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Challenges in xVA Pricing and RiskAndrew McClelland, PhDDirector of Quantitative ResearchNumerixMatlab Computational Finance ConferenceNYC 2017September 28, 2017Andrew McClelland, NumerixxVA Pricing and Risk1/13

Presentation OutlineQuick overview of Numerix and why we prioritize xVAWhat is xVA? Some intuition for how things come togetherComputational di culties encountered in xVAMulti-factor hybrid models and American Monte Carlo1Matlab for complicated exposures modeling and aggregations1Used when referring to Least-Squares Monte Carlo, a la Longsta andSchwartz ('01), in the context of exposures calculations.Andrew McClelland, NumerixxVA Pricing and Risk2/13

Numerix Overview 12 founded in '96, headquartered in NYC, 250 employeesNumerixIt has origins as a pricing & risk library for derivativesOriented towards exotic & multi-factor derivatives,eg .PRDCsManaging risk for exotics is a very complicated processRequires sophisticated & delicate models, methods, calibrations2https://www.numerix.com.Andrew McClelland, NumerixxVA Pricing and Risk3/13etc .

Numerix Overview 13 founded in '96, headquartered in NYC, 250 employeesNumerixIt has origins as a pricing & risk library for derivativesOriented towards exotic & multi-factor derivatives,eg .PRDCsManaging risk for exotics is a very complicated processRequires sophisticated & delicate models, methods, calibrations3https://www.numerix.com.Andrew McClelland, NumerixxVA Pricing and Risk3/13etc .

Pricing Work owrate curvesMarketDatavol surfacesFX levelsforward ratesCalibratedModelshort rate factorslocal/stochastic volExoticValuationMonte CarloPDElattice/treeStandard pricing work ow. First, market data such as volatility surfaces aresourced, transformed and cleansed. Second, a pricing model is calibrated to the marketdata. Third, a pricing method such as Monte Carlo is invoked to compute the price.Figure:Andrew McClelland, NumerixxVA Pricing and Risk3/13

Risk Work owrate curvesMarketMarketDataData'vol surfacesFX levelsforward ratesCalibratedCalibratedModelModel'short rate factorslocal/stochastic volExoticExoticValuationValuation'Figure:Monte CarloPDElattice/treeStandard risk work ow. Risk factors among market data and model elements,eg . yield nodes and model parameters, are identi ed as risk factors. Sensitivities againstthese factors are computed and are hedged using vanilla instruments.Andrew McClelland, NumerixxVA Pricing and Risk3/13

Risk Work owrate ation'vol surfacesFX levelsforward ratesCalibratedCalibratedModelModel'short rate factorslocal/stochastic volExoticExoticValuationValuation'Figure:Monte CarloPDElattice/treeStandard risk work ow. Risk factors among market data and model elements,eg . yield nodes and model parameters, are identi ed as risk factors. Sensitivities againstthese factors are computed and are hedged using vanilla instruments.Andrew McClelland, NumerixxVA Pricing and Risk3/13

Risk Work owrate ation'vol surfacesFX levelsforward ratesCalibratedCalibratedModelModel'short rate factorslocal/stochastic volExoticExoticValuationValuation'Figure:Monte CarloPDElattice/treeStandard risk work ow. Risk factors among market data and model elements,eg . yield nodes and model parameters, are identi ed as risk factors. Sensitivities againstthese factors are computed and are hedged using vanilla instruments.Andrew McClelland, NumerixxVA Pricing and Risk3/13

Numerix Overview 2After crisis exotics volumes dropped but xVAs,eg .CVA, emergedBig challenge for banks and managing xVA resembles managing exoticsNice use case for how our clients use a mix of NX and MatlabContent should be interesting for this audience alsoSits within larger issue ofholisticAndrew McClelland, Numerixmodeling of trade impactxVA Pricing and Risk4/13

Numerix Overview 2After crisis exotics volumes dropped but xVAs,eg .CVA, emergedBig challenge for banks and managing xVA resembles managing exoticsNice use case for how our clients use a mix of NX and MatlabContent should be interesting for this audience alsoSits within larger issue ofholisticAndrew McClelland, Numerixmodeling of trade impactxVA Pricing and Risk4/13

Numerix Matlab InterfaceFigure: Example of a CVA calculation using the Matlab interface for NX in combinationwith native Matlab functionality.Andrew McClelland, NumerixxVA Pricing and Risk4/13

Numerix Overview 2After crisis exotics volumes dropped but xVAs,eg .CVA, emergedBig challenge for banks and managing xVA resembles managing exoticsNice use case for how our clients use a mix of NX and MatlabContent should be interesting for this audience alsoSits within larger issue ofholisticAndrew McClelland, Numerixmodeling of trade impactxVA Pricing and Risk4/13

Numerix Overview 2After crisis exotics volumes dropped but xVAs,eg .CVA, emergedBig challenge for banks and managing xVA resembles managing exoticsNice use case for how our clients use a mix of NX and MatlabContent should be interesting for this audience alsoSits within larger issue ofholisticAndrew McClelland, Numerixmodeling of trade impactxVA Pricing and Risk4/13

What is xVA?Traditional valuations involve the risk-neutral valuation formulaClassical Valuationh RTie t e t r (u) du C(T )V (t) E45Many ways to derive this formula, equilibrium , arbitrage ,etc .6 is a bit restrictive but mimics bank practiceHedging approachClassical hedging assumptions ignore certain frictions eg .counterparty credit risk, funding spreads & collateral4eg . Cox, Ingersol and Ross ('85).eg . Harrison and Kreps ('79).6eg . Black and Scholes ('73) or Burgard and Kjaer ('11).5Andrew McClelland, NumerixxVA Pricing and Risk5/13

What is xVA?Traditional valuations involve the risk-neutral valuation formulaClassical Valuationh RTie t e t r (u) du C(T )V (t) E78Many ways to derive this formula, equilibrium , arbitrage ,etc .9 is a bit restrictive but mimics bank practiceHedging approachClassical hedging assumptions ignore certain frictions eg .counterparty credit risk, funding spreads & collateral7eg . Cox, Ingersol and Ross ('85).eg . Harrison and Kreps ('79).9eg . Black and Scholes ('73) or Burgard and Kjaer ('11).8Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartySwap HedgeCounterpartyVanillaSwapsStructured ianClearing HouseFigure:Basic considerations within hedging derivation of valuation rule.Andrew McClelland, NumerixxVA Pricing and Risk5/13

What is xVA?Traditional valuations involve the risk-neutral valuation formulaClassical Valuationh RTie t e t r (u) du C(T )V (t) EMany ways to derive this formula, equilibrium10 , arbitrage11 ,etc .12 is a bit restrictive but mimics bank practiceHedging approachClassical hedging assumptions ignore certain frictions eg .counterparty credit risk, funding spreads & collateral10eg . Cox, Ingersol and Ross ('85).eg . Harrison and Kreps ('79).12eg . Black and Scholes ('73) or Burgard and Kjaer ('11).11Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartySwap HedgeCounterpartyVanillaSwapsStructured ianClearing HouseModern considerations within hedging derivation of valuation rule, includingcounterparty credit risk, funding costs, initial margin requirements.Figure:Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartySwap HedgeCounterpartyVanillaSwapsStructured anClearing HouseModern considerations within hedging derivation of valuation rule, includingcounterparty credit risk, funding costs, initial margin requirements.Figure:Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartyCDSContractsSwap HedgeCounterpartyVanillaSwapsStructured anClearing HouseModern considerations within hedging derivation of valuation rule, includingcounterparty credit risk, funding costs, initial margin requirements.Figure:Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartyCDSContractsSwap HedgeCounterpartyVanillaSwapsStructured SwapBankTreasuryFundingCustodianClearing HouseClienttounccgAdinFuninitiInalrgMaModern considerations within hedging derivation of valuation rule, includingcounterparty credit risk, funding costs, initial margin requirements.Figure:Andrew McClelland, NumerixxVA Pricing and Risk5/13

Valuations via Hedging ArgumentsCDS HedgeCounterpartyCDSContractsSwap HedgeCounterpartyVanillaSwapsStructured SwapBankTreasuryFundingCustodianClearing HouseClienttounccgAdinFuninitiInalrgMaModern considerations within hedging derivation of valuation rule, includingcounterparty credit risk, funding costs, initial margin requirements.Figure:Andrew McClelland, NumerixxVA Pricing and Risk5/13

What is CVA?Counterparty credit risk credit valuation adjustment"13 or CVABasically upfront cost of dynamically hedging counterparty defaultMust be added to classical value, or cooked into deal paymentsCredit Valuation AdjustmentCVA(t)et E Lifetime Discounted Counterparty Loss(t, T ) IFRS13 (accounting standard) requires CVA be recorded in statementsBasel III extended capital framework to include a CVA charge13eg . Du e and Canabarro ('04) and Gregory ('15).Andrew McClelland, NumerixxVA Pricing and Risk6/13

What is CVA?Counterparty credit risk credit valuation adjustment"14 or CVABasically upfront cost of dynamically hedging counterparty defaultMust be added to classical value, or cooked into deal paymentsCredit Valuation AdjustmentCVA(t)et E Lifetime Discounted Counterparty Loss(t, T ) IFRS13 (accounting standard) requires CVA be recorded in statementsBasel III extended capital framework to include a CVA charge14eg . Du e and Canabarro ('04) and Gregory ('15).Andrew McClelland, NumerixxVA Pricing and Risk6/13

What is CVA?Counterparty credit risk credit valuation adjustment"15 or CVABasically upfront cost of dynamically hedging counterparty defaultMust be added to classical value, or cooked into deal paymentsCredit Valuation Adjustment ZetCVA(t) ET Discounted Counterparty Loss(s) dstIFRS13 (accounting standard) requires CVA be recorded in statementsBasel III extended capital framework to include a CVA charge15eg . Du e and Canabarro ('04) and Gregory ('15).Andrew McClelland, NumerixxVA Pricing and Risk6/13

What is CVA? Counterparty credit risk credit valuation adjustment"16 or CVABasically upfront cost of dynamically hedging counterparty defaultMust be added to classical value, or cooked into deal paymentsCredit Valuation Adjustment ZetCVA(t) ETe Rstr (u) du λ(s)(V (s)) dstIFRS13 (accounting standard) requires CVA be recorded in statementsBasel III extended capital framework to include a CVA charge16eg . Du e and Canabarro ('04) and Gregory ('15).Andrew McClelland, NumerixxVA Pricing and Risk6/13

What is CVA? Counterparty credit risk credit valuation adjustment"17 or CVABasically upfront cost of dynamically hedging counterparty defaultMust be added to classical value, or cooked into deal paymentsCredit Valuation Adjustment ZetCVA(t) ETe Rstr (u) du λ(s)(V (s)) dstIFRS13 (accounting standard) requires CVA be recorded in statementsBasel III extended capital framework to include a CVA charge17eg . Du e and Canabarro ('04) and Gregory ('15).Andrew McClelland, NumerixxVA Pricing and Risk6/13

CVA Complications: Counterparty AggregationCrucial to note that CVA is computed at theAssume swaps with valuesV1swapandV2swapcounterpartylevelwith counterpartyPortfolio exposure is not the sum of individual swap exposuresPortfolio Exposure Aggregation(V pfolio ) (V1swap V2swap ) 6 (V1swap ) (V2swap ) ClearlyV2swap 0Thus thecould o setincrementalV1swap 0,andvica versaCVA involves whole counterparty portfolioWhy xVA calcs are so expensive. some xVAs involve many c/parties!Andrew McClelland, NumerixxVA Pricing and Risk7/13