes the additional multiplication by -1 to attain the mirrored sample value. The performance of the quasi-Monte Carlo simulation depends strongly on the precision at which the low-discrepancy sequence is introduced. For comparison, at full precision the simulation takes 11,8 seconds to complete, when N = 100000 . I use here a precision of 10 digits, which has no discernible effect on the results.

I used Wolfram Mathematica 9 for modelling and running these simulations on a late-2012 MacBook Pro Retina laptop computer running OS X 10.8.3. For reference, the machine is equipped with a 2.7GHz quad-core i7 processor, 16GB of 1600MHz DDR3 SDRAM memory and an Nvidia GeForce GT 650M graphics processor8 with 1GB of GDDR5 memory. Performance-wise it represents the top-end of laptops at the time of writing. Vastly improved performance could be achieved with more powerful desktop computers or by using distributed computing.

8 Mathematica 9, and most other modern computational software, make use of the excess processing

power provided by graphics cards. This results in a major performance increase in numerical tasks.

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5 SUMMARY

The Monte Carlo methods continue to be popular and valuable tools in finance and oth-er sciences. Originally developed during the Second World War they have been contin-uously perfected to attain faster convergence of results and better estimation of error bounds. In light of their computative intensity the methods gain from the exponential development of processing power we continuously witness. They remain subject to ac-tive research, and one would not be overly speculative in expecting more sophistication and more widespread application of the Monte Carlo methods in future.

I demonstrated three flavours of Monte Carlo models applied to option pricing. Of these, the quasi-Monte Carlo model based on low-discrepancy sequences was clearly superior. However, I find the crude Monte Carlo model improved with the antithetic variate method to be valuable tool due to its relative simplicity and better results com-pared to the most basic crude model. My results are consistent with the literature I have reviewed for this thesis.

Finally, I review some processing time statistics for these simulations. I find that it is possible to achieve a good level of accuracy combined with very reasonable running times using standard commercially available tools. This is an encouraging result and further strengthens my view that the Monte Carlo methods are only at the beginning of their journey.

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