A UNIFIED PDE MODELLING FOR CVA AND FVA

Transcription

1 AWALEE A UNIFIED PDE MODELLING FOR CVA AND FVA By Dongli W JUNE 2016 EDITION

2 AWALEE PRESENTATION Chaper 0 INTRODUCTION The recen finance crisis has released he counerpary risk in he valorizaion of he derivaives The difference beween he evaluaions before and aferisis is compued as x value adjusmen (xva) Due o he possibiliy of he counerpary s defaul, he credi value adjusmen (CVA) is firsly inroduced ino xva In order o cover his risk, he endency is he rading wih collaeral The cos of he collaeral is referenced as he liquidiy value adjusmen (LVA) 1 in xva Anoheonsequence of he crisis is he absence of a unified funding rae (risk free) The refunding rae of he bank causes he funding valuaion adjusmen (FVA) as he hird componen of xva Consequenly, we need o inegrae he xva (CVA+FVA+LVA) ino he classic pricing PDE In general, he counerpary s defaul can be modeled wih hree levels (independen, immersion and beyond immersion) of he dependency on he underlying price In he chaper 1 and 2, we inroduce briefly he general modelling of he defaul ime and xva modelling on a porfolio under he immersion case The annex gives more deails on he adequaeness of he immersion case In he las chaper, we give some numeric resuls under he dynamic Gaussian copula model In he defaul-free period (before crisis), any derivaive produc wih payoff/claim Ψ(X T ) a mauriy T has he fair value a ime, wih X defined in (1) Acually, i is he probabilisic soluion of he following classic Black-Sholes EDP, In he filraion G, he behavior of he (F, Q)-Brownian moion B will be changed under he influence of he informaion on he defaul ime Here, we assume The (F, Q) -Brownian moion B remains a semi-maringale wih he canonical decomposiion, where W is a (G, Q)-Brownian moion So, he dynamics can be wrien in in G (2) (3) Chaper 1 F/G FILTRATIONS We consider a defaul free filraion F generaed by a Brownian moion The filraion G is enlarged progressively from F by a defaul ime τ wih he indicaor process H = 1 ha is, {τ } G = F σ( s H s ) We assume ha i exiss a pricing measure Q on he filraion G in G The defaul ime τ admis a G predicable inensiy I can be decomposed in wo par: an F predicable process λ (pre-defaul) before he ime of he defaul and 0 afer he defaul Under he wo previous assumpions, he relaionship beween he filraion F and he filraion G can be defined according o he hree differen cases In he filraion F, we suppose ha he dynamics of he underlying price follows in F (1) where B is he (F, Q)-Brownian moion From he no-arbirage condiion, he process r in (1) should be he repo rae of he underlying under he pricing measure Q Under he hypohesis ha he OIS rae is equal o he repo rae as before crisis, we denoe he OIS rae by r o represen he repo rae Independen case, ha is, he pre-defaul inensiy process λ is deerminisic and γ Immersion case, ha is, γ Beyond immersion case, ha is λ, is a funcion of (, B ) and γ r For he non independen case, he impac bilaeral beween τ and B is given respecively by λ (no deerminisic) and Δ Under he pricing measure Q, he choice of he immersion case, ha is Δ 0, will be jusified in he nex chaper and he credi risk/ counerpary risk quan focus on he modeling of he λ o rea he dependence beween τ and B 1 I is also referenced as he collaeral rae adjusmen for some praciioner AWALEE NOTES 1

3 Chaper 2 XVA(CVA/LVA/FVA) PDE MODELING 1 CVA PART Le τ be he defaul ime of he counerpary on he derivaive produc wih he defaul free fair value M defined in (2) The -ime value U is he counerpary risky fair value for he bank The no-arbirage condiion and he compleeness of he marke 2 give he (G, Q) -maringale condiion on U(, X ), ha is, 3 THE CHOICE OF γ Recall ha under he pricing measure Q, he drif par γ in he dynamics of X should be he repo rae Furhermore, we assumed he repo rae is always close o he OIS rae, ha is, in G or equivalenly γ In his case, he XVA-PDE is given by (4) where Ū is he derivaive value jus afer he defaul of he counerpary For he no collaeral case, Ū is given by wih R he recovery rae (assumed consan) In he following par, he bank s funding rae and he collaeral rae will be involved in he fair value U 2 FVA/LVA PART A CSA (cash) collaeralizaion schemas is used beween he bank and he counerpary in order o miigae CVA We denoe he collaeral value posed by he bank before he defaul ime τ by he process Γ and he xva-pde (5) can be derived 4 from (4) wih Ū = R (M Γ ) + (M Γ ) + Γ afer he defaul and U before he defaul (5) In he righ of (5), he OIS rae r is used as a reference for all he C F oheunding raes The r and r are he exra remuneraion of he collaeral and he refunding rae of he bank relaed o r CVA FVA LVA V(T, ) = 0 (6) This is a linear PDE, hus he probabilisic soluion is (7), where M, Γ, and λ are he funcion of he (, W ) wih dx / X = r d + σdw For he independen case, he immersion case for γ r holds obviously For he beyond immersion case, we show he exisence of he pricing measure such ha he process λ does no change and γ holds in Annex Consequenly, he immersion case is adequae for xva modelling as he dependence is expressed in erms of λ In he nex chaper, we will give he numeric resuls associaed o xva value compued on he dynamized Gaussian copula defaul ime model (DGC model) under he pricing measure The xva value V is given by V = U M, where U and M are respecively he soluion of he linear EDP in (3) and (5) We remark ha if γ holds in (5), he xva-edp on V will be simplified subsanially Acually, his is he case if we consider he underlying can be refunded by he repo marke 2 The lef par in (4) is derived from Io s formulas and he righ par is he (G-Q)-incremen rae of he auo financed porfolio 3 x = x + x 4 The righ par in (5) is he (G-Q) -incremen rae of he auo financed porfolio in ouinding environemen (See Vladimir Pierbarg) AWALEE NOTES 2

4 Chaper 3 NUMERICS : In his chaper, we use he dynamized Gaussian copula model o model a defaul ime and τ an underlying price process X They are respecively generaed by a bi-variae Brownian moion B = (B 1, B 2 ) wih a consan correlaion ρ The deails of DGC model is given in Annex The call opion on he underlying can be evaluaed by he Black-Sholes formulas before crisis, for example M 0 = 10266, for X 0 = 2, Srike = 1, r = 0 σ = 05 Maury = 1 We will show he xva value on his call opion λ 0 is given as a DGC model parameeor he level of he inensiy of τ The CVA value is deermined by he parameer λ 0 and he correlaion ρ beween he underlying price (exposure) and he defaul risk as show in figure (8) Γ = M Wih r = 0 and, we have he following LVA resul of λ 0 The Mone-Carlo simulaion of (7) needs: The markovian process in (7) is a bi-variae Brownian moion B = (B 1, B 2 ) in DGC model The pre-defaul inensiy process λ is in funcion of (, B 1 ) given in (10) The underlying price M is given by he Black-Sholes formulas The collaeral Γ is aken by wo case Γ = M and Γ = 0 and he OIS, he collaeral and he funding rae are consan in his numeric es Γ = 0 Wih ouunding environmen r = = = 0 and recovery of counerpary risk is assumed a 40%, we have he following CVA resul of = Correlaion ρ λ CVA V_ = λ_0=400bp λ_0=200bp λ_0=100bp λ_0=50bp Figure (8): CVA versus correlaion parameer ρ in DGC (Mone Carlo wih scenarios) Tables (9): LVA in fully collaeral case (Mone Carlo wih scenarios) AWALEE NOTES 3

5 The xva value Vo is dominaed by he collaeral rae rc The funding rae rf is remuneraion of he par U M, which has he minuscule impac The effecive mauriy T τ decreases wih he increasemen of he inensiy of he defaul ime, he absolue value of xva value will also decrease CONCLUSION In his noe, we have shown ha he modelling of he counerpary s defaul ime involves hree levels of dependence wih he underlying price Under he pricing measure chosen by he repo marke, he immersion case is suiable for xva PDE The dependence is only modeled by he inensiy process In paricular, he correlaion parameer ρ of he DGC model plays an imporan role o rea he wrong way risk on CVA Acually, he case beyond immersion focus on wha happens afer he defaul ime see N El Karoui, so ha i is suiable o deal wih credi porfolio or gap risk For he fully collaeral rading, we can see he xva is dominaed by he LVA par while he CVA and FVA pars have lile impac on he xva value AWALEE NOTES 4

6 A: Densiy framework01 ANNEX As he defaul ime τ is a posiive random variable, i is fully characerized by is survival funcion, ha is, P(τ > ), or equivalenly o he densiy funcion P(τ ϵ d), when i exiss In order o esablish he relaionship beween τ and filraion F, we use he F-condiional survival process F (v) P(τ > v F ) or he condiional densiy process f (v) P(τ ϵ dv F ) o model he defaul ime The inensiy process of he defaul ime can be deduced by he densiy process (see (10)) Consequenly, he densiy modeling of he defaul ime is a framework more general han our assumpion in he firs par In paricular, he (G, P)-Radon-Nikodȳm densiy L = 1 Ls 0 Δ s dw s defines a new probabiliy measure Q on G such ha, where W is he (G, Q)-Brownian moion Furhermore, he (F, P)-Brownian moion B remains he (F, Q)-Brownian moion, as he projecion of (G, P)-Radon-Nikodȳm densiy L on he filraion F is 1, ha is E[L F] 1 (See dongli wu) Consequenly, he (G,Q)-Brownian moion W given by The pricing filraion G is produced by he filraion F progressively enlarged by he defaul ime τ, so for every, The predicable represenaion heorem in he filraion G is generaed by wo fundamenal maringales: remains a (F,Q)-Brownian moion B, so he immersion holds beween F and G under he pricing measure Q Furhermore, he compensaed maringale N does no change as i is orhogonal of he W, so he inensiy does no change moion under he new probabiliy measure Q 1 The (G, P)-Brownian moion W defined 2 The compensaed maringale N defined as, wih he pre-defaul inensiy of τ (10) So, any (G, P)- Radon-Nikodȳm densiy L can be wrien as a Doléans-Dade exponenial driven by he (G, P)-Brownian moion W and he compensaed maringale N where he processes α and β are G predicable AWALEE NOTES ANNEX 5

7 B: A beyond immersion case wih dynamized Gaussian copula model One considers a bi-variae Brownian moion B = (B 1, B 2 ) wih pairwise correlaion ρ in is own compleed filraion F under a probabiliy measure P Le h be a differeniable increasing funcion from R + o R wih lim s 0 h(s) = and lim s h(s) = + We defined he defaul ime, where (W 1, W 2 ) are (G, P)-Brownian moions Precisely, we have and he compensaed maringale, wih f is a square inegrable funcion wih uni L 2 -norm In our numeric es, we ake f 1 and he horizon by 5 years The ime horizon plays a role of he volailiy of he inensiy, wih he pre-defaul inensiy of τ We denoe and, so The pricing measure Q such ha he immersion propery holds: The F-condiional survival process of τ is given as Le L = 1 0 Ls _ Δ s 1 dw 1 be a (G, P)-Radon-Nikodȳm densiydefining a new probabiliy measure Q From Girsanov heorem, we can verify he, wih Ф is he sandard Gaussian survival funcion The (F, P)-densiy process of τ is given as i,where W are he (G, Q)-Brownian moion Consequenly, he (F, P)-Brownian moion (B 1, B 2 ) given by Le F be he Azéma supermaringale of τ, ha is, F = P(τ> F ) The dynamics of f(v) and F are given by, remain he Brownian moion under he probabiliy Q in he filraion G We recall ha he inensiy does no change moion under he new probabiliy measure Q wih REFERENCE The fundamenal (G, P)-maringales: The (F, P)-Brownian moions (B1, B2) remain he semi-maringale wih he following G-canonical decomposiion of he B, for i=1,2, 1 Valdimir Pierbarg : Funding beyond discouing : collaeral agreemens and derivaives pricing 2 Dongli WU : Thesis < Densiy models and applicaions o counerpary credi risk> hp://wwwhesesfr/2013evry N El Karoui, M Jeanblanc, and Y Jiao Wha happens afer a defaul: he condiional densiy approach Sochasic Processes and heir Applicaions AWALEE NOTES ANNEX 6