Barr cc slides

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AWS in Academic FinanceHigh Performance Modeling

Stephen J. [email protected]

University of Rochester

March 24, 2012

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Welcome

Welcome!

I Glad to be here at Rochester Cloud Camp 2012

I I am a PhD student at the Simon School of Business,University of Rochester

I Presenting a bit of my research, and how I use AWS daily forfinancial models

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Welcome

Welcome!




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Welcome

Welcome!




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Quick Overview

First 5-7 minutes

I My research question

Remaining time

I How I use AWS daily in my research

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Quick Overview

First 5-7 minutes

I My research question

Remaining time

I How I use AWS daily in my research

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Research Question

I My area of interest is financial capital structure, meaninghow firms use debt and equity issuance to financethemselves

I Main quantities of interest are leverage and investment rates

Leverage =net debt

total assets

The main goal of this line of research is to explain the dynamicsof leverage and investment in the face of uncertainty.

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Research Question

I My area of interest is financial capital structure, meaninghow firms use debt and equity issuance to financethemselves

I Main quantities of interest are leverage and investment rates

Leverage =net debt

total assets

The main goal of this line of research is to explain the dynamicsof leverage and investment in the face of uncertainty.

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Research Question

Rather than talk, lets just put together a simple model.Modeling Principles

I Take as many simplifications as possible

I Based on economic principles

I Rather than forcing the firm to follow rules about investmentor leverage, the firm chooses the value-maximizing behaviourgiven reasonable incentives

I Firm’s incentive: maximize equity value

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Research Question






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Research Question






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Research Question






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Research Question - Math - I

e(k , k ′, b, b′, z) = (1)

zkθ Profit (2)

+ p′ − (1− r(1− τc)p) New debt + repay old (3)

+ k ′ − (1− δk) Investment (4)

+ A(k, k ′) Capital Adjustment Cost (5)

The function e(·) shows the firm’s financing status.e(·) ≥ 0 the firm make a distribution to shareholders

e(·) < 0 the firm is issuing equity(6)

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Research Question - Math - II

But, to issue equity, the firm goes to an investment bank. Thisisn’t free.

φ(e(·)) = λ1e(·) + λ2(e(·))2 (7)

The firm wants to make the choices of k ′ and p′ to maximizecurrent plus expected future discounted returns. In math, this iscalled a dynamic programming problem and is expressed as aBellman equation (recursively defined).

V (Kt ,Pt , z∗t ) = (8)

maxKt+1,Pt+1∈K×P

e(Kt ,Kt+1,Pt ,Pt+1, z

∗t ) + φ(e(Kt ,Kt+1,Pt ,Pt+1, z

∗t ))

(9)

+ β

∫V (Kt+1,Pt+1, z

∗)dΓ(z∗t , dz∗t+1)

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Phew!

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How do we solve this?

The math and algorithms to solve these programs fall under“dynamic programming”.HUGE ISSUE:

The curse of dimensionality!!!

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Why is this an issue? - I

For 3 k’s 5, 10, 15 and 2 p’s 2, 4

k =

5 5 5 5 5 55 5 5 5 5 510 10 10 10 10 1010 10 10 10 10 1015 15 15 15 15 1515 15 15 15 15 15

k ′ =

5 5 10 10 15 155 5 10 10 15 155 5 10 10 15 155 5 10 10 15 155 5 10 10 15 155 5 10 10 15 15

p =

2 2 2 2 2 24 4 4 4 4 42 2 2 2 2 24 4 4 4 4 42 2 2 2 2 24 4 4 4 4 4

p′ =

2 4 2 4 2 42 4 2 4 2 42 4 2 4 2 42 4 2 4 2 42 4 2 4 2 42 4 2 4 2 4

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Why is this an issue - II

1. For sufficient granularity, I need u 71 k ’s and u 29 p’s

2. The above leads to a 2059× 2059 matrix

3. I need one of those for each shock z

4. My z is a combination of two processes...I have 323 of them.

5. I have a (2059× 323)× 2059 matrix of double precisionnumbers sitting in memory

6. This is a 665057× 2059 = 1, 369, 352, 363 element matrix.

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How to solve

maxK ′,P′∈K×P

e(K ,K ′,P,P ′, z∗) + φ(e(K ,K ′,P ,P ′, z∗)) (10)

+ β

∫V (K ′,P ′, z∗)dΓ(z∗, dz∗′)

(11)

Algorithm: Value Function Iteration

1. Compute eq. (10)

2. Iteratively guess eq.(11)

3. Get a policy function, giving the value-maximizing choice forevery state

4. The policy function is a 323× 2059 matrix giving the best ofk ′, p′ for every z , k , p

5. Use a cc2.8xlarge

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How to solve


e(K ,K ′,P,P ′, z∗) + φ(e(K ,K ′,P ,P ′, z∗)) (10)

+ β

∫V (K ′,P ′, z∗)dΓ(z∗, dz∗′)

(11)


1. Compute eq. (10)





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How to solve


e(K ,K ′,P,P ′, z∗) + φ(e(K ,K ′,P ,P ′, z∗)) (10)

+ β

∫V (K ′,P ′, z∗)dΓ(z∗, dz∗′)

(11)


1. Compute eq. (10)





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How to solve


e(K ,K ′,P,P ′, z∗) + φ(e(K ,K ′,P ,P ′, z∗)) (10)

+ β

∫V (K ′,P ′, z∗)dΓ(z∗, dz∗′)

(11)


1. Compute eq. (10)





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How to solve


e(K ,K ′,P,P ′, z∗) + φ(e(K ,K ′,P ,P ′, z∗)) (10)

+ β

∫V (K ′,P ′, z∗)dΓ(z∗, dz∗′)

(11)


1. Compute eq. (10)





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Software - Old Version

I Original solver written in Python / Numpy

I Pros: beautifully general, modular,

I Cons: takes 15 mins to solve. Global interpreter lock makesthings hard to parallelize

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Software and Hardware - Version 2 - I

I Reimplemented in C++ using Eigen - high performancetemplating engine

I Eigen links against Intel’s Math Kernel Library (optimizedBLAS, LAPACK)

I Intel’s icpc C++ compiler, using -xavx

I cc2.8xlarge have Intel Xeon Sandy Bridge-EP’s, which haveAVX

I OpenMP to use all 32 threads

I Result....

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I Result....

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I Result....

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I Result....

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I Result....

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I Result....

Model Solves in 90 seconds

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Software and Hardware - Version 2 - II

I cc2.8xlarge’s are fast

I coding towards MKL / AVX makes things gives a substantialspeedup

I at this scale, optimizations such as row-major vs columnmajor are trivial. Eigen makes this easy.

typedef Eigen : : Matrix<double ,Eigen : : Dynamic , Eigen : : Dynamic ,Eigen : : RowMajor> MatrixRMXd ;

I I can iterate on a 665057× 2059 about once per second

I OpenMP is easy

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while ( ( err > iterpar . TOLERANCE )&& ( itercount < iterpar . MAXIT ) ) EV_prev . noalias ( ) = beta ∗ TMAT ∗ V_prev ;#pragma omp parallel for

for ( int i = 0 ; i < Nshocks ; i++) // p a r a l l e l copyfullmat . middleRows ( i∗nstates , nstates ) =staticmat . middleRows (i∗nstates , nstates ) ;// p a r a l l e l b r o adca s tfullmat . middleRows ( i∗nstates , nstates ) . rowwise ( )

+= EV_prev . row (i ) ;// p a r a l l e l new va l u e fnV_new . row (i ) = fullmat . middleRows (i∗nstates , nstates )

. rowwise ( ) . maxCoeff ( ) . transpose ( ) ;// p a r a l l e l ge t minsdiffs (i ) = ( V_new . row (i )

− V_prev . row (i ) ) . array ( ) . abs ( ) . maxCoeff ( ) ;lbounds (i ) = ( V_new . row (i )

− V_prev . row (i ) ) . minCoeff ( ) ;ubounds (i ) = ( V_new . row (i )

− V_prev . row (i ) ) . maxCoeff ( ) ; // end p a r a l l e l f o r

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lbound = beta ∗ double ( lbounds . minCoeff ( ) ) ;ubound = beta ∗ double ( ubounds . maxCoeff ( ) ) ;

double adjfactoriter = 0.5 ∗ ( ( beta/(1−beta ) )∗ lbound + ( beta/(1−beta ) )∗ ubound ) ;

double diff = diffs . maxCoeff ( ) ;

#pragma omp parallel for

for ( int ii = 0 ; ii < Nshocks ; ii++) V_new . row ( ii ) . array ( ) += adjfactoriter ;

err = diff ;

// UPDATE ITER COUNT AND VALUE FUNCTIONcout << "i: " << itercount++ << ": " << err << endl ;

V_prev = V_new ;

// END WHILE

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How I Use AWS - Solving the Model A LOT

I GOAL: See how the model behaves under random sets ofparameterizations

I HOWTO: - Small Python (using mostly boto) ScriptFor i in 1:N

1. Generate random parameterization Ωi = [ri , θi , δi , εP,i , ...]2. Assign a UUID to Ωi

3. Push Ωi SQS

I SOLVER AMI

1. Get latest solver binary2. Check SQS queue3. While (queue nonempty)

3.1 Get Ωi and solve3.2 Simulate and gather statistics3.3 Push all back to S3

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3. Push Ωi SQS

I SOLVER AMI



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1. Generate random parameterization Ωi = [ri , θi , δi , εP,i , ...]

2. Assign a UUID to Ωi

3. Push Ωi SQS

I SOLVER AMI



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3. Push Ωi SQS

I SOLVER AMI



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3. Push Ωi SQS

I SOLVER AMI



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3. Push Ωi SQS

I SOLVER AMI



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3. Push Ωi SQS

I SOLVER AMI

1. Get latest solver binary

2. Check SQS queue3. While (queue nonempty)


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3. Push Ωi SQS

I SOLVER AMI

1. Get latest solver binary2. Check SQS queue

3. While (queue nonempty)


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3. Push Ωi SQS

I SOLVER AMI



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3. Push Ωi SQS

I SOLVER AMI


3.1 Get Ωi and solve

3.2 Simulate and gather statistics3.3 Push all back to S3

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3. Push Ωi SQS

I SOLVER AMI


3.1 Get Ωi and solve3.2 Simulate and gather statistics

3.3 Push all back to S3

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3. Push Ωi SQS

I SOLVER AMI



Identification!

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No Identification

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AWS Advantages

I For each plot, each point takes at least 90 seconds to generate

I Allows me to work in real time - start a group of cc2.8xlarge’s

I Paid for by AWS in Education grant

I cc2.8xlarge’s are FAST!

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AWS Advantages





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AWS Advantages





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AWS Advantages





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Next Research Step - Simulated Method of Moments

MAIN IDEA:

I Reality has some true parameterization Ω∗ which we don’tknow.

I Reality has some statistics X ∗ which we do know.

I A model’s Ωi implies simulated statistics Xi .

Xi = f (Ωi )

I To find Ω∗, we can define

g(Ωi ) = sum(abs(X ∗ − f (Ωi )))

and then find the Ωi which minimizes g(Ωi ).

I This amounts to a multidimensional optimization, which iswell understood (e.g. Nelder Mead, Simulated Annealing)

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MAIN IDEA:




Xi = f (Ωi )





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MAIN IDEA:




Xi = f (Ωi )





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MAIN IDEA:




Xi = f (Ωi )





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MAIN IDEA:




Xi = f (Ωi )





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I Pick a large amount of starting values of the optimizers

I Each cc2.8xlarge can handle 2 optimizers simultaneously

I Should take about 500 iterations to find a solution

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Thank you

References:

I My blog: www.econsteve.com

I My code: www.bitbucket.com/stevejb

I Twitter: @stevejb

I AWS in Eduaction: http://aws.amazon.com/education/

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Thank you

References:



I Twitter: @stevejb


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Thank you

References:



I Twitter: @stevejb


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Thank you

References:



I Twitter: @stevejb
