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Multi-level Stochastic Valuations

14 March 2016

High Performance Computing in Finance Conference 2016

Grigorios Papamanousakis

Quantitative Strategist, Investment Solutions

Aberdeen Asset Management

1

• PFE Modelling explained

• Multi-Curve Modelling and stochastic basis for collateral management

• Cash vs. No-Cash Collateral – The impact of haircut on the valuation

• High Performance Computing

• Q & A

Contents

2

Consider the problem of calculating the potential Future Collateral Requirements of a large derivative book for a global asset manager

• Purpose: Provide a probabilistic approach of what may be the collateral requirements, on various granularity levels (fund level, counterparty level, division level, company level, etc.) and different time horizons, of clients portfolio

• Additional information could be extracted as well. Collateral decomposition (Cash Collateral, Gov. Bonds, HY bonds, etc.), counterparty exposure, etc.

• Typically the portfolio will include interest rate swaps, swaptions, inflation swaps, cross currency swaps, equity options, CDSs, etc.

Sketching the problem

Worse Case Scenario

Time Horizon Collateral Requirement

Fund level Division level Company level

90% 1w 1m 1y … … £60mln … … £45mln … … £40mln

70% 1w 1m 1y … … £30mln … … £23mln … … £20mln

Average case 1w 1m 1y … … £18mln … … £12mln … … £10mln

3

• Potential Future Exposure (PFE) is defined as the maximum expected credit exposure over a specified period of time calculated at some level of confidence. PFE is a measure of counterparty credit risk.

• Expected Exposure (EE) is defined as the average exposure on a future date

• Credit Valuation Adjustment (CVA) is an adjustment to the price of a derivative to take into account counterparty credit risk.

• Collateral can be considered any type of valuable liquid property that is pledged by the recipient as security against credit risk.

Basic Definitions

4

What do we need to built a ‘standard market’ PFE Model?

• A consistent simulation framework to project forward in time the interest rate curves. This will typically have 100-120 time-steps and around 100k -1mln scenarios

• Multiple stochastic processes to simulate the FX exposure for every different currency in the portfolio, aligned with the time steps and the number of simulations of the interest rate scenarios. The same idea applies for the inflation rates but with less time steps

• Correlate all the Brownian motions calibrating on the market data and use the Cholesky decomposition to form multivariate normal random variables and project forward in time on every scenario

• Re-evaluate every position, on every time step, for every scenario and aggregate per netting set to calculate the Expected Exposure - EE

• Calculate the collateral changes based on the CSA agreement with every counterparty and available collateral on our pool

PFE Modelling Explained

5

Example – Portfolio Value / Collateral Exposure over time

Jan14 Jan16 Jan18 Jan20 Jan22-100

-80

-60

-40

-20

0

20

40

60

80

100

Total MTM Portfolio Value for All Scenarios

Simulation Dates

Po

rtfo

lio

Va

lue (

£m

ln)

Jan14 Jan16 Jan18 Jan20 Jan220

20

40

60

80

100

120

Collateral Exposure for All Scenarios

Simulation Dates

Ex

po

su

re (

£m

ln)

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Example – Cash Collateral Exposure Distribution 95% & 80% Confidence level

Jan14 Jan16 Jan18 Jan20 Jan220

5

10

15

20

25

30

35

40

45

Simulation Dates

Ex

po

su

re (

£m

ln)

Portfolio Collateral Exposure Profiles

Max Exp

Collateral Exp (95%)

Collateral Exp (80%)

Exp Exposure (EE)

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• For every interest rate curve we will typically have 120+ simulation steps (50 year horizon)

• To construct the full term structure on every simulation step we need 20-25 tenor points

• We simulate different forward and discount curves in order to capture the stochastic

credit spreads between the two curves (e.g. Sonia and Libor)

• The same applies for every currency in our portfolio along with a stochastic process to

simulate the FX exposure

• Depending the degree of convergence we want to achieve we simulate 100k-1mln

scenarios

So for the simulation the dimensionality of the problem is:

nSim x nSteps x nTenors x nCurves x nCurrencies

1mln x 120 x 25 x 15

For valuation purposes we have to take under consideration on top of that the size of the

portfolio as well as the NO-Cash collateral

Dimensionality of the problem The need for HPC

8

• PFE Modelling explained

• Multi-Curve Modelling and stochastic basis for collateral management

• Cash vs. No-Cash Collateral – The impact of haircut on the valuation

• High Performance Computing

• Q & A

Contents

9

• Before the 2007 credit crunch the interest rates showed typical textbook behaviour

– 𝐹𝐷𝑒𝑝𝑜 = 𝐹𝐹𝑅𝐴. The forward rate implied by two deposits coincides with the corresponding FRA rate

– Compounding two consecutive 3m Libor rates yields to the corresponding 6m forward Libor rate

• Differences were still observable on the market but generally regarded as negligible

• Then 2007 arrived and the crisis widened the basis between OIS and LIBOR rates

Multi-Curve Modelling – Before 2007…

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• The LIBOR-OIS basis spread is neither deterministic nor negligible

GBP 3month Libor – OIS spread:

Multi-Curve Modelling – After 2007

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• Exactly the same for the forward OIS – LIBOR basis spread:

GBP 1Yx2Y FRAs vs 1Yx2Y forward OIS spread:

Multi-Curve Modelling – After 2007

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• What do we get when we model the basis spread stochastically:

– Accurate pricing. Swap rates depend on the OIS discount factors

– Better fitting since we are able to calibrate each model separately

– Better correlation structures

– We can quantify the risk sources as we are able to decompose on different risk factors

– More accurate estimation of the portfolio future collateral requirements and consequently more efficient collateral management

– More realistic modelling…

Is it easy? No….

Multi-curve modelling

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During the last 5 years or so there were significant efforts to define a consistent multi-curve framework that would be broadly acceptable.

We are still trying…

We have two main ways to model the basis spread and on each of them sub categories

• Implicit modelling, where we model the joint evolution of OIS rates and curve rates for different market tenors (Ref. [4], [7], [8], [10]).

• Explicit modelling, where we have three main categories

– Additive spreads, as in [1], [8], [10]

– Multiplicative spreads, as in [5], [6]

– Instantaneous spreads, as in [2]

The multi-curve world

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• The impact of the multi curve modelling is highly dependent on the portfolio

• Although for the valuation of an IRS portfolio we discount using SONIA (for both legs), for the forward curve we use LIBOR.

• In that case the impact on the value of the portfolio is subject to:

– The discount curve we use for discounting both legs of the swaps (SONIA)

– The forward curve we use for the floating leg of the swap (LIBOR)

– The discount curve we use for discount the liabilities (LIBOR)

• In case we are using IRSs within a LDI portfolio the LIBOR-OIS curve difference has a significant impact on the hedge effectiveness.

Portfolio Impact of Multi – Curve Modelling

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Portfolio Impact of Multi – Curve Modelling

Jan14 Jan15 Jan16 Jan17 Jan18 Jan19 Jan20 Jan21 Jan22 Jan230

5

10

15

20

25

30

35

40

45

Simulation Dates

Dis

co

un

ted

Exp

osu

re (

£m

ln)

Portfolio level Discounted Collateral Requirements

Discounted 95% Exp Col Single

Discounted 95% Exp Col Dual

Discounted 80% Exp Col Single

Discounted 80% Exp Col Dual

16

• PFE Modelling explained

• Multi-Curve Modelling and stochastic basis for collateral management

• Cash vs. No-Cash Collateral – The impact of haircut on the valuation

• High Performance Computing

• Q & A

Contents

17

Introducing the collateral pool on a multi-curve framework

– We assume that the fund on the previous example has a collateral pool with

• £15mln Cash

• £10mln of Gilts with 10% haircut

• £10mln of AAA Bonds with 20% haircut

• £20mln of HY Bonds with 30% haircut

– We now model a dynamic collateral portfolio that changes over time

– We re-evaluate the collateral on every time step, assuming there is no credit migration on the collateral posted

– Reconstruct the new collateral portfolio and simulate forward (additional computational complexity)

Cash vs Bond Collateral

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Bond Collateral Exposure Profiles - Results

Jan14 Jan16 Jan18 Jan20 Jan220

10

20

30

40

50

60

Simulation Dates

Exp

osu

re (

£m

ln)

Bond Collateral Exposure Profiles

Max Bond Collateral Exp (95%)

Bond Collateral Exp (95%)

Max Cash Collateral Exp (95%)

Cash Collateral Exp (95%)

Bond Collateral Exp (80%)

Cash Collateral Exp (80%)

Exp Exposure (EE)

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• When we include No-Cash collateral instruments in our collateral pool the complexity of

problem increases exponentially

• We now have a dynamic portfolio that changes continuously, adding an optimization

depending collateral decision on every step we re-evaluate the portfolio

• The moment that the first No-Cash Collateral is been posted either from us or our

counterparty, we need to start simulating the credit migration risk associated with that

bond

This is clearly an computational intensive exercise, ideal for parallelization and

acceleration using HPC

Computational Impact

20

• PFE Modelling explained

• Multi-Curve Modelling and stochastic basis for collateral management

• Cash vs. No-Cash Collateral – The impact of haircut on the valuation

• High Performance Computing

• Q & A

Contents

21

NVIDIA GPU Computing: Graphical Processing Units

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Data-Flow Computation on FPGA

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• Preprocessing the portfolio

– Fetch all the interest rate curve / FX data live from Bloomberg / Thomson Reuters

– Read the portfolio and construct all the asset classes and projected cash-flows

– Preprocess the portfolio for different calendars, payment/receive/reset frequencies

Mainly a task we use only CPUs for portfolios with less than 10000 derivatives. Banks with larger portfolios may consider parallelize this step as well.

• Monte Carlo simulation

– Perform the simulation for the various stochastic processes and construct the full term structure for every simulation step

• Post process the portfolio

– Re-Evaluate the portfolio on every scenario and simulation step adjusting for the collateral posted from either us or our counterparties

Acceleration using HPC is essential!

Decomposing the problem

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• Each thread calculates the forward interest rate for the i-th period

• Each thread block calculates a Monte Carlo scenario with different tenor dates

GPU Implementation

T1 T2 T3 … TN-2 TN-1 TN r1(T1) r2(T1) r2(T2) r3(T1) r3(T2) . . .

rk-1(T1) rk-1(T2) rk-1(T3) … rk-1(TN-2) rk-1(TN-1) rk-1(TN) rk(T1) rk(T2) rk(T3) … rk(TN-2) rk(TN-1) rk(TN)

Monte Carlo simulation evolution – N time steps

Te

rm s

tru

ctu

re

k –

Tenor

Poin

ts

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• CPU execution times & Speedup

• Test Environment:

– CPU: an Intel Xeon X5650 CPU with 6 cores 12 threads running at 2.67GHz

– an NVIDIA Tesla K40 (2880 cores) with the CUDA version 6.5

– a Xilinx Vertex-6 V-SX4757 FPGA card with Maxeler Max-Compiler

Acceleration Results

Number of paths

CPU GPU GPU/CPU FPGA FPGA/CPU

50K 21.5s 20.2s 1x 1.1s 19.5x 100K 43.3s 22.8s 2x 1.1s 39x 500K 251.8s 20.2s 12.5x 2.4s 102x 1M 503.6s 22.2s 25x 4.1s 120x

26

• Multi-curve modelling is here to stay

– Impact varies, but dual curve discounting is essential for accurate pricing

• No-Cash Collateral management is not trivial

– Multiple broker/fund specific CSA agreements

– Haircuts do have a significant impact on collateral and fund’s liquidity

– The dynamic portfolio grows the computational complexity exponentially

– Monte Carlo framework provides a sensible solution

• High Performance Computing using is essential

– Significant increases the performance of Monte-Carlo simulation

• Memory allocation problems have to be taken under consideration

– Total cost of ownership

– Solution depends on the business case

– In-house development has significant Person Risk

• Outsourcing?

Conclusions

27

Thank you / Any questions?

28

1. Amin, A. (2010). Calibration, simulation and hedging in a Heston Libor market model with stochastic basis

2. Andersen, L. and Piterbarg, V. (2010). Interest rate modelling, in three volumes. Atlantic Financial Press

3. Brigo, D. and Mercurio, F. (2006). Interest Rate Models, Theory and Practice. Springer Finance.

4. Crepey, S. Grbac, Z. Ngor, N. Skovmand, D. A Levy HJM multiple-curve model with application to CVA computation

5. Henrad, M. (2007) The irony in derivatives discounting. Wilmott Magazine

6. Henrad, M. (2013). Multi-curves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options. OpenGamma Quant Research

7. Kenyon, C. (2010). Post-shock short-rate pricing. Risk

8. Mercurio, F. and Xie, Z. (2012). The basis goes stochastic basis. Risk, Dec 2012

9. Mercurio, F. (2010). A LIBOR market model with stochastic basis

10.Moreni, N. and Pallavicini, A. (2010). Parsimonious HJM modelling for multiple yield-curve dynamics. Working paper

References

29

Past performance is not a guide to future returns. The value of investments, and the income from them, can go down

as well as up and your clients may get back less than the amount invested.

The views expressed in this presentation should not be construed as advice on how to construct a portfolio or whether

to buy, retain or sell a particular investment. The information contained in the presentation is for exclusive use by

professional customers/eligible counterparties (ECPs) and not the general public. The information is being given only

to those persons who have received this document directly from Aberdeen Asset Management (AAM) and must not be

acted or relied upon by persons receiving a copy of this document other than directly from AAM. No part of this

document may be copied or duplicated in any form or by any means or redistributed without the written consent

of AAM.

The information contained herein including any expressions of opinion or forecast have been obtained from or is based

upon sources believed by us to be reliable but is not guaranteed as to the accuracy or completeness.

Issued by Aberdeen Asset Managers Limited which is authorised and regulated by the Financial Conduct Authority in

the United Kingdom.

For professional investors only Not for public distribution

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Appendix

31

Setup of the Hull-White single factor interest rate model:

This is a short rate model and is defined as:

d𝑟 𝑡 = 𝜃 𝑡 − 𝛼𝑟 𝑡 𝑑𝑡 + 𝜎𝑑𝑧

Where 𝑟, is the short rate at time t, 𝜶 the mean reversion speed and σ is the volatility. The drift function 𝜽 𝒕 is defined as:

θ 𝑡 = 𝐹𝑡 0, 𝑡 + 𝛼𝐹 0, 𝑡 +

𝜎2

2𝛼(1−𝑒−2𝛼𝑡)

Where:

𝐹 0, 𝑡 : is the instantaneous forward rate at time t

𝐹𝑡 0, 𝑡 : is the partial derivative of 𝐹 with respect to time

• By applying the equations above we define the short rates and the corresponding forwards on every scenario one every time steps (nSim x nSteps x nCurves)

Interest Rate Modelling

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Once we have simulated the short rate path we can generate the full curve at each simulate date using the formulas:

𝑅 𝑡, 𝑇 = −

1

𝑇 − 𝑡ln𝐴 𝑡, 𝑇 +

1

𝑇 − 𝑡𝐵 𝑡, 𝑇 𝑟(𝑡)

ln 𝐴 𝑡, 𝑇 = ln 𝑃(0,𝑇)

𝑃(0,𝑡)+ 𝐵 𝑡, 𝑇 𝐹 0, 𝑡 − 1

4𝑎3𝜎2 𝑒−𝑎𝑇 − 𝑒−𝑎𝑡 2(𝑒2𝑎𝑡 − 1)

𝐵 𝑡, 𝑇 = 1−𝑒−𝑎(𝑇−𝑡)

𝑎

Where 𝑅(𝑡, 𝑇) is the zero rate on time step t and 𝑃(𝑡, 𝑇) is the price of the zero coupon bond at t that pays one pound at time maturity T.

• Using the above formulas we construct the a full term structure for every simulation point (nSim x nSteps x nTenors x nCurves) and we are able to evaluate our derivative portfolio and the corresponding collateral.

Deriving the full yield curve

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• Additive spreads:

𝑆𝑘𝑥 𝑡 = 𝐿𝑘

𝑥 𝑡 − 𝐹𝑘𝑥 𝑡

Where x are the different market tenors, k the simulation time steps, 𝑺 the spread, 𝑳 is the forward Libor rate and 𝑭 the forward OIS rate.

• Multiplicative spreads:

1 + τ𝑘

𝑥𝑆𝑘𝑥 𝑡 =

1 + τ𝑘𝑥𝐿𝑘𝑥 𝑡

1 + τ𝑘𝑥𝐹𝑘𝑥 𝑡

• Instantaneous spreads: 𝑃𝐿 𝑡, 𝑇 = 𝑃𝐷(𝑡, 𝑇)𝑒

𝑠 𝑢 𝑑𝑢𝑇𝑡

Where 𝑃𝐷(𝑡, 𝑇) denotes the Libor discount factor at t with maturity T

Explicit Modelling of the basis spread

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We follow Mercurio’s & Xie’s [8] additive spread approach by slightly twisting their

parameterisation. We simulate the OIS curve and on top of that the credit spread that

provide us with the Libor rates to simplify our calibration.

For each tenor x and time interval [𝑇𝑘−1𝑥 𝑡 , 𝑇𝑘

𝑥(𝑡)] we assume that forward Libor curve 𝐿𝑘𝑥 𝑡

is a function of the OIS forward rate 𝑅𝑘𝑥 𝑡 , the forward basis spread 𝑆𝑘

𝑥 𝑡 and an

independent martingale 𝑋𝑘𝑥 𝑡 ∶

Using a simple affine function we derive the Libor rates formula:

𝐿𝑘𝑥 𝑡 = 𝐿𝑘

𝑥 0 + 1 + 𝛼𝑘𝑥 𝑅𝑘

𝑥 𝑡 − 𝑅𝑘𝑥 0 + 𝛽𝑘

𝑥[𝑋𝑘𝑥 𝑡 − 𝑋𝑘

𝑥 0 ],

where 𝛼𝑘𝑥 and 𝛽𝑘

𝑥 are real constant parameters, for every k and x.

Modelling the Basis Spread

35

• 𝛽𝑘𝑥 models the volatility of basis spreads and 𝛼𝑘

𝑥 the correlation between OIS rates and the corresponding spreads. So we have:

𝑎𝑘𝑥 = 𝐶𝑜𝑟𝑟(𝑅𝑘

𝑥 𝑇𝑘−1𝑥 , 𝑆𝑘

𝑥 𝑇𝑘−1𝑥 )

𝑉𝑎𝑟[𝑆𝑘𝑥 𝑇𝑘−1

𝑥 ]𝑉𝑎𝑟[𝑅𝑘

𝑥 𝑇𝑘−1𝑥 ]

𝛽𝑘𝑥 = [1 − 𝐶𝑜𝑟𝑟 𝑅𝑘

𝑥 𝑇𝑘−1𝑥 , 𝑆𝑘

𝑥 𝑇𝑘−1𝑥 ]

𝑉𝑎𝑟[𝑆𝑘𝑥 𝑇𝑘−1

𝑥 ]𝑉𝑎𝑟[𝑋𝑘

𝑥 𝑇𝑘−1𝑥 ]

The basis factors 𝑋𝑘𝑥 may be different on every time step and tenor point. For the simulation

of 𝑋𝑘𝑥 we will follow a simple geometric Brownian motion.

Under this framework we are able to simulate on every time step both the Libor and OIS curves and price our derivatives using dual curve discounting

Modelling the Basis Spread