Pricing Financial Derivatives Using Grid Computing

1. 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

2. 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

3. 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

4. Motivation • 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 2009 Financial Derivatives Proposal

5. Types of options • Standard options • Call, put • European, American • Exotic options (non standard) • More complex payoff (ex: Asian) • Exercise opportunities (ex: Bermudian) 2009 Financial Derivatives Proposal

6. Black Scholes Equation & Stochastic Processes • Integration of statistical and mathematical models • For example in the standard Black-Scholes model, the stock price evolves as • dS = 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

7. Monte Carlo method • 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 methods • often give better results. For large dimensional integrals, Monte Carlo methods converge • to 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.” 2009 Financial Derivatives Proposal

8. Binomial Method 2009 Financial Derivatives Proposal

9. Grid Computing 2009 Financial Derivatives Proposal

10. Monte Carlo Vs. Difference Method 2009 Financial Derivatives Proposal

11. MATLAB program for Monte Carlo 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 simulations S_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 < 0 S_old(:,1) = S_new(:,1); % % end of generation of all data for all simulations % for this timestep end % timestep loop 2009 Financial Derivatives Proposal

12. MATLAB program for Asian Options 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 step for j = 1:nn; St = St .* exp(Drift + vSqrdt * Epsilon(:,j)); pathSt(:,j)=St; end SS = cumsum(pathSt,2); Pvals = exp(-r*T) * max(CallPut * (SS(:,nn)/nn - K), 0); % Pvals dimension: nSimulations x 1 Pmean = mean(Pvals); width = 1.96*std(Pvals)/sqrt(nSimulations); Elapsed time is 115.923847 seconds. price = 6.1268 2009 Financial Derivatives Proposal

13. Data Management • 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 2009 Financial Derivatives Proposal

14. Data Processing System http://www.gemstone.com/pdf/GIFS_Reference_Architecture_Grid_Data_Management.pdf 2009 Financial Derivatives Proposal

15. Tentative RoadMap • 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. 2009 Financial Derivatives Proposal

16. Conclusion • 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 2009 Financial Derivatives Proposal

17. Reference • Peter Forsyth, “An Introduction to Computational Finance Without Agonizing Pain” • Guangwu Liu , L. Jeff Hong, "Pathwise Estimation of The Greeks of Financial Options” • John Hull, “Options, Futures and Other Derivatives” • Kun-Lung Wu and Philip S. Yu, “Efficient Query Monitoring Using Adaptive Multiple Key Hashing” • Denis Belomestny, Christian Bender, John Schoenmakers, “True upper bounds for Bermudan products via non-nested Monte Carlo” • Desmond J. Higham, “ An Introduction to Financial Option Valuation” 2009 Financial Derivatives Proposal

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19. Thank You 2009 Financial Derivatives Proposal