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San Jose, California | September 30 – October 2, 2009
Computational Finance on the GPU
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Objective
• Develop understanding of what
“Computational Finance” means
• Learn who uses Computational Finance
– Why do the users need it?
– What do they do with the information?
• Take a look at some typical algorithms
• Consider the challenges and benefits of adoption
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Financial Instruments
Cash Instruments
Equities
Commodities
Fixed Income
Foreign Exchange
Derivatives
Exchange-traded
Over-the-counter
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Equities
• Share ownership
• Value determined by market
• Dividends
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Commodities
• Raw resource
– Agriculture: corn, rice
– Livestock: pork bellies
– Energy: oil, gas
– Metals: precious, industrial
• Supply and demand
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Fixed Income
• Also: Credit
• Loans and bonds
• Different rates according to duration
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Foreign Exchange
• Also: Forex or FX
• Take advantage of changes in rates
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Derivatives
• Based on one or more underlying assets
– Equities, FX, credit
• Many types of contract
– Forwards and futures
– Options
– Swaps
• Exchange-traded or Over-the-Counter (OTC)
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Example: Options
• Holder has the right to buy (call) or sell (put) the underlying asset
– By a certain date
– At a certain price
-500
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60
Pro
fit
($)
Stock price ($)
pricespot
price strike where
)0,max(
S
K
SKPayoffput
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Traders
• Trading:
– Standardised instruments
– New (often complex) instruments
• Requires models:
– Pricing
– Prediction
– Risk analysis Traders Backoffice Quants Developers
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Backoffice
• Monitoring the banks exposure
• Model all trades
– Value
– Risk
• Value-at-Risk (VaR) required for regulation
Traders Backoffice Quants Developers
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Quants
• Develop and implement models for traders
• Develop independent models for validation
• Research
• Modelling exposure and capital
Traders Backoffice Quants Developers
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Developers
• Implement models from quants
• Integrate into larger applications
– Interface to other models
– Interface to database
– Interface to user
Traders Backoffice Quants Developers
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Inputs/Outputs
Instrument parameters
Market data
Value Confidence Sensitivities
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Traders
• Determine price for trade
– Negotiate over the phone, require results fast
– Minimize out-trades (errors)
• Run positions
– Sensitivities allow trader to predict response to changes in underlying assets
– Run often to allow trader to react quickly
Traders Backoffice Quants Developers
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Backoffice
• Manage risk and capital reserves for regulation
– Large runs can take days to complete
– Accurate results allow greater control, and hence more trades
• Monitor traders’ exposure
– Run intra-day
• Greater accuracy requires longer run times
Traders Backoffice Quants Developers
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Analytic
• Some derivatives have an analytic solution
• Compare analytical result with numerical result
– Provides a Control Variate
OptionPut European for Price
)2()ln(
)2()ln(
)()(
Formula Scholes-Black2
1
1
2
02
2
01
102
2
222
TdT
TrKSd
T
TrKSd
dSdKef
rfS
fS
S
frS
t
f
rT
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Binomial Trees
• Represent possible paths of stock
• Assumptions:
– Stock has a probability p of moving up
by a certain percentage u
– Stock has a probability (1-p) of moving
down by a certain percentage d
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Binomial Trees
• Construct the tree
– Create a branch for each
time step
– At each node the stock
can either go up u% or
down d%
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Binomial Trees
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Binomial Trees on the GPU
• Work backwards in time
– Compute value at each
node
• Partition the work across
the SMs
– Overlap input data
– Fit input partition in
smem
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Finite Differences
• Solve the Partial Differential Equation iteratively
– Divide the life of the derivative into equal intervals
of length Δt
– Divide the range of stock prices [0,Smax] into equal
intervals of size ΔS
• Work backwards in time, compute the value at
each node
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Finite Differences - Implicit
• Relationship:
– Three values at t and one
value at t + Δt
• Always converges to
solution
• Requires solving
simultaneous equations
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Finite Differences - Explicit
• Relationship:
– One value at t and three
values at t + Δt
• Compute nodes in
parallel
• Can diverge from
solution
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Explicit Finite Differences on the GPU
• Partition the grid across
the SMs
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Explicit Finite Differences on the GPU
• Partition the grid across
the SMs
• Each SM requires a
“halo”
– Halo size determines how
many time steps in batch
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Explicit Finite Differences on the GPU
• Partition the grid across
the SMs
• Each SM requires a
“halo”
– Halo size determines how
many time steps in batch
– After each batch,
distribute new halos
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Monte Carlo
• Sample a random walk for the asset(s)
• Calculate the payoff of the derivative
• Repeat to get many sample payoff values
• Calculate the mean payoff
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Monte Carlo
Expected
payoff
-10
-8
-6
-4
-2
0
2
4
6
8
10
90
92
94
96
98
100
102
104
106
108
110
0 T
Pay
off
($
)
Ass
et
pri
ce (
$)
Time
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Monte Carlo on the GPU
GenerateRandom Numbers
• Distribution• Covariance
Compute payoff for each path
• Scenarios are independent
Compute statistics
• Reduction
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Monte Carlo – Multiple Kernels
parfor [0..T)
for [0..N)
genrand()
parfor [0..N)
for [0..T)
genpath()
parfor [0..N)
payoff()for [0..lg2N)
reduce()Sam
ple
s
Path
s
Payoff
s
RNG Paths Payoffs Reduce
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Monte Carlo – Single Kernel
parfor [0..N)
for [0..T)
genrand()
addtopath()
payoff()
reducestep()
for [0..lg2N)
reduce()Payoff
s
ReduceMC
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Software legacy
• Millions of lines of code
• Complex relationships between code blocks
– E.g. Primary algorithm generates paths which are
reused in multiple payoff models
• Hundreds of man-years of work
• Significant refactoring of application required
– Support/feed the parallelized algorithms
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Education
• Parallel programming is paradigm shift
– Quants are starting to rethink algorithms
– Designing or reusing different algorithms/strategies
• Reuse of libraries and frameworks
– Concentrate on the core algorithm
– Increasing number of libraries for random numbers,
linear algebra, reduction etc.
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Case Study: Equity Derivatives
2 Tesla S1070 500 CPU Cores
2.8 KWatts 37.5 KWatts
$24 K $250 K
16x Less Space
13x Lower Power
10x Lower Cost
15x Faster15 1
Source: BNP Paribas, March 4, 2009
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Case Study: Equity Derivatives
2 Tesla S1070 500 CPU Cores
2.8 kWatts 37.5 kWatts
$24 K $250 K
16x Less Space
13x Lower Power
10x Lower Cost
15x Faster15 1
Source: BNP Paribas, March 4, 2009
190x Lower
Power in Total
No need to
compromise
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Case Study: Real-time Options
3 Tesla S870 600 CPU Cores
$140 K $1,200 K
$42 K $262 K
9x Less Space
9x Lower Annual Cost
6x Lower Cost
Same Performance1 1
Figures assume:
• NVIDIA Tesla S870s with one 8-core host server per unit
• CPUs are 8-core blade servers; 10 blades per 7U
• $1,800/U/month rack and power charges, 5-year depreciation
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Case Study: Security Pricing
Source: Wall Street & Technology, September 24, 2009
48 Tesla S1070 8000 CPU Cores
2 hours 16 hours
10x Less Space
8x Faster
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AGENDA
Adoption
Algorithms
Modeling
Who’s Who
Financial Instruments
Summary
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Conclusion
• Computers are used to model financial instruments for
price, sensitivity and risk
• Algorithms include Finite Differences and Monte Carlo
• Parallelising algorithms requires structural support from
the application
– Benefit is substantial on all measures
– GPUs are transforming the industry
• Opportunities for algorithmic development
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Resources
• GTC presentations
– Finance presentations, Thursday from 2pm, Atherton Room
– 3D Finite Differences on GPU, Friday 10.30am, Empire Room
– Tridiagonal solvers on GPU, Friday 2pm, Atherton Room
• SDK examples
– binomialOptions, 3DFD, MonteCarlo/MonteCarloMultiGPU
• NVIDIA finance page (links to online resources)
– http://www.nvidia.com/object/computational_finance.html