RNGs in options pricing

Presented by Yu Zhang

OutlineOptions

What is option? Kinds of options Why options?

Options pricing ModelsMonte Carlo Methods

Apply to European options Disadvantages Speed up

Least-Squares Monte Carlo methodQuasi-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 optionsEuropean options: can only be

exercised at the expiration date T.

American options: can be exercised at any time up to and including the expiry date, T. So, it is much more difficult to value.

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.

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 modelsThe 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 MethodWith the Monte Carlo technique, we

try to evaluate the value of E[f(YT)]. (Expectation of a function of a random variable)

The quality of the random number generator typically determines the quality of the simulation.

)]([)(1

1lim T

N

nn

N

YfEYfN

Apply to European call optionsGet n trajectories of the form St+1, …, ST, where

each period corresponds to one quarter. Path 1: S1

t+1, S1t+2, …, S1

T

Path 2: S2t+1, S2

t+2, …, S2T

… Path n: Sn

t+1, Snt+2, …, Sn

T

where Z is a standard random variable, i.e. Z~N(0,1).

])2/exp[( 2 ZttrSS ttt

)0,max()( XSSV nT

nT

Get n terminal values V(ST)

Apply to European call options

Apply to options European call options

N

n

nT

tTrT SV

NetSV

1

)( )(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.

DisadvantageRequires running many simulations

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

The convergence of Monte Carlo methods is slow and it is hard to determine the error terms.

Speed up[3]

Speed upBox-Muller Random Number Generatora simulation core that provides

computational resources for iteration,a stochastic volatility computing module

based on the GARCH* modela 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.

LSMAfter 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~