A Complete XVA Valuation Framework - iruizconsulting.com

1 A Complete XVA Valuation FrameworkWhy the Law of One Price is deadIgnacio Ruiz January 2015 Version a book of derivatives has become quite a complicated task, evenwhen those derivatives are simple in nature. This is the effect of the newtrading environment, highly dominated by credit, funding and capital this paper the author formally sets up a global Valuation Framework thataccounts for market risk (risk neutral price), credit risk (CV A), fundingrisk (FV A) of self-default potential hedging (LV A), collateral (CollV A) andmarket hedging positions (HV A), as well as tail risk (KV A). These pricingmetrics create a Framework in which we can comprehensively value tradingactivity. An immediate consequence of this is the emergence of a potentialdifference between fair value accounting and internal accounting.

2 This pieceof work also explains the difference between both of them, and how to performcalculations in both worlds in a realistic and coherent manner, demonstratingvia arbitrage-impossibility arguments that an XVA frameworks should be usedin both the past few years we have witnessed the birth of a number of price value ad-justments. It all started with CVA, but then we moved to DVA, FVA, CollVA, KVA, Founding Director, iRuiz Consulting, London. Ignacio is an independent consultant in quantitative riskanalytics with a special focus in XVA. Prior to this, he was the head strategist for counterparty riskand exposure measurement at Credit Suisse, and Head of Market and Counterparty Risk Methodologyfor equities at BNP Paribas. Contact: [7, 3, 4, 13, 12, 10, 1, 9, 2, 6, 11]. The emergence of these pricing metrics has beensomewhat irregular, naturally creating some degree of confusion in the goal of this piece of work is to set up a formal Framework for all those value ad-justments, as well as to introduce some other ones generally left out in the are going to formally calculate the value of a book of derivatives, accounting for allthe associated cash flows that come not only from the derivative itself, but from the actof hedging and managing default risk, funding risk and capital costs.

3 To the author sknowledge, this is the first time that a global Valuation Framework , that accounts forCV Aasset,CV Aliab(DV A),CV Across,FV A,LV A,CollV A,HV AandKV Ain a solidand comprehensive manner, is put to it, we are going to see how a key concept to understand derivative valuationis the difference between Price and Value. By Price it is meant the exit price that goesinto a balance sheet for fair value accounting. By Value we mean how much a bookof derivatives is worth to an institution. Those two concepts, being highly related, paper demonstrates that the value of a book of derivatives is not the same for allmarket players, and any useful Valuation Framework should reflect so. We are going to seethat the idea of risk-neutral Valuation for fair value accounting, being a good theoreticalframework, cannot be currently used because it is based in the very idea of derivativearbitrage, that cannot be exercise in the real market as that theory are going to start with the well known risk neutral pricing with CVA.

4 Many readerswill be familiar with it, but it is good to refresh it here to ensure the Valuation frameworkused in subsequent sections is well understood. Further to it, we are going to extendthat Framework to a general XVA one, in whichFV A, viaLV A,CollV AandHV A,together withKV Aare set up. Once the XVA Valuation structure is set, we are going todiscuss each term to understand what they do and don t mean, as well as implicationsfor pricing, Valuation and risk management. Finally, we will compare the results tothe classic risk-neutral Valuation Framework , and discuss why the XVA one is moreappropriate at Valuation FrameworkDuring the recent years, there has been a profusion of adjustments to the risk-neutralprice of an OTC derivative, often referred to as X-Value Adjustments (XVA).We are going to introduce the idea of XVA in two steps. First considering CVA pricingfrom a risk-neutral standpoint, and then introducing a derivative Valuation through afull XVA Framework .

5 By CVA we are going to mean the bilateral pricing and Valuation frameworks are based in the fundamental theorem of assetpricing, that states that the fair value of a financial product today is the expectationA Complete XVA Valuation Framework2of the present value (PV) of its future cash ( iPV(future cash flowi))(1)We assume that there exists a risk-free interest rate (ru) between the time pointsuandu+du, so that the present value of a generic future cash flow (Xt) att, to be deliveredby a default-free entity, is given byPV0=e t0ruduXt(2)In other words,DF0,t=e t0ruduis the riskless discount we are trying to value a derivative that is going to have a future cash flowsXt=xtdtbetweentandt+dt, then, iPV(future cash flowi) = T0e t0ruduxtdt(3)whereTis the maturity of the derivative. Consequently,P0=E( T0e t0ruduxtdt)(4)Given that so far we live in the risk-neutral world, this price should be the same asthat obtained by the Black-Scholes-Merton model.

6 That model equates the price ofa derivative (a set of future cash flows contingent on some external risk factors like,typically, interest rates, FX prices, etc) to the price of hedging out its risk and producingon this way a risk-less portfolio. In that context, the price of a derivative and its evolutionis given by Equations 5 and Pt=rPt rS Pt S(5)L= t+ 2S22 2 S2(6)Risk-neutral pricing with counterparty riskRisk neutral pricing with counterparty risk has been well explained in the literature[7]. Let s introduce it here to set up the subsequent Valuation Framework , as well as forA Complete XVA Valuation financial derivative is a contract between two entities, the counterparties , to exchangea number of cash flows up to the maturity date. During this time, one or both of thesecounterparties may , the classic risk-neutral pricing theory does not contemplate that any marketplayer can default.

7 In order to incorporate this, let s say that Any counterparty of the derivative can buy credit protection insurance on the othercounterparty defaulting (typically in the form of a Credit Default Swap (CDS) ). That these credit protection contracts have unlimited liquidity and no transactioncosts. That the external entity selling them cannot it will be well known by the reader, the expectation of a generic quantityZin thefuture can be obtained by summing the product of the value ofZin each possible eventby the probability of each event happening. In other words,E(Z) = iPiZi(7)wherePiis the probability of eventiandZiis the value ofZif eventitakes we have a bilateral derivative contract with a counterparty, there are four events thatmay happen in the future interval fromttot+dt, subject to both counterparties havingsurvived up to the time pointt:1. That both counterparties are survived att+dt,2.

8 That we survive up tot+dt, but our counterparty defaults during the interval(t,t+dt),3. That we default during the interval (t,t+dt), but our counterparty survives up tot+dt,4. That both counterparties default during the interval (t,t+dt).Let s say that there is a default intensity so that the default probability of an entityin the intervalt+dtis given by tdt. In this Framework , the survival probability of thatentity up to the time pointt, subject to being alive att= 0, is given byS0,t=e t0 udu(8)1 Those readers familiar with CVA may want to skip this Complete XVA Valuation Framework4A snap shot of the four possible events we are facing, with their probabilities (Pi)2andthe cash flows that would occur in each of them, is shown in the following tableEventPiin (t,t+dt)Cash flow1 Sourt,t+dtScptyt,t+dtxt dt2 Sourt,t+dt cptytdt (1 RRcptyt)P+t3 ourtdtScptyt,t+dt (1 RRourt)P t4 ourt cptytdt (1 RRcptyt)P+t (1 RRourt)P twherextdtis the cash flow that takes place in the derivative in the interval (t,t+dt)if no default happens,Ptis the price of the derivative at timet3,P+t= max(Pt,0),P t= min(Pt,0)

9 AndRRrepresents the recovery rate obtained by the surviving partywhen a default we say that the survival probability in an infinitesimal time stepSt,t+dt'1, andnoting also that the probability of all these events must be multiplied by the probabilityof both counterparties having survived attSour0,tScpty0,t=e t0( ouru+ cptyu)du(9)then the price of the derivative that accounts for counterparty risk is given byPCptyRisk0=E( T0e t0(ru+ ouru+ cptyu)duxtdt) (10)E( T0e t0(ru+ ouru+ cptyu)du cptyt(1 RRcptyt)P+tdt) (11)E( T0e t0(ru+ ouru+ cptyu)du ourt(1 RRourt)P tdt) (12)E( T0e t0(ru+ ouru+ cptyu)du ourt cptyt((1 RRcptyt)P+t (1 RRourt)P t)dt)(13)2 Remembering that thesePiare subject to both counterparties having survived up to speaking,Ptshould be the replacement value of an equivalent derivative should a default replacement trade would be with a counterparty with equivalent credit quality of the defaultedentity.

10 This leads to two problems: firstly, what credit quality should we use? one second beforethe company defaults? one year before? This is not clear. Secondly, this creates a mathematicalrecursive loop, asPtshould contain also a counterparty risk adjustment. It is market practice toignore this refinement in the calculation because it is very difficult to solve and, importantly, it hardlymakes any relevant difference in most practical should be noted that both cash flows in terms 2 and 3 must have a negative sign. In the case of2, because it is a loss that we could incur. In the case of 3, it is a net gain, butP tis a negativenumber, and so it needs a negative sign to counteract Complete XVA Valuation Framework5where we have sum across all possible time points int5. Each of those terms can becalledP 0(Eq. 10),CV Aasset(also known as CVA, Eq. 11),CV Aliab(also known asDVA, Eq.)