www.sungard.com/adaptiv

Risk Management and Operations Solutions

Derivative Pricing for Risk Calculations – Challenges and Approaches

Research Workshop on Fast Financial Algorithms and Computing

Dan Travers

Product Manager

SunGard Adaptiv

4th July 2007

Introduction

Capital Markets and Investment Banking

Adaptiv Product Suite – Enterprise Risk Management & Operations

Different perspective on similar problems

Challenges

Challenges

Increasing size of Portfolios Volumes are expanding exponentially

Increasing complexity of Portfolios Mix of exotic derivative instruments is increasing

Requirements and Incentives to use more risk-sensitive Risk Measurement techniques Basel II allows much more risk-sensitive treatment of risks Usually involve simulation techniques

Push for greater consistency and rigor in risk Basel II required more validation and internal oversight

Market Problems 1 – Credit PFE Simulation

Potential Future Exposure (PFE) Simulation-based Credit risk measure Model portfolio over the lifetime of the deals “Age” the portfolio Apply Netting and collateral Key metrics:

Portfolio Exposure at Confidence level Expected Exposure

Example of how many valuations would be required per second 100,000 trades, 50 timepoints / trade, 5000 simulations --> 25Billion valuations In a 5 hour window --> 1.4 Million valuations / second

Analytic Approximation – MC2

Analytical Approximation of the portfolio value Approximate each deal by quadratic polynomial in

the Risk Factor driver space

Where xi are normal variates

Aggregate payoffs to portfolio level Transform onto orthogonal set of risk factors &

use PCA analysis to reduce dimensionality Calculate quantiles from the payoff surface as a

function of these independent normal variables

,

( , ) ( ) ( ) ( ) ( ) ( ) ( )k k k ki i ij i j

i i j

U x t a t b t x t c t x t x t

Analytic Approximation – MC2

Fitting the quadratic models Use Taylor expansion where possible For non-linear instruments, fit a quadratic to risk factor

shifts at defined level of shift

Expected Exposure: More complicated:

1 0

1 1[ )] ( ) ( )sin ( ) ( ) cos ( )

2

n

Q q jj

E Q a c r y A y y B y y dy

I

MC2 – Shortcomings & Challenges

Instrument and Portfolio Factors Highly non-linear instruments provide difficulties Path dependent instruments are similarly challenged to fit

into analytic framework Netting and Collateral

Hybrid approach developed Model “acceptable” part of portfolio as a quadratic surface,

with the other parts of the portfolio full-priced Apply simulations to the quadratic surface & full-priced

deals Ensure scenario consistency Handle Netting & Collateral

Retain the quadratic approximation at low enough level to get under the netting agreements

MC2 - Challenges

-4,000

-3,500

-3,000

-2,500

-2,000

-1,500

-1,000

-500

0

500

1,000

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91

time since Ref date

Ab

solu

te D

iffe

ren

ce

1M 3M 6M 1Y

Tested Hybrid Approach Good, but not accurate enough to supplant full simulation Majority of instruments have some form of path-

dependency Greatly complicated by the Ageing

Brute Force?

If we cannot use clever technique to reduce the load, then we must distribute the work Grid Computing becomes the only solution Many systems distribute, but often with little efficiency Scalability must be excellent – 90%+ efficiency

Implemented distribution to Minimise the data passed around the grid Maximise the work done on individual grid nodes

Achieved results hoped for

Scalability

Increasing Portfolio size with Increasing Grid Size

0

50

100

150

200

250

300

350

20,000 / 2 40,000 / 4 80,000 / 8 160,000 / 16

No. of Trades / No. of Grid Nodes (expanding Grid)

Tim

e (

min

s)

Actual with constant grid

Actual with increasing grid

Expected with constant gridExpected with increasing grid

Increasing Portfolio Size with Constant Grid Size

0

50100

150200

250300

350

20,000 40,000 80,000 160,000No. of Trades

Tim

e (m

ins)

ActualExpected

Increasing Portfolio Complexity with Increasing Grid Size

0

5

10

15

20

25

30

35

40

25% / 2 50% / 4 75% / 8 100% / 16

% Complex Trades / No. of Grid Nodes (increasing Grid)

Tim

e (

min

s)

Actual with constant grid Actual with increasing grid

Expected with constant gridExpected with increasing grid

Increasing Scenarios Numbers

0

200

400

600

800

1000

1,000 5,000 10,000No. of Scenarios

Tim

e (m

ins)

Actual

Expected

Increasing trade volumes – constant Grid Increasing volumes – Increasing Grid

Increasing portfolio complexity –

increasing GridIncreasing number of scenarios

Product Coverage & Consolidation of Pricing

What about product coverage?

Consolidation of Pricing Driver: Combined Market and Credit Risk Driver: One set of models for Front – to – Back

One validation of models

Differences: Front Office models can be slow and accurate, but risk

models are fast with less accuracy Credit Models will need to Age

Multi-grade of models should be available in same framework Multi-grade of Market data and simulation models

Extensibility

Model library must be

Multi-grade Market, Credit and Front-office

Extensible Extensible by users and by quant / developers

Transparent Easily verifiable by outside source

Extensibility 2

Extensibility Framework must be strong & flexible Allow anyone to add models Externally added models execute with the same speed as

native models Models must have “Ageing” embedded in the pricing

function – for Credit pricing

What about: Path-dependent, Callable products Many custom derivatives – infinitely customizable products

across all institutions – not possible to add a generic model

Scripting Framework

“Scripting” framework

Model the payoff and behaviour of the instrument in a “Script”

Accompany the framework with a library of Stochastic models Numerical solvers

Finite Difference Grid Monte Carlo Tree Pricing

Relatively common feature in Front Office systems, but bringing this to risk is more difficult Ageing is a problem Need enough power in the scripting and solving environment to

allow performance, while keeping flexibility

www.sungard.com/adaptiv

Risk Management and Operations Solutions

Derivative Pricing for Risk Calculations – Challenges and Approaches

Research Workshop on Fast Financial Algorithms and Computing

Dan Travers

Product Manager

SunGard Adaptiv

4th July 2007