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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