Esimaion of Fuure Iniial Margins Muli-Curve Ineres Rae Framework Marc Henrard Advisory Parner - OpenGamma Visiing Professor - Universiy College London
March 2016 2 Esimaion of Fuure Iniial Margins 1 Iniial margin, muli-curve and collaeral framework 2 Raional model 3 IM dynamics 4 Margin value adjusmen
March 2016 3 Margins 1 Iniial margin, muli-curve and collaeral framework 2 Raional model 3 IM dynamics 4 Margin value adjusmen
March 2016 4 Variaion Margin - Iniial Margin - MPR 15 Pas Fuure 10 IM Value 5 VM 0-5 -10-20 -15-10 -5 0 5 10 Time in days
March 2016 5 Variaion Margin - Iniial Margin - MPR 15 Pas Fuure Cash flows / Value 10 5 0 IM -5-10 -20-15 -10-5 0 5 10 Time in days
March 2016 6 Regulaory ime able 2013 Mandaory clearing (USA) 2016 2019 Mandaory Cenral Clearing (Europe) EMIR Caegory 1: 21 June 2016 Fron-loading as of 21 February 2016 2016 2019 Mandaory Bilaeral margin Caegory 1: 1 Sepember 2016 for VM and IM
March 2016 7 Cash collaeral pricing formula ( ) N c = exp c τ dτ 0 Theorem (Collaeral wih cash price formula) In presence of cash collaeral wih rae c, he quoe a ime of an asse wih price V c u a ime u is V c = N c E Q [ (N c u) 1 V c ] u F for some measure Q (idenical for all asses, bu poenially currency-dependen). Noe ha he resul refers o hree objecs : V u, c and Q. This formula is also called collaeral accoun discouning.
March 2016 8 Collaeral: pseudo-discoun facors Definiion (Collaeral pseudo-discoun facors) The collaeral (pseudo-)discoun facors for he collaeral rae c paid in currency X are defined by P c (, u) = N c E Q [ (N c u) 1 F ]. In he sequel we will work wih OIS discouning and use he noaion D u = (N c u) 1. Change of numeraire is sill possible in his framework. In paricular we will inroduce a differen measure, called M, and use he noaion (D ) 1 E Q [ D u V c ] u F = (h ) 1 E M [ h u V c ] u F
March 2016 9 Muli-curve framework wih collaeral I The value of a j floaing coupon in currency X wih collaeral a rae c is an asse for each enor j, each fixing dae θ and each collaeral rae c. Definiion (Forward index rae wih collaeral) The forward curve F c,j (θ, u, v) is he coninuous funcion such ha, P c (, v)δf c,j (, u, v) is he quoe a ime of he j-ibor coupon wih fixing dae θ, sar dae u, mauriy dae v ( 0 u = Spo( 0 ) < v) and accrual facor δ collaeralised a rae c.
March 2016 10 Margins 1 Iniial margin, muli-curve and collaeral framework 2 Raional model 3 IM dynamics 4 Margin value adjusmen
March 2016 11 Raional model Macrina, A. (2014), Crépey, S. e al. (2015) Formulas for he discouning and forward in he raional model are P c (, u) = Pc (0, u) + b 1 (u)a (1) P c (0, ) + b 1 ()A (1) L c,j (; u, v) = Lc,j (0; u, v) + b 2 (u, v)a (1) + b 3 (u, v)a (2) P c (0, ) + b 1 ()A (1) where A (i) = exp A (i) ( is a maringale ) in a M-measure wih a i X (i) 1 2 a2 i 1 where j is an Ibor index and v u = Tenor(j). Noe: The L c,j (; u, v) in he raional model corresponds o he P c (, v)f c,j (, u, v) in he sandard muli-curve framework.
March 2016 12 Raional model - spread Le s define L c (; T i 1, T i ) = 1 δ i ( P c (, T i 1 ) P c (, T i ) ) wih δ i he accrual facor for he period [T i 1, T i ]. The dynamics for his quaniy is described by L c (; T i 1, T i ) = Lc (0; T i 1, T i ) + (b 1 (T i 1 ) b 1 (T i ))/δ i A (1) P c (0, ) + b 1 ()A (1) L c,j (; u, v) = Lc (0) + (b 1 (u) b 1 (v))/δ i A (1) P c (0, ) + b 1 ()A (1) + (Lc,j (0) L c (0)) + (b 2 b 1 )A (1) + b 3 (u, v)a (2) P c (0, ) + b 1 ()A (1)
March 2016 13 Raional model - calibraion Calibraion one-facor model, erm srucure. 90 80 Marke Calibraed Black vol (%) 70 60 50 40 30 20 10 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 P10YxP3Y P10YxP4Y P10YxP5Y P10YxP7Y
March 2016 14 Raional model - calibraion Calibraion one-facor model, smile. 60 Marke 5Yx5Y Calibraed 5Yx5Y Marke 5Yx10Y Calibraed 5Yx10Y 55 Black vol (%) 50 45 40 35-1 -0.5 0 0.5 1 Moneyness (%)
March 2016 15 Margins 1 Iniial margin, muli-curve and collaeral framework 2 Raional model 3 IM dynamics 4 Margin value adjusmen
March 2016 16 Iniial margin Iniial margin are usually compued as Value-a-Risk (VaR) or Expeced Shorfall (ES) using hisorical or Mone-Carlo approaches. Direc brue force calculaion of fuure IM and MVA migh be expensive. Hisorical VaR usually implies full revaluaion Some simplifying mehods neglec sochasic ineracions beween IM and marke. Nesed Mone Carlo ieraions are very expensive. Our approach: Characerize he iniial margin process in erms of he dynamics of he underlying processes and a condiional risk measure.
March 2016 17 Absrac formulaion Iniial Margin The iniial margin process {IM } 0 T associaed o he porfolio is a process such ha where IM = λ (Z,+δ ) Z,+δ are he cash flows given defaul associaed wih he porfolio beween imes, + δ (loss given defaul) δ is he margin period of risk (ime o close ou ) {λ } 0 T is a family of condiional risk measures
March 2016 18 Risk measure Condiional risk measures: λ maps any F measurable r.v. o an F -measurable r.v. wih finie expecaion. Examples: If P denoes he subjecive probabiliy observed by a CCP: VaR α,p [X] = ess inf {Θ : P [X Θ F ] α a.s., [ Θ is F -measurable} ] ES α,p [X] = VaR α,p [X] + 1 1 α EP (X VaR α,p [X]) + F Noaion: λ (X) = λ ( X)
March 2016 19 Risk measure Properies: We use several properies saisfied by VaR α,p and ES α,p ; Le X, Y be wo F measurable r.v. and le Θ be an F -measurable r.v. wih E P [Θ] <. We assume: Normalizaion: λ (0) = 0 P-a.s. Monooniciy : X Y(P a.s.) λ (X) λ (Y). (P-a.s.) Condiional posiive homogeneiy: if Θ 0(P a.s.) λ (ΘX) = Θλ (X). Condiional ranslaion: λ (X + Θ) = λ (X) + Θ
March 2016 20 Risk measure Cashflow decomposiion: V = 1 D E Q [ n i=1 D Ti C Ti F ] Value given defaul: (signs for member, assuming closing ou exacly a δ ): Z,+δ = 1 D(, + δ) V +δ }{{} close ou value Cashflows missed in [, + δ) {}}{ n 1 + D(, T i ) C T i 1 {Ti [,+δ)} i=1 V }{{} Variaion margin a D: discouning erm associaed o he funding of CCP (may differ from D)
March 2016 21 Risk measure Cashflow decomposiion: V = 1 D E Q [ n i=1 D Ti C Ti F ] Value given defaul: (signs for member, assuming closing ou exacly a δ ): Z,+δ = 1 D(, + δ) V +δ }{{} close ou value Cashflows missed in [, + δ) {}}{ n 1 + D(, T i ) C T i 1 {Ti [,+δ)} i=1 V }{{} Variaion margin a D: discouning erm associaed o he funding of CCP (may differ from D)
March 2016 22 Iniial margin compuaion ( 1 IM = λ D(, + δ) V +δ V + n i=1 ) 1 D(, T i ) C T i 1 {Ti [,+δ)} Raional framework leads o complexiy reducion: Profi from he explici/semi-explici expressions available for mos common IR derivaives wih raional framework under he M measure: V = V (A (1), A (2) ) ; C = C (A (1), A (2) ) Exploi numerically he simplify seup Inroduce hisorically esimaed change of measure beween M and P (CCP scenario informaion)
March 2016 23 Iniial margin compuaion IM = κes α,p ( h +δ h V +δ V + n i=1 h Ti h C +δ 1 {Ti [,+δ)} Raional framework leads o complexiy reducion: Profi from he explici/semi-explici expressions available for mos common IR derivaives wih raional framework under he M measure: V = V (A (1), A (2) ) ; C = C (A (1), A (2) ) Exploi numerically he simplify seup Inroduce hisorically esimaed change of measure beween M and P (CCP scenario informaion) )
March 2016 24 Iniial margin compuaion IM = κes α,p ( h +δ h V +δ V + n i=1 h Ti h C +δ 1 {Ti [,+δ)} Raional framework leads o complexiy reducion: Profi from he explici/semi-explici expressions available for mos common IR derivaives wih raional framework under he M measure: V = V (A (1), A (2) ) ; C = C (A (1), A (2) ) Exploi numerically he simplify seup Inroduce hisorically esimaed change of measure beween M and P (CCP scenario informaion) )
March 2016 25 Change of measure beween M and P Change of measure a ime δ: Comparing he model and hisorical disribuion of P&L 1 0.8 0.6 0.4 0.2 10-4 PnL CCP vs. model model CCP daa 1 0.8 0.6 0.4 0.2 10-4 PnL CCP vs. adj. model model CCP daa 0-2 -1 0 1 2 10 4 0-2 -1 0 1 2 10 4 Change of measure afer ime δ: Using independen incremens assumpion and ieraing
March 2016 26 IM: Ineres rae swaps Assume no paymens in [, + δ]. Then: ( IM = κes α,p Sw h +δ Sw +δ h Since he swap price processes saisfy Sw = c 0() + c 1 ()A (1) + c 2 ()A (2) P(0; ) + b 0 ()A (1), he properies of he risk measure and he assumed model imply IM = C 0 + C 1 v(r ) + C 2 v(r ) where C 0, C 1, C 2, R are F -meas. (depend only on, A (1), A (2) ( ), ) v(x) := κes α,p Y (1) δ + xy (2) δ ; Y (i) δ := A (i) δ + 1 for i = 1, 2. )
March 2016 27 Funcions v, v The funcions v, v we jus defined are smooh and increasing and can be approximaed by a grid wih few elemens. 20 VaR and ES, α=0.003 35 VaR and ES, α=0.997 10 30 0 25-10 20-20 15-30 10-40 5-50 0-60 VaR ES -70-30 -20-10 0 10 20 30 R *Values obained wih 4 10 7 MC ieraions -5 VaR ES -10-15 -10-5 0 5 10 15 R
March 2016 28 Numerical esimaion (IM) IM for a 5 years fuure swap wih enor of 6 monhs 3.5 3 2.5 2 1.5 1 0.5 4 104 IM - ES 5 x 5 swap- M 0 0 2 4 6 8 10 Mean value (yellow) and 90% and 10% perceniles (red and blue)
March 2016 29 Numerical ess: speed CPU ime in prooype implemenaion of MC 2 and he refined mehod Pahs Grid MC 2 Refined Mehod (s) Accel. (s) poins Iniial Evoluion Toal facor 1250 120 28.52 5.73 0.20 5.93 4.81 1250 520 126.32 5.76 0.76 6.52 15.83 2500 120 94.21 5.72 0.23 5.95 19.37 2500 520 423.37 5.73 0.92 6.65 63.66 1250 pahs 5yrs hisory; 2500 10yrs hisory. 120 monhly periodiciy; 520 weekly periodiciy.
March 2016 30 Margins 1 Iniial margin, muli-curve and collaeral framework 2 Raional model 3 IM dynamics 4 Margin value adjusmen
March 2016 31 Cos of IM: Funding rae Beween imes u and δ u, a clearing member pays on average he funding cos: funding rae {}}{ IM u δ u r f u Assumpion: The reasury of a clearing member funds all liquidiy requiremens by securing a baske of funds wih bes-maching mauriies. r f := M k=1 ( ) γ k L, T k i (), T k i k k ()+1 + A (3) r where 1,..., M R + are mauriies γ 1,..., γ M [0, 1] wih M k=1 γ k = 1 weighs A (3) is an idiosyncraic facor.
March 2016 32 MVA Then, he (Follmer Schweizer) price of MVA can be modeled as [ ] MVA = 1 T E Q D u r f u IM u du D F + H [ ] = 1 T E M h u r f u IM u du h F + H where H is a maringale orhogonal o he radable asses. IM and r f are Markovian: funcions of he underlying facors. In he swap porfolio case and A (3) 0, he MVA is hedgeable and H 0.
March 2016 33 Numerical MVA MVA for a 5 years fuure swap wih enor of 6 monhs (A (3) 0) 600 MVA wih IM - ES 5 x 5 swap- M 500 400 300 200 100 0 0 2 4 6 8 10 Mean value (yellow) and 90% and 10% perceniles (red and blue)
March 2016 34 Conclusion Proposed a mehod based on explici compuaion of he IM as a dynamic process by iself. Numerical implemenaion can become efficien as we remove one layer of numerical effor. The implemenaion is basically porfolio size invarian: only he cash flow descripion (c i ()) is porfolio composiion dependen. The approach clearly differeniae beween pricing and risk measures. In paricular he IM compuaion includes he ail used in he acual CCP margin compuaion. The dynamic IM can be used as a base o MVA compuaions.
March 2016 35 Presenaion based on he paper C. A. Garcia Trillos, M. P. A. Henrard, A. Macrina (2016) Esimaion of Fuure Iniial Margins in a Muli-Curve Ineres Rae Framework hp://ssrn.com/absrac=2682727 Conac OpenGamma Web: www.opengamma.com Email: info@opengamma.com Twier: @OpenGamma Europe OpenGamma 185 Park Sree London SE1 9BL Unied Kingdom Norh America OpenGamma 125 Park Avenue New York, NY 10017 Unied Saes