RNGs in options pricing
Presented by Yu Zhang
Outline

Options



What is option?
Kinds of options
Why options?
Options pricing Models
 Monte Carlo Methods




Apply to European options
Disadvantages
Speed up
Least-Squares Monte Carlo method
 Quasi-Monte Carlo methods

Option?
Option maybe a good choice.
Option[1]

Call options: 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.
C(S, T)=max(S-X, 0)

Put options: 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.
P(S, T)=max(X-S, 0)
Kinds of options

European options: can only be exercised at
the expiration date T.

American options: can be exercised at any

Asian options: the strike price is the average
time up to and including the expiry date, T. So, it
is much more difficult to value.
price of the asset over a period of time,
computed by collecting the daily closing price
over the life of the option.
Why option?

Options for hedging

Options for speculating
What determine the value 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, .
 The annual rate of return for risk-free
investment, r.
Option pricing models

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. (Fastest)

Binomial Tree
Memory-intensive because it requires an iterative computing
process.

Monte Carlo models
flexible computational tools to calculate the value of options
with multiple sources of uncertainty or with complicated
features.

The Monte Carlo method was first
suggested as a way to price options in
1977 by Phelim Boyle in his paper:
“Options: A Monte Carlo Approach”[2]
Monte Carlo Method

With the Monte Carlo technique, we try to
evaluate the value of E[f(YT)]. (Expectation of a
function of a random variable)
1
N
N
lim  f (Y )  E[ f (Y )]
N  

n 1
n
T
The quality of the random number generator
typically determines the quality of the
simulation.
Apply to European call options

Get 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
St t  St exp[( r   2 / 2)t   t Z ]
where Z is a standard random variable, i.e. Z~N(0,1).
Apply to European call options
Get n terminal values V(ST)
V ( STn )  max( STn  X ,0)
Apply to options European call options
V ( ST , t )  e  r (T  t )
1
N
N
n
V
(
S
 T)
n 1
 Average the cumulative results and discount 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.
Disadvantage
Requires running many simulations based on
random series of events, so it is the most timeconsuming.
 The convergence of Monte Carlo methods is
slow and it is hard to determine the error
terms.

Speed up[3]
Speed up
Box-Muller Random Number Generator
 a simulation core that provides computational
resources for iteration,
 a stochastic volatility computing module based
on the GARCH* model
 a post processing module.
e.g. for averaging intermediate option prices.

GARCH* is a model for error variance, which is
widely used in Financial Forecast.
Apply to American options[4]

The optimal exercise strategy is fundamentally
determined by the conditional expectation of the
payoff from continuing to keep the option alive.

Monte Carlo simulation for an American option
has a “Monte Carlo on Monte Carlo” feature
that makes it computationally complex.
Least-Squares Monte Carlo method[5]

At each exercise time point, option holders
compare the payoff for immediate exercise with
the expected payoff for continuation.

If the payoff for immediate exercise is higher,
then they exercise the options. Otherwise, they
will leave the options alive.

The expected payoff for continuation is
conditional on the information available at that
time point.
LSM

After get the option value V, we perform
regression of V as a function of the polynomials
X, X2, …, Xm for some small value of m which is
called basic function;
i.e. approximate Vk by a least squares fit of these
polynomials in X. Hence we use this regressed
value in deciding whether to exercise early.

It performs better than other Monte Carlo
methods in high dimensional cases.
Quasi-Monte Carlo[6]

The use of low discrepancy sequences (Sobol
sequences) in Monte Carlo method leads to
what is known as quasi-Monte Carlo method.

Advantage:

It is more accurate than traditional Monte Carlo
methods, and has better convergence property.

QMC simulations are well suited to parallel computing.
So, it can provide rapid solutions for financial market.
Quasi-Monte Carlo vs. Monte Carlo
The accurate value of the option is 22.772,
which was computed using a finite difference
lattice.
Parallel QMC algorithm[7]
Result of parallel algorithm
References
[1] Random numbers in Financial Mathematics:Valuing Financial Options. Peter Duck,
University of Manchester, ENGLAND. September 11, 2007.
[2] Options: A Monte Carlo approach. Boyle, Phelim P. Journal of Financial Economics, 4
(1977), P 323-338.
[3] Design and implementation of a high performance financial Monte-Carlo simulation
engine on an FPGA supercomputer. Xiang Tian Benkrid, K. ICECE Technology,
2008. Dec. 2008, P 81-88
[4] Pricing American Options using Monte Carlo Methods. Johan Tysk. Jun 2009.
Department of Mathematics. Uppsala University.
[5] Valuing American Options by Simulation: a Simple Least-Squares Approach. Longstaff,
F. A., Schwartz, E. S., 2001. Review of Financial Studies 14 (1).
[6] Multi-asset derivative pricing using quasi-random numbers and Monte Carlo
simulation. George Levy, Numerical Algorithms Group. Oct. 2002.
[7] 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.
Thanks~