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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, USA Zhang Zhenhua Yang Le Chinese Academy of Sciences, Beijing , China. Pricing Financial Derivatives Using Grid Computing.

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Pricing Financial Derivatives Using Grid Computing

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  1. Vysakh Nachiketus Melita Jaric College of Business Administration and School of Computing and Information Sciences Florida International University, Miami, FL, USA Zhang Zhenhua Yang Le Chinese Academy of Sciences, Beijing , China Pricing Financial Derivatives Using Grid Computing

  2. Road Map • Motivation • Why financial derivatives • Why the pricing of financial derivatives is complex • Why distributed environment • Why Monte Carlo / Binomial Method / Finite Difference Method • Proposed Work • For a given criteria continuously update a diversified portfolio of European, American, Asian and Bermuda Options • Given current price, estimate the future stock option value by implementing • Monte Carlo or Binomial Method in grid computing environment • 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 parameter 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. Binomial Method 2009 Financial Derivatives Proposal

  8. Binomial Method • The value of the replicating portfolio at time T, with stock price ST, is ΔST + erT B where Δ is the no of share and B is cash invested • At the prices ST = Sou and ST = Sod, a replicating portfolio will satisfy (Δ * Sou * erT ) + (B * erT ) = Cu (Δ * Sod * erT ) + (B * erT ) = Cd where Sou is the stock price when price goes up • Solving for Δ and B and substitute in C = ΔS + B to get the option price 2009 Financial Derivatives Proposal

  9. 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. • 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

  10. Monte Carlo method St = S0e(a – 0.5 σ 2)t + σ√tZ If V(St,t) is the option payoff at time t, then the time-0 Monte Carlo price V(S0,0) is where ST1, … , STn are n randomly drawn time-T stock prices 2009 Financial Derivatives Proposal

  11. Option Pricing Sensitivities The Greeks 2009 Financial Derivatives Proposal

  12. Grid Computing 2009 Financial Derivatives Proposal

  13. Data Management • Define Stock Input as a 7-tuple • ( Ticker, Price, Low, High, Close, Change, Volume) • Implement FAST decompression to get the actual data • Select the stocks 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. 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 Peter Forsyth 2008 2009 Financial Derivatives Proposal

  16. 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 www.fbe.hku.hk/doc/courses/tpg/mfin/2007-2008/mfin7017/Chapter_2.ppt 2009 Financial Derivatives Proposal

  17. Monte Carlo Parallelism • Iterations are independent - Embarrassingly parallel • Parallel Random Number Generation: Master Processor • SPRNG (Scalable Pseudorandom Number Generation) library function MC (S, r, T, sigma, nSamples, nP) dataRand[nSamples] = rand(); start = idP*p/nSamples; end = (idP+1)*p/nSamples; for each idP mean_value[idP] = findMC(S,E,r,T,sigma,dataRand[start,end)); end reduce(mean_value[idP]); endfunction 2009 Financial Derivatives Proposal

  18. Binomial Parallelism function Bin (S, E, r, T, sigma, nTime ) … for step = nTime:-1:1 for i=1:nStep+1 value(i) = (pUp*value(i+1)+pDown*value(i))*exp(-r*dt); price(i) = price(i+1)*exp(-sigma*sqrt(dt)); value(i) = max(value(i), price(i)-E) endfunction 2009 Financial Derivatives Proposal

  19. Binomial Parallelism 2009 Financial Derivatives Proposal

  20. Finite Difference Parallelism • Careful: dt and dS are not independent • Use Ghost points to minimize the communication • Communication comes from one side only Explicit Finite Differences 2009 Financial Derivatives Proposal

  21. System Analysis • Number of processors vs. time for each method • Computation and Communication Overlap • Communication Overhead • Design and implement an algorithm for optimizing performance for any combination of: • number of stocks • number of processors • latency 2009 Financial Derivatives Proposal

  22. Tentative Road Map • Provide this system to individual investors through cloud computing. • Implement advanced financial hedging techniques for Fixed Income, • Future Exchanges … • Address system Reliability by checking for failure, introducing • redundancy and error recovery • Address system Security by implementing encryption algorithms • Introduce different sources of information (news, internet) and trigger • warning alerts to support automated trading. 2009 Financial Derivatives Proposal

  23. 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

  24. 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” • Alexandros V. Gerbessiotis, “Architecture independent parallel binomial tree option price valuations” • Bernt Arne Odegaard, “Financial Numerical Recipes in C++” • Paul Wilmott, “Introduces Quantitative Finance” 2009 Financial Derivatives Proposal

  25. Questions, Comments, Suggestions

  26. ThankYou

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