Pricing Financial Derivatives Using Grid Computing

Vysakh Nachiketus

Melita Jaric

College of Business Administration and

School of Computing and Information Sciences

Florida International University, Miami, FL

Zhang Zhenhua

Yang Le

Chinese Academy of Sciences, Beijing

ROAD MAP

Motivation

★Why financial derivatives

★Why the pricing of financial derivatives is complex

★Why distributed environment

★Why Monte Carlo or Binomial Method

Proposed Frame Work

★Implement Monte Carlo and Binomial Methods for

European, American, Asian and Bermuda Options in grid computing environment

★Given current price, estimate the future stock option value by implementing

Monte Carlo or Binomial Method

★Provide a framework for correlating the processing speed with the portfolio performance

Conclusion

2009 Financial Derivatives Proposal

Motivation

Why Financial Derivatives?

★Building block of a portfolio

★Current Importance/Relevance

★Complexity of algorithms

★Spreading the market risk and control

Why is pricing of financial derivatives complex?

★Uncertainty implies need for modeling with Stochastic Processes

★High volume, speed and throughput of data

★Data integrity cannot be guaranteed

★Complexity in optimizing several correlated parameters

2009 Financial Derivatives Proposal

Why distributed environment?

★Time is money ★Grid computing is more economical than supercomputing★Exploit data parallelism within a portfolio★Exploit time and data precision parallelism for a given algorithm

Why Monte Carlo or Binomial Method?

★Ability to model Stochastic Process ★Ubiquitous in financial engineering and quantum finance★They have obvious parallelism build into them, since they use two dimensional

grid (time, RV) for estimation★For higher dimensions Monte Carlo Method converges to the solution more quickly

than numerical integration methods ★Binomial Method is more suitable for American Options

Motivation

2009 Financial Derivatives Proposal

Standard options

★Call, put

★European, American

Exotic options (non standard)

★More complex payoff (ex: Asian)

★Exercise opportunities (ex: Bermudian)

Types of options

2009 Financial Derivatives Proposal

Black Scholes Equation & Stochastic Processes

★Integration of statistical and mathematical models

★For example in the standard Black-Scholes model, the stock price evolves asdS = μ(t)Sdt + σ(t)SdWt.

where μ is the drift parameter and σ is the implied volatility

★To sample a path following this distribution from time 0 to T, we divide the time interval into M units of length δt, and approximate the Brownian motion over the interval dt by a single normal variable of mean 0 and variance δt.

★The price f of any derivative (or option) of the stock S is a solution of the following partial-differential equation:

2009 Financial Derivatives Proposal

★In the field of mathematical finance, many problems, for instance the problem of finding the arbitrage-free value of a particular derivative, boil down to the computation of a particular integral.

★When the number of dimensions (or degrees of freedom) in the problem is large, PDE's and numerical integrals become intractable, and in these cases Monte Carlo methodsoften give better results. For large dimensional integrals, Monte Carlo methods convergeto the solution more quickly than numerical integration methods, require less memory , have less data dependencies and are easier to program.

★The idea is to use the result of Central Limit Theorem to allow us to generate a random set of samples as a valid representation of the previous value of the stock.“The sum of large number of independent and identically distributed random variables will be approximately normal.”

Monte Carlo method

2009 Financial Derivatives Proposal

Binomial Method

2009 Financial Derivatives Proposal

Grid Computing

2009 Financial Derivatives Proposal

Monte Carlo Vs. Difference Method

2009 Financial Derivatives Proposal

drift = mu*delt;sigma_sqrt_delt = sigma*sqrt(delt);S_old = zeros(N_sim,1);S_new = zeros(N_sim,1);S_old(1:N_sim,1) = S_init;for i=1:N % timestep loop% now, for each timestep, generate info for% all simulationsS_new(:,1) = S_old(:,1) +...S_old(:,1).*( drift + sigma_sqrt_delt*randn(N_sim,1) );S_new(:,1) = max(0.0, S_new(:,1) );% check to make sure that S_new cannot be < 0S_old(:,1) = S_new(:,1);%% end of generation of all data for all simulations% for this timestepend % timestep loop

MATLAB program for Monte Carlo

2009 Financial Derivatives Proposal

function [Pmean, width] = Asian(S, K, r, q, v, T, nn, nSimulations, CallPut) dt = T/nn;Drift = (r - q - v ^ 2 / 2) * dt;vSqrdt = v * sqrt(dt);pathSt = zeros(nSimulations,nn); Epsilon = randn(nSimulations,nn);St = S*ones(nSimulations,1);% for each time stepfor j = 1:nn; St = St .* exp(Drift + vSqrdt * Epsilon(:,j)); pathSt(:,j)=St;endSS = cumsum(pathSt,2);Pvals = exp(-r*T) * max(CallPut * (SS(:,nn)/nn - K), 0); % Pvals dimension: nSimulations x 1Pmean = mean(Pvals);width = 1.96*std(Pvals)/sqrt(nSimulations);

Elapsed time is 115.923847 seconds.price = 6.1268

MATLAB program for Asian Options

2009 Financial Derivatives Proposal

★Define Stock Input as a 7-tuple

( Ticker, Price, Low, High, Close,

Change, Volume)

★Select the ones that satisfy specified criteria

★Use hashing to assign each stock to a particular

processor

★Create a dynamic storage management database

★Collect and correlate data

★Update portfolio

Data Management

2009 Financial Derivatives Proposal

Data Processing System

2009 Financial Derivatives Proposal

http://www.gemstone.com/pdf/GIFS_Reference_Architecture_Grid_Data_Management.pdf

★Provide this system to individual investors through cloud computing.

★Provide not only option pricing, but also the information about the option that

comes from different sources (Internet, Bloomberg, Wall Street journal) . This

information will be used to in conjunction with the Monte Carlo method to create

new estimate for the particular stock.

★Implement more advanced algorithms, such as Time Warping, and develop data

structures that would be dynamic and flexible to accommodate storage and

searches on streaming data.

Tentative RoadMap

2009 Financial Derivatives Proposal

We propose to develop a software system for scientific applications in finance

with following characteristics:

★Runs in distributed environment

★Efficiently processes and distributes data in real time

★Efficiently implements current financial algorithms

★Modular and scales well as the number of variables increases

★Processes multivariable algorithms better than a sequential time system

★Expends logically for more complex systems

★Scales well for cloud computing so that even a small investor can afford to use it

★Provides an efficient and easy to use infrastructure for evaluation of current

research

Conclusion

2009 Financial Derivatives Proposal

1. Peter Forsyth, “An Introduction to Computational Finance Without Agonizing Pain”

2. Guangwu Liu , L. Jeff Hong, "Pathwise Estimation of The Greeks of Financial

Options”

3. John Hull, “Options, Futures and Other Derivatives”

4. Kun-Lung Wu and Philip S. Yu, “Efficient Query Monitoring Using Adaptive Multiple

Key Hashing”

5. Denis Belomestny, Christian Bender, John Schoenmakers, “True upper bounds for

Bermudan products via non-nested Monte Carlo”

6. Desmond J. Higham, “ An Introduction to Financial Option Valuation”

Reference

2009 Financial Derivatives Proposal

2009 Financial Derivatives Proposal

Thank You

2009 Financial Derivatives Proposal