albanese - coherent global market simulations for counter party credit risk (icbi global derivatives...

8/4/2019 Albanese - Coherent Global Market Simulations for Counter Party Credit Risk (ICBI Global Derivatives Paris 2011)

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Coherent Global Market Simulations forCounterparty Credit Risk

Claudio Albanese

Presented at the ICBI Global Derivatives Conference, April 13,

2011

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References and Thanks

Based on the paper Coherent global market simulations forcounterparty credit risk, appeared in the January 2011 issueof Quantitative Finance.

Co-authors: Toufik Bellaj, Guillaume Gimonet and GiacomoPietronero

Additional references at www.albanese.co.uk

Opinions expressed in this presentation are personal and

they are the authors sole responsibility.

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Outline

Our objectives

Regulatory environment

Global valuation and single node architectures

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PART I. High Throughput Portfolio Analysis

The objective is to analyse global portfolios of netting setsover long time horizons by means of global marketsimulations under the risk neutral measure

All credit and market factors are modelled dynamically Current instrument coverage currently includes derivatives

that can be valued by backward induction such as (callable)swaps, FX options and CDSs

Basket options, CDO tranches, etc.., require nestedsimulations and are work in progress

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Global Valuation (i.e. Arbitrage Free Picing)

In 1931, de Finetti first derived the Fundamental Theorem ofFinance according to which, assuming absence of arbitrage,the relative price of all assets At with respect to a chosennumeraire gt can be expressed in the form

At

gt= EQt

AT

gT

(1)

for some probability measure Q, globally defined across all

assets in the economy.

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Global Valuation

According to the Fundamental Theorem, global valuation is

the only sensible and theoretically justified procedure forportfolio valuation, hedging and replication

But theres more: theres no fully rigorous alternative tovaluing portfolios globally and as a whole!

In the presence of netting agreements and funding costs,arbitrage free pricing is not additive across transactions

The price of a transaction in isolation is not equal to themarginal price of a transaction as it is added to a portfolio

Rigor and efficiency go together: Pricing portfolios as a

whole on a single node and under a globally definedmartingale measure allows one to share computationalbuilding blocks and achieve a markedly sub-linearperformance ratios (the more instruments are there theshorter the execution time on a per instrument basis)

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Analytics for global portfolios of netting sets (andsub-portfolios thereof)

Point-in-time loss distributions Cumulative loss distributions for cash waterfalls including

losses arising from counterparty defaults funding costs

collateral (modeled dynamically) multi-tiered liabilities

Sensitivities of loss distributions and macro hedge ratios

Risk resolutions of outliers in the tail of the loss distributionand risk concentration hedging

Risk resolution at the individual instrument level

EPE and CVA statistics are a biproduct, not the mainfocus of the analysis

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Loss distribution for a CCR portfolio of 302 netting sets.

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A different view on the same loss distribution emphasizingthe equity tranche

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Tranched CVA for a portfolio of 302 netting sets.Tranches are given as percentages of expected exposures.(Already shown)

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Loss distribution for the EUR denominated sub-portfolio

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Risk resolution and stress testing: Zooming on an outlierfor the EUR denominated sub-portfolio

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Risk resolution and stress testing: Zooming on an outlierfor the EUR denominated sub-portfolio

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Loss distribution a EUR denominated fix-for-float swap

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Loss distribution for a cross-currency EUR-USD swap withnominal exchange at maturity

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Loss distribution for a portfolio of 62 netting sets forcounterparties in the financial sector.

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T h d CVA f f li f 62 i f


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Tranched CVA for a portfolio of 62 netting sets forcounterparties in the financial sector.

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C i f l di ib i f h USD d EUR


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Comparison of loss distributions for the USD and EURdenominated subportfolios.

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S i i i i ll l hif i h USD f h


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Sensitivities to a parallel shift in the USD curve for theloss distributions corresponding to the entire portfolio.

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C l ti l di t ib ti f th ti CCR tf li


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Cumulative loss distribution for the entire CCR portfoliowith 302 counterparties

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C l ti l di t ib ti f th t CCR


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Cumulative loss distribution for the corporate CCRsub-portfolio

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Cumulative loss distribution for the sovereign CCR


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Cumulative loss distribution for the sovereign CCRsub-portfolio

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Outline


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Outline

Our objectives

Regulatory environmentGlobal valuation and single node architectures

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PART II Regulatory environment


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PART II. Regulatory environment

Current capital adequacy requirements are predicated on

Expected Positive Exposure (EPE) statistics and anapproximate version of portfolio level Credit ValuationAdjustment (CVA)

Historically, the CVA was introduced in the early 1990s to

analyze valuation adjustments for counterparty credit risk atthe single instrument level

After the crisis, the CVA was elevated to the rank of riskmeasure for global portfolios of netting sets in an approximateversion whereby the CVA figure is fudged from the EPE

The EPE and the CVA are perhaps the very worst and thesecond worst risk measures for counterparty risk at theportfolio level that can possibly be conceived

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Reminder: definition of portfolio level CVA


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Reminder: definition of portfolio level CVA

The CVA of a derivative position is defined as the discountedexpected loss due to counterparty default risk. More precisely, theCVA of a portfolio of netting sets is

CVA = E0n

0

et0rsds(Pn

t

)+dn

t (2)

where (x)+ denotes max(x, 0), n is an index for netting sets, Pnt is

the price process of the portfolio corresponding to the netting setn, rt is the process for the domestic short rate and

nt is the

process followed by the cumulative probability of default up to timet for netting set n.

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Additivity: The key to computability by trivial


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Additivity: The key to computability by trivialparallelization

The CVA can be decomposed as follows:

CVA = n

CVAn (3)

where

CVAn = E0

0

et0rsds(Pnt )+d

nt

, (4)

i.e. the CVA of a portfolio of netting sets is given by the sums ofthe CVAs of each individual netting set.

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Shortcomings of additive risk measures


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Shortcomings of additive risk measures

CVA is additive across netting sets. Only the 1988 version ofBasel I used additive risk measures, but that choice was thenhighly criticized as additive risk measures are blind todiversification.

The 1994 Amendment for market risk and in Basel II forcredit risk are based on VaR which is a measure for tail riskand is not additive.

CVA-based capital rules revert back to a 1980s type of

situation

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The disincentive to diversify in CVA-based rules


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The disincentive to diversify in CVA based rules

Consider a Good Bank and a Bad Bank, both holding the

same derivatives portfolio P. Suppose that Bad Bank holds all the exposures in P against a

single counterparty.

Suppose also that Good Bank holds the portfolio P but is welldiversified across thousands of counterparties.

Who pays more capital under the CVA-based rules?

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The disincentive to diversify in Basel III


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The disincentive to diversify in Basel III

Consider a Good Bank and a Bad Bank, both holding the

same derivatives portfolio P. Suppose that Bad Bank holds all the exposures in P against a

single counterparty.

Suppose also that Good Bank holds the portfolio P but is well

diversified across thousands of counterparties. Who pays more capital under the CVA-based rules?

Good Bank!

Thats because the benefit of diversification is not recognized

and the only effective way of reducing exposure is by netting. In the presence of netting, using an additive risk

measure results in an incentive to reduce the degree of

diversification and concentrate risk!

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Striking an impossible balance


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Striking an impossible balance

By using the CVA as a gauge to stipulate capital adequacyrequirements, regulators are playing an impossible balancingact

The capital requirements computed in terms of the CVA forundiversified Bad Banks is insufficient

The capital requirements for well diversified Good Banks isexcessive

The very concept of using the CVA as a risk metric for capital

adequacy is wrong both ways!

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The CVA is the wrong concept even at the single


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The CVA is the wrong concept even at the singleinstrument level

The CVA was originally meant to be a price adjustment forderivative valuations whereby an instrument is modeledindividually as opposed to being part of a portfolio. Even thisis wrong!

In the presence of netting agreements and multi-tiered fundingcosts, the marginal value of a derivative contract in thebackground of a portfolio of current holdings is not equal tothe value of the contract in isolation.

To rigorously value the counterparty risk adjustment for asingle instrument, one needs nothing short of a global marketsimulation for the entire portfolio of holdings and toimplement background pricing.

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CVA Hedging and Optimization gives rise to Systemic Risk


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g g p g y

When one attempts to optimize or hedge the CVA, theoreticalinconsistencies lead to undesirable behaviours

What is the most effective way of reducing the CVA of aglobal bank portfolio?

Transfer first loss to investors and lever up This

creates a moral hazard situation transfers counterparty credit exposure from the body of the

loss distribution to the tail, inducing systemic risk

To mitigate systemic risk one should give an incentive to thetransfer of tail risk instead

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Our suggestion to regulators


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gg g

Include credit-credit and market-credit correlations within thecontext of coherent global market simulation over long timehorizons

Penalize tail risk as opposed to first loss risk

Penalize risk concentrations on few counterparties

Recognize the existence of a multi-tiered hierarchy of capitalbuffers

Create a securitization framework for counterparty credit riskportfolios based on the use of detailed information on loss

distributions and simulations of cash flow waterfalls Current technology allows us to do things right. Theres no

merit or excuse to compromise on rigour!

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Outline


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Our objectives

Regulatory environment Global valuation and single node architectures

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PART III. Global valuation and single node architectures


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g

Technology comes first (and always did)

The Mathematical Framework is adapted to technology

Models are formulated by means of the mathematicalframework

Regulations and business models are organized aroundmodels

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From grid computing to the CVA


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Technology: grid computing

Mathematical Framework: based on stochastic calculus andaimed at constructing partially solvable processes

Models: tailored to individual instruments in isolation while

ignoring the Fundamental Theorem at the portfolio level

Regulations and business models: based on the EPE-CVAmethodology and single instrument processing

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From single node technologies to coherent global market


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simulations

Technology: large computing boards capable to processentire global bank portfolios at once

Mathematical Framework: based on an algebraic formalismaimed at bypassing the memory bottleneck of current

microprocessor architectures Models: flexibly specified and econometrically realistic, solved

numerically and calibrated globally, suitable for highthroughput portfolio processing

Regulations and business models: based on portfolio

processing and marginal instrument valuation, real time accessto coherent global market simulations, 3d risk visualizationtools, risk resolution and portfolio level hedging

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Its all about achieving sublinear scaling without


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uncontrollable approximations

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Single node technologies


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The system architecture is based on a single node designpattern whereby large portfolios are loaded entirely on a largeboard with with dozens of CPU cores, several GPUcoprocessors and TB scale memory

Having bypassed the network bottleneck at the root, we

implement a number of portfolio level algorithmicoptimizations to achieve sublinear scaling, such as

globally defined, high quality models for all risk factors sharing of proxy models across minor currencies and credit

processes

dictionaries of elementary building blocks for derivativevaluation that is possible to compute just once and shareglobally at the portfolio level

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Heritage Grid Farms


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Current boards: lots of memory, many cores and narrow


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data paths

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Outline of a Global Valuation System Architecture


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Performance


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Global Calibration of FX Models


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Global multi-currency portfolio are an example of failure of localmodels for individual crosses as these ignore triangular andpolygonal relations.

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A Minimum Spanning Tree for Global Currency Modeling


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Calibration of Global FX Models


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A consistent model can still use local models but as parts in ahierarchy of controllable approximations.

The tree is devised in such a way to achieve stable historicalcorrelations and to minimize volatilities of crosses in the tree

The USD-EUR-JPY triangle needs to be calibrated separatelyby fixing the interest rate processes in the three currencies andoptimizing the ROOT centered crosses

All other crosses in the minimum spanning tree are calibratedseparately

Crosses not in the tree are implied

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Calibration of Global Credit Models


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Global credit correlations are estimated to achievesimultaneous consistency with the CDX-IG, ITX-IG and theCDX-HY index tranche spreads

Recognizing the great importance of credit vegas, the creditprocesses are calibrated also against CDS index options

The credit model assumes stochastic interest rates andcorrelates recovery rates to the interest rate and credit cycle

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Cumulative loss distribution for the CDX-IG index CDO


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Cumulative loss distribution for the CDX-HY index CDO


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Conclusions


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We propose to resolve counterparty credit risk of global

portfolios of netting sets against a multi-tiered capitalstructure by means of coherent global market simulationsunder the risk neutral measure

We achieved sub-linear scaling and real time performanceby leveraging on a rigorous application of the FundamentalTheorem of Finance and single node technologies

Our approach goes beyond the EPE/CVA methodology

Banking is the very last industry sector that didnt make ahigh tech transition yet. This is now overdue!

We believe that the impact of technology innovation onbanking practices, business models, mathematical models andsoftware architectures will soon occur

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