Topics from the xVA desk •xVA with Threshold and Independent Amount •Netting Intuition •Standalone/Incremental/Marginal xVA •Other challenges on the xVA desk [email protected], Counterparty Credit & Funding Risk, Danske Bank, Markets xVA with Threshold and Independent Amount Threshold and Independent Amount •If the ISDA (Master Agreement) is supported by a CSA (Credit Support Annex) the counterparty credit risk will be mitigated to a certain extend that depends on the specific details of the CSA. •The CSA stipulates that Collateral must be exchanged when the exposure of the derivatives portfolio covered exceeds a given Threshold, and when the difference between collateral exchanged and current exposure exceeds a given Minimum Transfer Amount. •In addition to the collateral exchanged to cover the exposure, an Independent Amount may be exchanged, and sometimes delivered by both parties at the same time. •If the Threshold is zero (or very low), and Minimum Transfer Amount is very low, and Collateral can be called for on a daily basis, the exposure is reduced significantly to be a matter of Close-Out risk (not covered further in this talk). •For any significant Threshold level the exposure below contributes to the CVA. •If an Independent Amount is received, only the exposure above contributes to the CVA. •Depending on the right to rehypothecate, the Independent Amounts delivered and received should be handled carefully for DVA and FCA. Source: www.danskebank.com/CI 2 xVA with Threshold and Independent Amount (cont.) Threshold and Independent Amount •With V(t) denoting the portfolio value at time t, and H the level of the threshold, the exposure driving CVA (and FCA) is given by: Exposure(t) max(min(H,V (t)),0) •If in addition IA denotes the independent amount received, the exposure driving CVA is given by: Exposure(t) max(min(H IA,V (t) IA),0) Graphically this may be expressed by: Exposure Comparison, H=30, IA=10 60 50 40 30 20 10 0 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 max(V,0) max(min(H,V),0) max(min(H-IA,V-IA),0) Source: www.danskebank.com/CI 3 xVA with Threshold and Independent Amount (cont.) Credit Valuation Adjustment - recap of LS-MC approach •CVA is defined as the following (ignoring recovery R for simplicity): T CVA E V (t) ( t)dt 0 •If V can be computed in closed form or through a quick model we are done (but have to take the pain of deriving closed form expressions or implement quick models). •We can do LS-MC on V to get a proxy (with the tilde), and evaluate CVA as: T ~ CVA E V (t) ( t)dt 0 •This puts a high demand on the proxy, which needs to be very close for all states of the underlying variables, even in the extremes. Source: www.danskebank.com/CI 4 xVA with Threshold and Independent Amount (cont.) Credit Valuation Adjustment - recap of LS-MC approach •To reduce the dependency on the proxy the following alternative CVA calculation is used: T CVA E[ V (t) ( t)dt] 0 T E[ V (t) 1~ ( t)dt] 0 V(t)0 regression proxy T T E[ Et [ c(u) du]1~ ( t)dt] 0 t V (t)0 future cashflow T T E[ c(u)1~ ( t)dudt] 0 t V (t)0 T u E[ 1~ ( t)dt c(u)du] 0 0 V (t)0 CVA notional •We now only depend on the proxy close to zero. Source: www.danskebank.com/CI 5 xVA with Threshold and Independent Amount (cont.) Credit Valuation Adjustment with Threshold •To take a threshold H into account we can modify the CVA slightly: T CVA E[ max(min(V (t), H ),0) ( t)dt] 0 T H E[ V (t) min(1, ) ( t)dt] 0 V (t) T H E[ V (t)1~ min(1, ~ ) ( t)dt] 0 V (t)0 V (t) T T H E[ Et [ c(u)du]1~ min(1, ~ ) ( t)dt] 0 t V (t)0 V (t) •The dependence on the proxy is now stronger, but still most important around zero and around and above H! •Same trick can be applied to FCA and DVA, taking into account that Threshold may be different for counterparty and investor (us). Source: www.danskebank.com/CI 6 xVA with Threshold and Independent Amount (cont.) Credit Valuation Adjustment with Threshold and Independent Amount •To take a threshold H and an independent amount IA received into account we can modify the CVA slightly more (assume H >> IA) : T CVA E[ max(min(V (t) IA, H IA),0) ( t)dt] 0 T IA H IA E[ V (t) max(min(1 , ),0) ( t)dt] 0 V (t) V (t) T IA H IA E[ V (t)1~ max(min(1 ~ , ~ ),0) ( t)dt] 0 V (t)0 V (t) V (t) T T IA H IA E[ Et [ c(u)du]1~ max(min(1 ~ , ~ ),0) ( t)dt] 0 t V (t)0 V (t) V (t) •The dependence on the proxy is now even stronger, but still most important around zero , around IA, and around and above H-IA! •If IA received may not be rehypothecated (used for funding) the exposure used for FCA is Source:unchanged. www.danskebank.com/CI 7 xVA with Threshold and Independent Amount (cont.) Credit Valuation Adjustment with Threshold and Independent Amount •Graphically we can compare the impact on the CVA Notional: Value Decomposition, H=30, IA=10 CVA Notional Comparison, H=30, IA=10 60 1.20 50 1.00 40 0.80 30 0.60 20 0.40 10 0.20 0 0.00 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 IA max(min(H-IA,V-IA),0) Collateral CVA_Ntl CVA_Ntl(H) CVA_Ntl(H,IA) Source: www.danskebank.com/CI 8 xVA with Threshold and Independent Amount (cont.) Expected Positive/Negative Exposure with Threshold and Independent Amount •Example (i): 1B EUR 10Y IRS, Counterparty Pays Floating, Investor (us) Pays Fixed, IRS set ATM before XVA. Exposure, Threshold = Infinite, IA = 0 Exposure, Threshold = 50M EUR, IA = 10M EUR 70,000,000 70,000,000 60,000,000 60,000,000 50,000,000 50,000,000 40,000,000 40,000,000 30,000,000 30,000,000 20,000,000 20,000,000 10,000,000 10,000,000 0 0 -10,000,000 -10,000,000 -20,000,000 -20,000,000 -30,000,000 -30,000,000 -40,000,000 -40,000,000 EPE ENE EE EPE ENE EE Source: www.danskebank.com/CI 9 xVA with Threshold and Independent Amount (cont.) Expected Positive/Negative Exposure with Threshold and Independent Amount •Example (i): We can decompose the exposure into exposure captured by IA and collateral recevied/posted beyond the Threshold. Exposure (stacked), Threshold = 50M EUR, IA = 10M EUR 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000,000 0 -10,000,000 -20,000,000 -30,000,000 -40,000,000 Positive Collateral Positive Exposure Positive IA Negative Collateral Negative Exposure Negative IA Source: www.danskebank.com/CI 10 Netting Intuition Expected Positive Exposure (EPE) is an option on a portfolio •Most quants have a developed intuition for what impacts the value of a call option on a single underlying. Often this intuition is derived from careful study of the Black-Scholes model and the background theory. •This intuition is less developed for options on the sum or difference of several underlyings, say basket or spread options, nor for best-of types of options. •For analyzing Credit Valuation Adjustment and related xVAs, the ability to understand netting effects between different trades, or risk factors in a derivatives portfolio is crucial! •Most often this analysis is done on a before-and-after basis, hence the comparison is between the existing netting set and the netting set augmented with a new trade (alternatively reduced by a terminated trade). The resulting xVA impact is called the Incremental xVA. •When an xVA is decomposed into the contributions of different trades or risk factors we consider it a Marginal xVA. •Before analyzing how to compute these in the xVA framework, we consider some intuitive approaches for understanding the EPE and ENE impact. Source: www.danskebank.com/CI 11 Netting Intuition (cont.) Expected Positive Exposure (EPE) is an option on a portfolio •In practice a typical OTC derivatives netting set may consist of 1000s of trades with exposure across different asset classes and derivative types. •In a pricing situation it is close to impossible (within the time-frame given) to analyze the individual trades of the netting set, and their co-dependency structure in detail. •However, the derivative type and main risk factors of the single (or few) additional trade is known. Further analytics, such as exposure profiles (across time) for the netting set, the additional trade, and the augmented netting set allows heuristic reasoning for incremental xVA impacts. •Consider: n Net NetNew Net New V (t) X i (t) EPE (t) EPE (t) EPE (t) i1 and V New (t) X (t) n1 EPE NewNet (t) E V New (t) V Net (t) EPE Net (t) EV Net (t) Net Net New Net Net ENE (t) EV (t) E EV (t) V (t) V (t) EE Net (t) EV Net (t) EPE Net (t) ENE Net (t) Source: www.danskebank.com/CI 12 Netting Intuition (cont.) Expected Positive Exposure (EPE) is an option on a portfolio •Example (ii): Consider the following hypothetical exposure measures EPE, ENE, and EE observed for a single time point.