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Term Project Report

Monte Carlo methods and Quasi-Monte Carlo

methods in options pricing

CIS 5930 Random Number Generator

Yu Zhang

[email protected]

mailto:[email protected]

Monte Carlo methods and Quasi-Monte Carlo methods in options pricing

Abstract:

In this project, we introduced different kinds of options. In today’s financial market,

to estimate the price of options is a very important problem. However, options

pricing is complex to compute. One of the most effective methods in estimating

options price is Monte Carlo method. We presented several evaluation methods base

on this theory.

Keywords:

Options; Monte Carlo methods; Quasi-Monte Carlo methods; Options pricing.

1. Introduction

As the development of the financial market, option pricing [1] becomes a more

important issue in this area. Many different methods are used in evaluate the price

of options. At early time, the primary methods for pricing options are binomial trees

and other lattice methods (trinomial trees). However, in the real-world applications,

options pricing has a property of dynamics. It is difficult to evaluate because of some

restrictive assumptions. Under this circumstance, many complex computation

theories are developed to improve the efficiency and speed of pricing.

Monte Carlo simulation is one of the most popular tools in option pricing. This

method bases on the Monte Carlo integration theory in random number generation.

It simulates paths for asset prices and then estimates the options price with a fast

convergence rate with less memory. However, in some simpler situations, this

method is not the best solution because it is very time-consuming and

computationally intensive.

Many papers presented methods to improve the speed of Monte Carlo methods. For


example, researchers preferred to use hardware to speed up this method such as

using Field Programmable Gate Arrays. This idea can make the Monte Carlo method

run much faster. We will introduce this kind of methods in session 4.

However, another difficulty of using Monte Carlo to value the options is the “Monte

Carlo on Monte Carlo” problems when it used in American options. The optimal

exercise for American options strategy is fundamentally determined by the

conditional expectation of the payoff from continuing to keep the option alive. So, it

is much more complex. The Least Squares Monte Carlo (LSM) algorithm is an

effective way to solve this problem. It performs regression to value the price. Session

5, we will discuss this method in detail.

After Monte Carlo methods used in finance in 1977 by Boyle [2], Paskov and Traub [14]

used quasi-Monte Carlo methods to estimate the price of a collaterized mortgage

obligation. Since then, researchers derived many algorithms of using Quasi-Monte

Carlo methods to value the price. The convincing results show that Quasi-Monte

Carlo methods can provide better results than Monte Carlo methods, especially in

high dimension cases.

Besides that, another advantage of Quasi-Monte Carlo methods is they are well

suited to parallel computing. After distributing tasks to several processors, it can

speed up significantly. So, the method can provide rapid solutions for financial

market. This property makes the Quasi-Monte Carlo method popular in options

pricing.

A brief introduction of options and Monte Carlo and Quasi-Monte Carlo methods are

given in session 2 and session 3 respectively. Then, in session 4, we will mention

algorithm of Monte Carlo methods in European options. Session 5 is about Monte

Carlo methods used in American options. After that, we will present Quasi-Monte

Carlo methods in options pricing and give the conclusion.


2. Options

In finance, an option is a contract between buyer and seller. It gives the holder the

right, but not the obligation, to buy or to sell a particular property on or before the

expiration time, at certain price.

There are two options: call option and put option. Call option gives the holder the

right, but not the obligation to buy the underlying, S, at a certain date, T, for a certain

price, known as the exercise (or strike) price, X. On the other hand, put option gives

the holder the right, but not the obligation to sell the underlying, S, at a certain date,

T, for a certain price, known as the exercise (or strike) price, X.

There are many different kinds of options in the financial market. The most common

options are European options, American options and Asian options. The simplest

option is the European option; it can only be exercised at the expiration date T. The

value of the European call option is:

C(S, T) = max(S-X, 0).

The value of the European put options is:

P(S, T) = max(X-S, 0).

Figure 1 shows the relation between the underlying and the price of options.

Different from European options, the American options can be exercised at any time

up to and including the expiry date, T. So, it is much more difficult to value than

European options. For the Asian options, the strike price is the average price of the

asset over a period of time, computed by collecting the daily closing price over the

life of the option. Options can be used for hedging or speculating by the entities in

financial market.


a

b

Figure 1. the relation between the underlying and the price of options.

The value of an option, V, is determined by the granted price (strike price), X; the

current price, S; the time to the expiration date, T; the volatility of the underlying

asset, and the annual rate of return for risk-free investment, r.

The Black-Scholes, Binomial Tree and Monte Carlo models are the three model

people often use to value the price. The Black-Scholes only can be used for pricing

European options, because it does not have the flexibility to calculate pricing of

options that are exercised early such as American options. However, the advantage

of this method is it is the fastest one to compute the price. Binomial Tree can be used

in most of the options. But it is memory-intensive because it requires an iterative

computing process. Monte Carlo models are flexible computational tools to calculate

the value of options with multiple sources of uncertainty or with complicated

features. So, since it has been proposed in 1977, it became a popular method in this

area. Many various algorithms are derived based on this theory.


3. Monte Carlo methods and Quasi-Monte Carlo methods

The general problem for Monte Carlo methods is estimate the multidimensional

numerical integration:

μ=∫¿t

❑

f (u )du.

Monte Carlo methods approximate the value ofμ by choosing a point set Pn = {u0, u1,

…, un-1} belong to [0, 1) and calculate the average value of the function f over Pn.

Here, the set of points are independent and uniformly distributed over [0, 1)t.

The quality of the random number generator determines the accuracy of the

simulation. However, random sampling is almost impossible for a computer. Actually,

researchers use a pseudorandom number generator to choose these points, such as

linear congruential methods and Lagged Fibonacci generators. Box-Muller method is

also a popular method people used in generating pseudorandom numbers. We will

discuss and use the method in the later part of this paper.

Similar to Monte Carlo methods, the goal of Quasi-Monte Carlo methods is estimate

the integral value. However, the difference between the two theories is the way to

choose the points. Researchers want to use sequence which do not from random

origin and obtain the results which have the lowest possible extreme or mean square

discrepancy. So, instead of use pseudorandom numbers, it uses low-discrepancy

sequences such as Halton sequences and Sobol sequences.

The discrepancy of sequences used for the quasi-Monte Carlo method is bounded by

a constant time(logN )s

N, while for random sequence used in the Monte Carlo

method, the discrepancy has an order of convergence √loglogN2N

. Obviously, the


accuracy of the Quasi-Monte Carlo method increases faster than that of the Monte

Carlo method. So, Quasi-Monte Carlo method has more advantages if the integrand

is smooth, and the number of dimensions s of the integral is small. Besides that,

some important properties let it be an important technique in option pricing such as

easy to parallel.

4. Monte Carlo methods in European options

Now, let’s talk about the Monte Carlo methods in European options. Here, we just

use it in European call options to show the algorithm.

4.1 Monte Carlo methods in European call options [3]

First, it gets n trajectories of the form St+1, …, ST, where each period corresponds to

one quarter.

Path 1: S1t+1, S1

t+2, …, S1T;

Path 2: S2t+1, S2

t+2, …, S2T;

…

Path n: Snt+1, Sn

t+2, …, SnT ,

whereSt+∆t=St exp [( r−σ22 )∆ t+σ √∆ t Z ].

Z is a standard random variable, i.e. Z~N (0, 1). Figure 2 shows the Sample paths

generated by crude Monte Carlo.


Figure 2. Sample paths generated by crude Monte Carlo.

Second, it computes n terminal values V(ST) by:

V (Stn )=max (S tn−x ,0).

Finally, it averages the cumulative results and discounts the value to the present to

get an estimate for the value of the option. Here, the principle of the time value of

money is used. For example, if you want to receive $100 at T, then at an earlier time t

it is worth $100e-r(T-t). r is the compound rate.

Although this algorithm can provide satisfactory results, it requires running many

simulations based on random series of events, so it is the most time-consuming.

Besides that, the convergence of Monte Carlo methods is slow and it is hard to

determine the error terms.

4.2 Ways to speed up

Because the Monte Carlo methods have the flaw of large time-consuming,

researcher derived many methods to speed up this algorithm. Software

implementation is much lower than hardware implementation. One of the efficiency

methods is to use Field Programmable Gate Arrays (FPGA). It has a high performance


of a dedicated hardware solution of an algorithm.

Zhang [11] derived a FPGA-based Monte-Carlo simulation core. The results show that,

by using FPGA, we can get about 25 times speed-up compared to the results by using

a 1.5GHz computer. Thomas [12] implemented five hardware architectures for Monte-

Carlo based simulations. The results show that these implementations are 80 times

faster than running software on a 2.66GHz PC. Baxter [13] mentioned that by using

Maxwell, a supercomputer with 64 FPGA nodes, one can get an over 300 times faster

performance.

Figure 3. The Box-Muller method.

Tian [4] presented a new architecture using the Box-Muller method for the hardware

generation of Gaussian random numbers. The box-Muller method is a random

number generator which can provide two sets of statistic independent random

numbers by given two independent realizations (u0 and u1) of a uniform random

variable over the interval [0, 1), and a set of intermediate functions f, g1 and g2.

Here,


f (u0 )=√−2× ln (u0).

g1 (u1 )=sin (2 π u1 ).

g2 (u1 )=cos (2π u1 ).

Then, we can get two sample points x1 = f(u0)g1(u1), x2 = f(u0)g2(u1). Figure 3 shows the

architecture. Then, the author used the Box-Muller method into his Generic

architecture of a Monte-Carlo simulation engine. Figure 4 shows the hardware

architecture.

Figure 4. Generic architecture of a Monte-Carlo simulation engine.

This architecture includes the simulation core which provides computational

resources for iteration; a stochastic volatility computing module based on the GARCH

model and a post processing module (e.g. for averaging intermediate option prices).

Here, GARCH is a model for error variance, which is widely used in Financial Forecast.

Such architecture can provide a 600 times faster performance.


5. Monte Carlo methods in American options [5]

The optimal exercise for American options strategy is fundamentally determined by

the conditional expectation of the payoff from continuing to keep the option alive.

So, it is more complex than European options. Monte Carlo simulation for an

American option has a “Monte Carlo on Monte Carlo” feature that makes it

computationally complex.

The basic idea of this problem is compare the payoff for immediate exercise with the

expected payoff for continuation at each exercise time point. If the payoff for

immediate exercise is higher, the options holder exercises the options. Otherwise,

they will leave the options alive. The expected payoff for continuation is conditional

on the information available at that time point.

One of the most effective algorithms is LEAST SQUARES MONTE CARLO (LSM).

Tsitsiklis [6] and Longstaff [7] presented this algorithm to American options. Instead of

using the future expectation, this method uses a least squares interpolation. The

same as the crude Monte Carlo methods, the method starts with N random paths

(Skn, tn), where 1<=k<=N and tn = ndt. Then, it evaluates the price by “rolling-back” on

these paths. The point value is computed as follows:

Bases on the paths, we can get

Fn+1k =F (Sn+1

k , t n+1)

at point (Sn+1k , t n+1

). If X = Snk is the current options value,

Y=e−rdt F(Sn+1k ,t n+1)

is the value of deferred exercise. We can do regression of Y as a function of the


polynomialsX , X2 ,…, Xm, where m is a small value. Then, use the regressed value

we got as the expected value of deferred exercise. Now, we can decide whether it is

worth to exercise the options early. The experiment results show that LSM performs

better than other Monte Carlo methods especially in high dimensional cases.

6. Quasi-Monte Carlo methods in options pricing

6.1 Quasi-Monte Carlo methods in options pricing

Although the crude Monte Carlo methods are broadly used, they have several flaws

for option pricing. For example, the pseudorandom sequence may not have enough

statistical randomness. They only provide probabilistic error bounds and have errors

that decrease according to the square root of the number of samples. Actually,

researchers showed that the discrepancy of the samples may also be useful in

obtaining other bounds. Bases on the analysis above, many options pricing methods

are derived by using Quasi-Monte Carlo method, which uses the low-discrepancy

sequences.

Birge [8] presented an early quasi-Monte Carlo approach to options pricing and

demonstrated the improved estimates in simple options. These results showed that

quasi-Monte Carlo methods have advantage in simple model. Moreover, according to

the author, “the advantage may be even greater in more complex models which

require substantial computation for each function evaluation. They may also be

applicable to evaluating path based options due to their excellent average case

behavior which is an order of magnitude better than standard Monte Carlo”.

After that, many researchers followed this idea and do many different experiments.

The results showed that Quasi-Monte Carlo sequences can provide accurate option

price approximations for a variety of options. For example, Levy [9] did some

experiments between Monte Carlo methods and Quasi-Monte Carlo methods.


For the Quasi-Monte Carlo method, the author used Sobol sequences. He showed

that the Sobol sequence always gives better results than pseudo-random sequences.

The result shows in table 1. Here, the accurate value of the option is 22.772. From

the table we can see, when the number of points is small (500), the accuracy of

Quasi-random is larger than pseudo-random. As the number of points went large,

both of the two algorithms have better performance. However, the Quasi-random

method was always better than the pseudo-random method.

Table 1. Performance of Quasi-random method and Pseudo-random method.

6.2 Parallel Quasi-Monte Carlo methods

Besides the accuracy, the Quasi-Monte Carlo methods also have another advantage.

The QMC simulations are well suited to parallel computing. So, it can provide rapid

solutions for financial market.

Chen [10] implemented a distributed Quasi-Monte Carlo algorithm for options pricing.

According to the user, in order to gain an ideal parallel Quasi-Monte Carlo algorithm,

people should distribute the computation tasks to processors according to their

computation ability. So, the first step of the algorithm is information collecting about

the entire computing resources. If distributing the tasks blindly, the load of the


processors cannot be balanced. So, we cannot get the best performance. After that,

we can implement the parallel algorithm as shown in table 2.

Table 2. Algorithm of parallel Quasi-Monte Carlo Methods.

In the experiment, the author did 1,000,000 simulations on seven processors. The

result shows in table 3. From the table we can see, when the number of simulations

is not fairly large, the parallel algorithm does not have advantage. However, if there

are a large number of simulations, the algorithm has enormous superiority.

Table 3. Time to do Quasi-Monte Carlo simulations. (x=100000)


7. Conclusion

In this project, we want to learn the Monte Carlo methods in options pricing. By

studying the concepts of options, we know the options pricing is a useful but

complex problem in financial area.

Although there are many different kinds of models we can use the value the price of

options, Monte Carlo methods are the best choice. However, the crude Monte Carlo

method has some flaws such as the time-consuming. In order to solve this problem,

we can use hardware implementation.

Quasi-Monte Carlo methods is widely used in this area. It can produce satisfactory

results by using low-discrepancy sequences. Parallel Quasi-Monte Carlo methods can

speed up so significantly that can provide the results rapidly. The experiment results

show that these algorithms can provide satisfactory performance.


Reference

[1] Peter Duck. Random numbers in Financial Mathematics: Valuing Financial

Options. University of Manchester, ENGLAND. September 11, 2007.

[2] Boyle, Phelim P. Options: A Monte Carlo approach. Journal of Financial

Economics, 4 (1977), P 323-338. (I did not get the electronic version; I read this

paper in Strozier library.)

[3] Sibel KAPLAN. MONTE CARLO METHODS FOR OPTION PRICING. 2008

[4] Xiang Tian Benkrid, K. Design and implementation of a high performance

financial Monte-Carlo simulation engine on an FPGA supercomputer. ICECE

Technology, 2008. Dec. 2008, P 81-88

[5] Johan Tysk.Pricing American Options using Monte Carlo Methods. Jun 2009.

Department of Mathematics. Uppsala University.

[6] Tsitsiklis, J. A., and B. Van Roy. 1999. Optimal stopping of Markov processes:

Hilbert space theory, approximation algorithms, and an application to pricing

highdimensional financial derivatives. IEEE Transactions on Automatic Control

44:1840–1851.

[7] Longstaff, F., and E. Schwartz. 2001, Valuing American Options by Simulation: A

Simple Least-Squares Approach. The Review of Financial Studies 14 (1): 113–147.

[8] Birge, J.R. \Quasi-Monte Carlo approaches to option pricing," Technical Report 94-

19. Department of Industrial and Operations Engineering, University of Michigan,

Ann Arbor 1994.

[9] Levy, G F, Computational Finance, Numerical Methods for Pricing Financial

Instruments, To be published 2003


[10] Distributed Quasi-Monte Carlo Algorithm for Option Pricing on HNOWs Using

mpC. Gong Chen Thulasiraman, P. Thulasiram, R.K. Simulation Symposium,

2006. 39th Annual, 2-6 April 2006.

[11] Zhang, G.L., Leong, P.H.W., Ho, C.H., Tsoi, K.H.,Cheung, C.C.C., Lee, D.-U., Cheung,

R.C.C. and Luk,W., Reconfigurable Acceleration for Monte-Carlobased Financial

Simulation. In Field-Programmable Technology, 2005. Proceedings. 2005 IEEE

International Conference on, (2005), 215 - 222.

[12] Thomas, D.B., Bower, J.A. and Luk, W., Hardware architectures for Monte-Carlo

based financial simulations. In Field Programmable Technology, 2006. FPT

2006. IEEE International Conference on, (2006), 377 - 380.

[13] Baxter, R., Booth, S., Bull, M., Cawood, G., Perry, J., Parsons, M., Simpson, A.,

Trew, A., McCormick, A., Smart, G., Smart, R., Cantle, A., Chamberlain, R. and

Genest, G., Maxwell - a 64 FPGA Supercomputer. In Adaptive Hardware and

Systems, 2007. AHS 2007. Second NASA/ESA Conference on, (2007), 287-294

[14] S.P. Paskov. New methodologies for valuing derivatives. In S. Pliska and M.

Dempster, editors, Mathematics of Derivative Securities. Cambridge University

Press, Isaac Newton Institute, Cambridge, 1996.

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