Multi-Curve Interest Rate Framework Estimation of Future Initial Margins Marc Henrard Advisory Partner - OpenGamma Visiting Professor - University College London
March 2016 2 Estimation of Future Initial Margins Initial margin, multi-curve and collateral framework 1 Rational model 2 IM dynamics 3 Margin value adjustment 4
March 2016 3 Margins Initial margin, multi-curve and collateral framework 1 Rational model 2 IM dynamics 3 Margin value adjustment 4
March 2016 4 Variation Margin - Initial Margin - MPR Past Future 15 10 IM Value 5 VM 0 -5 -10 -20 -15 -10 -5 0 5 10 Time in days
March 2016 5 Variation Margin - Initial Margin - MPR Past Future 15 10 Cash flows / Value 5 IM 0 -5 -10 -20 -15 -10 -5 0 5 10 Time in days
March 2016 6 Regulatory time table 2013 Mandatory clearing (USA) 2016 – 2019 Mandatory Central Clearing (Europe) EMIR Category 1: 21 June 2016 Front-loading as of 21 February 2016 2016 – 2019 Mandatory Bilateral margin Category 1: 1 September 2016 for VM and IM
March 2016 7 Cash collateral pricing formula (∫ t ) N c t = exp c τ d τ 0 Theorem (Collateral with cash price formula) In presence of cash collateral with rate c, the quote at time t of an asset with price V c u at time u is V c t = N c t E Q [ ( N c u ) − 1 V c � ] � F t u for some measure Q (identical for all assets, but potentially currency-dependent). Note that the result refers to three “objects”: V u , c and Q . This formula is also called collateral account discounting.
March 2016 8 Collateral: pseudo-discount factors Definition (Collateral pseudo-discount factors) The collateral (pseudo-)discount factors for the collateral rate c paid in currency X are defined by P c ( t , u ) = N c ( N c u ) − 1 � t E Q [ ] � F t . In the sequel we will work with OIS discounting and use the notation D u = ( N c u ) − 1 . Change of numeraire is still possible in this framework. In particular we will introduce a different measure, called M , and use the notation D u V c � h u V c � ( D t ) − 1 E Q [ ] = ( h t ) − 1 E M [ ] � F t � F t u u
March 2016 9 Multi-curve framework with collateral I The value of a j floating coupon in currency X with collateral at rate c is an asset for each tenor j , each fixing date θ and each collateral rate c . Definition (Forward index rate with collateral) The forward curve F c , j t ( θ, u , v ) is the continuous function such that, P c ( t , v ) δ F c , j ( t , u , v ) is the quote at time t of the j-Ibor coupon with fixing date θ , start date u, maturity date v (t ≤ t 0 ≤ u = Spot ( t 0 ) < v) and accrual factor δ collateralised at rate c.
March 2016 10 Margins Initial margin, multi-curve and collateral framework 1 Rational model 2 IM dynamics 3 Margin value adjustment 4
March 2016 11 Rational model Macrina, A. (2014), Crépey, S. et al. (2015) Formulas for the discounting and forward in the rational model are P c ( t , u ) = P c (0 , u ) + b 1 ( u ) A (1) t P c (0 , t ) + b 1 ( t ) A (1) t L c , j ( t ; u , v ) = L c , j (0; u , v ) + b 2 ( u , v ) A (1) + b 3 ( u , v ) A (2) t t P c (0 , t ) + b 1 ( t ) A (1) t where A ( i ) is a martingale in a M -measure with t ( ) A ( i ) a i X ( i ) = exp 2 a 2 i t − 1 − 1 t t where j is an Ibor index and v − u = Tenor ( j ) . Note: The L c , j ( t ; u , v ) in the rational model corresponds to the P c ( t , v ) F c , j ( t , u , v ) in the standard multi-curve framework.
March 2016 12 Rational model - spread Let’s define L c ( t ; T i − 1 , T i ) = 1 ( P c ( t , T i − 1 ) − P c ( t , T i ) ) δ i with δ i the accrual factor for the period [ T i − 1 , T i ] . The dynamics for this quantity is described by L c ( t ; T i − 1 , T i ) = L c (0; T i − 1 , T i ) + ( b 1 ( T i − 1 ) − b 1 ( T i ))/ δ i A (1) t P c (0 , t ) + b 1 ( t ) A (1) t L c (0) + ( b 1 ( u ) − b 1 ( v ))/ δ i A (1) L c , j ( t ; u , v ) t = P c (0 , t ) + b 1 ( t ) A (1) t +( L c , j (0) − L c (0)) + ( b 2 −· b 1 · ) A (1) + b 3 ( u , v ) A (2) t t P c (0 , t ) + b 1 ( t ) A (1) t
Calibration one-factor model, term structure. Rational model - calibration Black vol (%) 10 20 30 40 50 60 70 80 90 0 P1YxP10Y P2YxP3Y P2YxP4Y P2YxP5Y P2YxP7Y P2YxP10Y P3YxP4Y P3YxP5Y P3YxP7Y P3YxP10Y P4YxP4Y P4YxP5Y P4YxP7Y P4YxP10Y P5YxP4Y P5YxP5Y P5YxP7Y P5YxP10Y P7YxP3Y P7YxP4Y P7YxP5Y P7YxP7Y P7YxP10Y P10YxP1Y P10YxP2Y Calibrated Market P10YxP3Y P10YxP4Y P10YxP5Y P10YxP7Y March 2016 13
March 2016 14 Rational model - calibration Calibration one-factor model, smile. Market 5Yx5Y 60 Calibrated 5Yx5Y Market 5Yx10Y Calibrated 5Yx10Y 55 Black vol (%) 50 45 40 35 -1 -0.5 0 0.5 1 Moneyness (%)
March 2016 15 Margins Initial margin, multi-curve and collateral framework 1 Rational model 2 IM dynamics 3 Margin value adjustment 4
March 2016 16 Initial margin Initial margin are usually computed as Value-at-Risk (VaR) or Expected Shortfall (ES) using historical or Monte-Carlo approaches. Direct brute force calculation of future IM and MVA might be expensive. Historical VaR usually implies full revaluation Some simplifying methods neglect stochastic interactions between IM and market. Nested Monte Carlo iterations are very expensive. Our approach : Characterize the initial margin process in terms of the dynamics of the underlying processes and a conditional risk measure .
March 2016 17 Abstract formulation Initial Margin The initial margin process { IM t } 0 ≥ t ≤ T associated to the portfolio is a process such that IM t = λ t ( Z t , t + δ ) where Z t , t + δ are the cash flows given default associated with the portfolio between times t , t + δ ( loss given default ) δ is the margin period of risk ( time to “close out” ) { λ t } 0 ≤ t ≤ T is a family of conditional risk measures
March 2016 18 Risk measure Conditional risk measures : λ t maps any F measurable r.v. to an F t -measurable r.v. with finite expectation. Examples : If P ∗ denotes the subjective probability observed by a CCP: [ X ] = VaR α, P ∗ t ess inf { Θ : P ∗ [ X ≤ Θ |F t ] ≤ α a . s ., Θ is F t -measurable } 1 − α E P ∗ [ ] [ X ] = VaR α, P ∗ [ X ] + ( X − VaR α, P ∗ [ X ]) + |F t ES α, P ∗ 1 t t t Notation : ¯ λ t ( X ) = − λ t ( − X )
March 2016 19 Risk measure Properties : We use several properties satisfied by VaR α, P ∗ t and ES α, P ∗ ; Let X , Y be two F− measurable r.v. and let Θ be t an F t -measurable r.v. with E P ∗ [Θ] < ∞ . We assume: Normalization: λ t (0) = 0 P -a.s. Monotonicity : X ≤ Y ( P − a.s. ) ⇒ λ t ( X ) ≤ λ t ( Y ) . ( P -a.s.) Conditional positive homogeneity: if Θ ≥ 0( P − a . s . ) ⇒ λ t (Θ X ) = Θ λ t ( X ) . Conditional translation: λ t ( X + Θ) = λ t ( X ) + Θ
March 2016 20 Risk measure Cashflow decomposition: [ n � ] � V t = 1 ∑ D T i C T i E Q � � F t D t � i =1 Value given default: (signs for member, assuming ”closing out” exactly at δ ): Cashflows missed in [ t , t + δ ) � �� � n 1 1 ∑ Z t , t + δ = D ( t , t + δ ) V t + δ D ( t , T i ) C T i 1 { T i ∈ [ t , t + δ ) } V t + − ¯ ¯ ���� i =1 � �� � Variation margin at t “close out” value D : discounting term associated to the funding of CCP ¯ (may differ from D )
March 2016 21 Risk measure Cashflow decomposition: [ n � ] � V t = 1 ∑ D T i C T i E Q � � F t D t � i =1 Value given default: (signs for member, assuming ”closing out” exactly at δ ): Cashflows missed in [ t , t + δ ) � �� � n 1 1 ∑ Z t , t + δ = D ( t , t + δ ) V t + δ D ( t , T i ) C T i 1 { T i ∈ [ t , t + δ ) } V t + − ¯ ¯ ���� i =1 � �� � Variation margin at t “close out” value D : discounting term associated to the funding of CCP ¯ (may differ from D )
March 2016 22 Initial margin computation ( n ) 1 1 ∑ D ( t , t + δ ) V t + δ − V t + D ( t , T i ) C T i 1 { T i ∈ [ t , t + δ ) } IM t = λ t ¯ ¯ i =1 Rational framework leads to complexity reduction: Profit from the explicit/semi-explicit expressions available for most common IR derivatives with rational framework under the M measure: V t = V t ( A (1) , A (2) ) ; C t = C t ( A (1) , A (2) ) Exploit numerically the simplify setup Introduce historically estimated change of measure between M and P ∗ (CCP scenario information)
March 2016 23 Initial margin computation ( n ) h t + δ h T i ∑ IM t = κ ES α, P ∗ V t + δ − V t + C t + δ 1 { T i ∈ [ t , t + δ ) } t h t h t i =1 Rational framework leads to complexity reduction: Profit from the explicit/semi-explicit expressions available for most common IR derivatives with rational framework under the M measure: V t = V t ( A (1) , A (2) ) ; C t = C t ( A (1) , A (2) ) Exploit numerically the simplify setup Introduce historically estimated change of measure between M and P ∗ (CCP scenario information)
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