XVAs Series: CVA. An Introduction. María R. Nogueiras March 2016

Transcription

1 XVAs Series: CVA An Introduction María R. Nogueiras March 2016

2 XVAs Before the crisis: the price of derivatives was computed evaluating the expected return and risk of the underlying asset (at a trade level) After the crisis: New Risks that not negiglible anymore are taken into account in pricing, risk managemant and regulation (at a counterparty, netting set, and/or portfolio level) Counterparty Credit risk (CVA) Own Credit risk (DVA) Funding Cost/Benefit (FVA) Capital Costs (KVA) Initial Margin on CCPs, Initial Margin on new CSA (MVA) and Friends Date: 24 de marzo de

3 Counterparty Risk The counterparty credit risk is defined as the risk that the counterparty to a transaction could default before the final settlement of the transactions cash flows. An economic loss would occur if the transactions or portfolio of transactions with the counterparty has a positive economic value at the time of default. Basel II. If we have an open position with a counterparty C, with value V; if the counterparty defaults we may loose the exposure E: E = máx(v,0) In fact, we may loose a percentage of the position value, the Loss Given Default, LGD Similarly, we recuperate a percentage of the portfolio value, the recovery rate R, LGD = 1 R... and Friends Date: 24 de marzo de

4 In Risk contexts we are used to answer the question (concept of Credit VAR): How much can I loose of this portfolio, within one year, at a some confidence level, due to default risk and exposure? LGD EAD PD where EAD denotes the exposure at default and the PD is the probability of default. Under Basel II, the risk of counterparty default and credit migration risk were addressed but mark-to-market losses due to credit valuation adjustments (CVA) were not. During the financial crisis, however, roughly two-thirds of losses attributed to counterparty credit risk were due to CVA losses and only about one-third were due to actual defaults. Basel Committee on Banking Supervision, BIS (2011). Counterparty Risk was measured before the crisis, but not for (risk-neutral) pricing. The perception of counterparty risk (probability of default) was low.... and Friends Date: 24 de marzo de

5 From[B-11-slides]... and Friends Date: 24 de marzo de

6 Credit Valuation Adjustment or CVA is CVA Definition The market price of credit/counterparty risk on a financial instrument that is markedto-market The price of a CDS on the counterparty with a contingent nominal equal to the exposure. The reduction in price we ask to our counterparty C for the fact that C may default... Market convection is: CVA = V(default free) V(risky)... and Friends Date: 24 de marzo de

7 Credit Risk versus Counterparty Risk Both terms are referred to the default risk. In general, we talk about credit risk when considering loans, bonds,... (or credit derivatives as CDSs, CDOs, etc) For a loan, for instance, the exposure/value is (almost) certain/deterministic and it is always positive E = máx(v,0) = V And the uncertainty about the instrument is mainly due to credit risk. When modelling, prices can expressed as functions of default probabilitities and LGD (and discounted cashflows). Counterparty risk is in this context a more general concept The exposure is uncertain and not always positive; in fact it is a nonlinear function of the derivative/position value.... and Friends Date: 24 de marzo de

8 In the general, when talking about counterparty risk and CVA: The exposure is an option on future (netting set/position with my counterparty) prices: Non linearity Even the most simple derivative will need a model specification to compute CVA (non static replication): Model dependent If fact, not only the (various, different asset classes) underlyings s volatilities are required, but also the probability of default of the counterparty, and correlation between underlyings and probability of default: Hybrid models required... and Friends Date: 24 de marzo de

9 Unilateral CVA Assuming that the bank B can not default, and the counterparty C is risky, we can derive the following expression: ] CVA(t) = E [Lgd C 1 τc TD(t,τ C )(V(τ C )) + F t The model is formulated under risk neutral measure, with numeraire the bank account with r t the risk free rate at time t, and dβ t = r t β t dt D(t,τ) = e τ t r sds : stochastic discount factor between t and τ. Lgd C : Loss given default of the counterparty τ C : Time of default of the counterparty... and Friends Date: 24 de marzo de

10 V(t): is the risk-free value of the portfolio for the Bank (Assumption on the close out!) F t : is the sigma-algebra with the information up to time t (market risk and counterparty credit) Idea of the proof Lets omit the notation C, and let us introduce R = 1 Lgd π(s 1,s 2 ): cash flows from s 1 to s 2 discounted at time s 1 ; they may be contingent but they are independent of the default of the counterparty By definition V(s) = E[π(s,T) F t ], and π(s,t)+d(s,t)π(t,u) = π(s,u) Let us introduce the corresponding (counterparty) risky quantities: ˆπ and ˆV... and Friends Date: 24 de marzo de

11 From the point of view of B we have the general risky payoff ˆπ(t,T) = 1 τ T π(t,t)+ 1 τ<t [π(t,τ)+ D(t,τ) ( Rec(V(τ)) + ( V(τ)) +)] Taking the risk neutral expectations, ˆV(t) = E[π(t,τ)1 τ<t F t ]+E[π(t,T)1 τ T F t ]+ E [ R D(t,τ)(V(τ)) + 1 τ<t F t ] E [ D(t,τ)( V(τ)) + 1 τ<t F t ] By using that (V(t)) + ( V(t)) + = V(t), V(t) = E[π(t,T) F t ], ˆV(t) = V(t) E[(1 R)D(t,τ)(V(τ)) + 1 τ<t F t ]... and Friends Date: 24 de marzo de

12 If we denote by G t the filtration to time t just containing the market state variable and not the credit worthiness of the counterparty we can write CVA(t) = T t f C (t,u)e[(1 R)D(t,u)(V(u)) + G t,τ = u]du with f C (t,u) the probability density function of variable time to default τ = τ C. with λ C the counterparty hazard rate f C (t,u) = λ C (u)e u t λ C(s)ds Notice that the expected value is conditioned to the default of the counterparty!... and Friends Date: 24 de marzo de

13 Unilateral CVA and Simplifications To get the standard CVA formula that is usually implemented, we have to make further assumptions. The recovery is a deterministic quantity CVA(t) = (1 R) T t f τ (u) E[D(t,u)V(u)) + G t,τ = u] du... and Friends Date: 24 de marzo de

14 Bucketing time t = t 0 < t 1 <... < t n = T and approximatting the integral CVA(t) = (1 R) or, also n 1 i=0 P(τ [t i,t i+1 ]) E[D(t,t i+1 )(V(t i+1 ) + G t,τ [t i,t i+1 ])] n 1 CVA(t) = (1 R) (G C (t,t i ) G C (t,t i+1 ) E[D(t,t i+1 )(V(t i+1 ) + G t,τ [t i,t i+1 ])] i=0 with G C (t,s) the probability of being solvent at time s (contidioned to be solvent at time t).... and Friends Date: 24 de marzo de

15 Credit risk is independent of any market risk factors CVA(t) = (1 R) n 1 i=0 P(τ [t i,t i+1 ]) E[D(t,t i+1 )(V(t i+1 ) + G t ] n 1 CVA(t) = (1 R) (G C (t,t i ) G C (t,t i+1 ) E[D(t,t i+1 )(V(t i+1 ) + G t ] i=0 Making explicit the hazard rates and the risk neutral rate r: n 1 CVA(t) = (1 R) (e t i t λ(s)ds e t i+1 t λ(s)ds )E[e t i+1 t r(s)ds (V(t i+1 ) + G t )] i=0 Warning Big Assumption: WWR, credit derivatives, etc... and Friends Date: 24 de marzo de

16 It λ = λ C is the counterparty hazard rate G C (t,s) = e s t λ(s)ds With this notation, the default probability at time s, F C (t,s) is F C (t,s) = 1 G C (t,s) = 1 e s t λ(s)ds And, consistently with the previous definitions, the probability density function of time to default τ C (for time s conditined to survival at t) is f C (t,s) = FC(t,s) s = λ(s)e s t λ(u)du... and Friends Date: 24 de marzo de

17 Unilateral DVA If we consider same assumptions from the counterparty point of view, i.e., the Bank is unrisky and the counterparty is risky, we will arrive to the symmetrical quantity called Debit Value Adjustment. More precisely, if we denote by V X, and ˆV X the risk neutral and risky prices from the point of view of X we have, V B (t) = V C (t), ˆVB (t) = ˆV C (t) ˆV C (t) = (V B (t) CVA B (t)) = V C (t)+cva B (t) where CVA B (t) = E[(1 R C )D(t,τ C )(V B (τ C )) + 1 τc <T F t ] Let us introduce the quantity DVA C (t) = E[(1 R C )D(t,τ C )( V C (τ C )) + 1 τc <T F t ] = CVA B (t) Therefore ˆV C (t) = V C (t)+dva C (t).... and Friends Date: 24 de marzo de

18 DVA is the increase in value I need to pay to enter a deal with a counterparty because I am default risky. DVA introduces symmetry in prices, (theoretically) the law of one price again. Controversial quantity Paradox: My DVA increases (and therefore my portfolio value) when my credit worthiness decreases Impossible to hedge: Selling protection on myself? We can not sell our own CDSs; we could trade on our own debt, or hedging with proxies, as CDSs indexes (but problems when actual defaults). Basel III does not consider DVA when computing CVA charges, but FASB aproved it. Double counting with FVA?... and Friends Date: 24 de marzo de

19 WWR Important (and common) assumption: Independency between Counterparty default and market value of the portfolio (exposure); it does not take into account the possible correlation between them. If fact, we may have cases (specific wrong way risk): with Wrong Way Risk: my exposure is higher when my counterparty is more likely to default Example: I have a put option on my counterparty with Right Way Risk: my exposure is higher when my counterparty is less likely to default Example: I have a call option on my counterparty... and Friends Date: 24 de marzo de

20 We may consider also systemic wrong way risk: Example: My portfolio exposure increases if the interest rates decrease; in general, in crisis periods the probability of defaul of any counterparty increases, and we have low interest rates. Example: During the crisis, wrong way risk were especially severe for monolines, and was not recognized in most credit models In general, default rates are highly correlated between them; therefore a credit derivatives are likely exposed to WWR. WWR is, in general, very difficult to model! Common practice is assuming independence, and analysing separately cases of WWR.... and Friends Date: 24 de marzo de

21 Bilateral CVA Now assuming that both the bank and the counterparty are risky, we can compute both CVA and DVA from the banks point of view CVA B (t) = E[(1 R C )D(t,τ C )(V B (τ C )) + 1 τc <τ B <T F t ] DVA B (t) = E[(1 R B )D(t,τ B )( V B (τ B )) + 1 τb <τ C <T F t ] And finally ˆV B (t) = V B (t) CVA B (t)+dva B (t)... and Friends Date: 24 de marzo de

22 With simplifications: deterministic R, bucketing, and independence between default and the rest of market risk factors, we obtain CVA(t) = (1 R) n 1 i=0 P(τ C [t i,t i+1 ],t i+1 < τ B ) E[D(t,t i+1 )(V(t i+1 ) + G t )] Modelling the dependence between default times τ C and τ B is difficult (1st to default modelling), not only because of the models themselves but also because a correlation parameter between defaults needs to be specified. Standard assumption is that they are independt, under which P(τ C [t i,t i+1 ],t i+1 < τ B ) = (G C (t,t i ) G C (t,t i+1 ))G B (t,t i+1 ))... and Friends Date: 24 de marzo de

23 Risk Mitigants: Netting and Collateral Netting:This is the agreement to net all positions towards a counterparty in the event of the counterparty default. CCR calculations are typically computed at a counterparty and a netting set level. Collateral:It is a guarantee that is deposited in a collateral account in favour of the investor party facing the exposure. If the depositing counterparty defaults, thus not being able to fulfill payments associated to the above mentioned exposure, collateral can be used by the investor to offset its loss Adding the collateral to the CVA formula: CVA B (t) = E[(1 R C )D(t,mín(τ C ))(V B (τ C ) M(τ C )) + 1 τc <τ B <T F t ] with M(t) the value of the collateral portfolio at time t.... and Friends Date: 24 de marzo de

24 Ideal CSA contract: Instantaneaous posting, no thresholds, no minimum transfer amounts, cash, etc. The collateral would inmediatly reduce the CVA to zero. Realistic CSA contract: The collateral is posted with certain frequency; there is a grace period that consideres the number of days to realise a counterparty has defaulted; we may have desagreements about the collateral to post, the post may be no cash etc. There is still GAP RISK and a remaining CVA. Even under daily collateralization there can be large mark to market swings that make collateral rather ineffective. This is called GAP RISK and is one of the reasons why Central Clearing Counterparties (CCPs) and the new standard CSA have an initial margin as well.... and Friends Date: 24 de marzo de

25 Gap Risk and MPoR The GAP RISK is the possibility of a market crash between the time the counterparty defaulted and the time when close-out of the positions is completed. Margin period of risk (MPoR) is the length of time between the default event and the time when positions with the counterparty are closed out If we consider a marging period of risk of δ M(t) = f(v(t δ),x 1,X 2,...) where X i are characteristics of the collateral contract, as threshold, minimum transfer amount etc In the case of 0 thresholds, for instance CVA B (t) = E[(1 R C )D(t,mín(τ C ))(V B (τ C ) V B (τ C δ)) + 1 τc <τ B <T F t ] Very recent MPoR model in [APS-16-paper].... and Friends Date: 24 de marzo de

26 Close-out value We have assumed that the value of the residual deal computed at the closeout time was the risk-neutral value. However, there is a debate on that: Risk neutral residual may generate a discountinuity in the mark to market prices We may have considered a replacement closeout, where the remaining deal is priced by taking into account the credit quality of the surviving party and of the party that replaces the defaulted one. Obviously, this second approach is more complicated Still a debate: What is the risk neutral rate? (when no collateral)... and Friends Date: 24 de marzo de

27 Particular cases: Analytical Exposures Equity Forward Let us consider an Equity forward which pays at time T the strike K and receives S(T). Assmuning E ] [ [D(t,T j )(V(T j )) + F t = E t D(t,T j ) (S(T ) ] j ) Ke r(t T + j) Ft which is the prime of a call option on S maturing at T j with strike Ke r(t T j).... and Friends Date: 24 de marzo de

28 IRS Let us consider a payer IRS, swaping fixed coupons K by floating coupons at times T a+1,...,t b, and let i = T i T i 1 [ ] E t D(t,T j )(V(T j,t)) + F t = E t D(t,T j ) (TS(T j ;T j,t b ) K) b D(T j,t k ) k j + F t which is the time t swaption prime of a swaption with maturity T j, underlying T b T j and strike K.... and Friends Date: 24 de marzo de

29 General case: Implementation Even with the simplified (unilateral) formula, but we still need to compute the expected exposure at a netting set level. Except from some particular, mainly theoretical, but very useful, exercises (see [SS-15- paper]), the expected exposure is computed by simulation: Monte Carlo simulation for generating different states of the world are calculated (risk factor simulation), Pricing of the instruments at scenario level. Simplifications in some of the pricers are needed (to avoid nested Monte Carlo simulations, for instance). Aggregation rules are performed (taking into account netting, collateral, etc).... and Friends Date: 24 de marzo de

30 Incremental CVA We are insterested in computing the impact of adding a new instrument to the netting set. It can only be calculated on a differential basis, that is through two calculations, one with the original portfolio and one with the new portfolio. From the computational efficiency point of view: We should be computing MC scenarios and intruments prices (all of them, both the portfolio and the new instrument) once; and then developing the aggregations twice: with the new instrument and without.... and Friends Date: 24 de marzo de

31 Allocated CVA For simplicity, we omit the time-dependency and the discount factors in the notation. quantities for i = 1,...,n portfolio instru- The objective is to compute CVA i atributed ments CVA = i CVA i atributed Let V i be the trade i value. The LGD and default probabilities are counterparty dependent and not trade-dependent. Therefore our problem is to compute EEallocated i such that EE = i EE i allocated However, in general, expected exposure EE = EE(V 1,...,V n ) = E[( i V i ) + ] is not additive in the trades.... and Friends Date: 24 de marzo de

32 Let us define auxiliary functions and variables: V i = α i u i f(α 1,...,α n ) = EE(α 1 u 1,...,α n u n ) = E[( i α i u i ) + ] Since f is homogeneous order 1, Eulers theorem gives EE(α 1 u 1,...,α n u n ) = i EE α i (α 1 u 1,...,α n u n ) In case α i = 1 Finally EE(V 1,...,V n ) = i EE α i (V 1,...,V n ) EE i allocated(v 1,...,V n ) = EE α i (V 1,...,V n ) One way to compute the above partial derivative is to change the size of a transaction by a small value and calculate the marginal EE using a finite difference.... and Friends Date: 24 de marzo de

33 Alternatively, it can be also computed via a conditional expectation ([PR-10-paper]): Using that ( E ( ) ( x i ) + = E ( ) x i )1 ( i x i) + >0 i i And for α i = 1 EE ( ) (α 1 u 1,...,α n u n ) = E u i 1 α ( i α iu i ) + >0 i EE ( ) (V 1,...,V n ) = E V i 1 α ( i V i) + >0 i The intuition behind the above formula is that the future values of the trade in question are added only if the netting set has positive value at the equivalent point.... and Friends Date: 24 de marzo de

34 Further discussion: Risk or Price when modelling CVA and Counterparty Risk Management Focused on risk management and capital. Model under real-world distributions. Mostly driven by regulations CVA and Counterparty Risk Pricing Focused on pricing and hedging. Model under risk-neutral distributions. Mostly driven by business... and Friends Date: 24 de marzo de

35 However, the debate about which measure/calibration to use is still open. See [HSW-14-paper] for a joint measure model!? Compared to VAR, for instance, in CCR (and CVA) we consider the whole life of the netting set, not just short term movements. In general In the short term: the drift componente is dominated by the volatility component In the long term: the drift term becomes dominant Therefore we are interested not only in measuring well the volatility but also the drift.... and Friends Date: 24 de marzo de

36 ... and the Regulation For a review on the regulation (Basel III and FRTB) see [G-16-slides]. Basell III: Current, 2 approaches Standarized: CVA capital charge as a function of EAD Advanced: Approval for CCR and Market Risk IMM required If full revaluation VAR: Formula for CVA similar to the unilateral CVA models (next slide) Only credit sensitivity considered (exposure remains constant in scenarios) Exposure calculated from CRR engines, in general historical calibration, IMM approval and backtesting required... and Friends Date: 24 de marzo de

37 Basel III regulation asks explicitly for market implied probability of default and LGD in the CVA ((for CVA VAR computations) as: where CVA = LGD mkt PD i 1,i = máx i ( ) EEi 1 D i 1 +EE i D i PD i 1,i 2 ( ( 0,exp s i 1t i 1 LGD mkt ) exp ( s it i LGD mkt is the approximation of the probability of default in the interval [T i 1,T i ], and LGD mkt the loss given default based on market expectations and not historical estimates. )) s j CDS spread for time T j EE j expected exposure for time T j D j discount factor for time T j... and Friends Date: 24 de marzo de

38 What is coming: new methodology allows for 3 approaches Basic CVA FRTB-CVA Standarized Approach: SA-CVA Internal Model approach: IMA-VA Important: for FRTB sensitivities are needed! Sensitivities for SA-CVA with finite differences; but in general the industry is moving towards AAD, LS-regression based methods, etc More details can be found in [G-16-slides]... or in another London Quants and Friends talk?... and Friends Date: 24 de marzo de

39 Some References [APS-16-paper] Rethinking Margin Period of Risk. Andersen, L. and Pykhtin, M. and Sokol, A. (2016) [BMP-13-paper] Counterparty Credit Risk, Collateral and Funding: With Pricing Cases For All Asset Classes. Brigo, D. and Morini, M. and Pallavicini, A. (2013) [B-11-slides] Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending. Brigo, D. (2011) [B-11-slides] Nonlinear valuation under credit gap risk, collateral margins, funding costs and multiple curves. Brigo, D. (2015) [BM-05-paper] A Formula for Interest Rate Swaps Valuation under Counterparty Risk in presence of Netting Agreements. Brigo, D. and Masetti, M. (2005) [G-16-book] XVA Credit, Funding and Capital Valuation Adjustments, Green, A. (2016)... and Friends Date: 24 de marzo de

40 [G-16-slides] FRTB, FRTB-CVA and implications for capital valuation adjustment (KVA), Green, A. (2016) [G-12-book] Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets, Gregory, J. (2012) [G-15-book] The XVAs chagenllenge: Counterparty Credit Risk, Funding, Collateral and Capital, Gregory, J. (2015) [HSW-14-paper] Modeling the Short Rate: The Real and Risk-Neutral Worlds,Hull, Sokol, White (2014) [M-15-slides] XVAs: Funding, Credit, Debit and Capital in pricing. Morini, M. (2015) [PR-10-paper] Pricing Counterparty Risk at the Trade Level and CVA Allocations, Pykhtin, M.and Rosen, D. (2010) [SS-15-paper] Potential Future Exposure (PFE), Credit Value Adjustment (CVA) and Wrong Way Risk (WWR) Analytic Solutions Syrkin, M. and Shirazi, A. (2015)... and Friends Date: 24 de marzo de