Define Stock Input as a 7-tuple ( Ticker, Price, Low, High

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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
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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

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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

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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

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Types of options

• Standard options
• Call, put
• European, American
• Exotic options (non standard)
• More complex payoff (ex Asian)
• Exercise opportunities (ex Bermudian)

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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 s(t)SdWt.
• where µ is the drift parameter and s 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 dt, and approximate the Brownian
  motion over the interval dt by a single normal
  variable of mean 0 and variance dt.
• The price f of any derivative (or option) of the
  stock S is a solution of the following
  partial-differential equation

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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.

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Binomial Method
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Grid Computing
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Monte Carlo Vs. Difference Method
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MATLAB program for Monte Carlo
drift mudelt sigma_sqrt_delt
sigmasqrt(delt) S_old zeros(N_sim,1) S_new
zeros(N_sim,1) S_old(1N_sim,1) S_init for
i1N timestep loop now, for each timestep,
generate info for all simulations S_new(,1)
S_old(,1) ... S_old(,1).( drift
sigma_sqrt_deltrandn(N_sim,1) ) S_new(,1)
max(0.0, S_new(,1) ) check to make sure that
S_new cannot be lt 0 S_old(,1) S_new(,1)
end of generation of all data for all
simulations for this timestep end timestep
loop
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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
Sones(nSimulations,1) for each time
step for j 1nn St St . exp(Drift
vSqrdt Epsilon(,j)) pathSt(,j)St end
SS cumsum(pathSt,2) Pvals exp(-rT)
max(CallPut (SS(,nn)/nn - K), 0) Pvals
dimension nSimulations x 1 Pmean
mean(Pvals) width 1.96std(Pvals)/sqrt(nSimula
tions) Elapsed time is 115.923847
seconds. price 6.1268
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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

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Data Processing System
http//www.gemstone.com/pdf/GIFS_Reference_Archite
cture_Grid_Data_Management.pdf
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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.

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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
  

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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

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• Thank You

2009 Financial Derivatives Proposal