Term Project Report
Monte Carlo methods and Quasi-Monte Carlo
methods in options pricing
CIS 5930 Random Number Generator
Yu Zhang
yz08@fsu.edu
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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
3. Monte Carlo methods and Quasi-Monte Carlo methods
The general problem for Monte Carlo methods is estimate the multidimensional
numerical integration:
μ = ∫[0,1)t f(u)du.
Monte Carlo methods approximate the value of μ 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
√loglogN
2N
. 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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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, S1t+2, …, S1T;
Path 2: S2t+1, S2t+2, …, S2T;
…
Path n: Snt+1, Snt+2, …, SnT ,
r−σ2
where St+∆t = St exp[(
2
) ∆t + σ√∆tZ].
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
Second, it computes n terminal values V(ST) by:
V(Stn ) = max(Stn − 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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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(2πu1 ).
g 2 (u1 ) = cos(2πu1 ).
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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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
k
k
Fn+1
= F(Sn+1
, t n+1 )
k
at point (Sn+1
, t n+1). If X = Snk is the current options value,
k
Y = e−rdt F(Sn+1
, t n+1 )
is the value of deferred exercise. We can do regression of Y as a function of the
polynomials X, X 2 , … , X m , 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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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)
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
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
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
[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.
Monte Carlo methods and Quasi-Monte Carlo methods in options pricing
[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.