INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES.

Transcription

1 INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES. Join work wih Ying JIAO, LPMA, Universié Paris VII 6h World Congress of he Bachelier Finance Sociey, June 24, This research is par of he Chair Financial Risks of he Risk Foundaion, he Chair Derivaives of he Fuure sponsored by he Fédéraion Bancaire Française, and he Chair Finance and Susainable Developmen sponsored by EDF and Calyon 1/ 29

2 OUTLINE OF THE TALK 1 INTRODUCTION 2 THE INFORMATIONAL STRUCTURE 3 PRICING UNDER THE HISTORICAL PROBABILITY 4 RISK NEUTRAL PRICING 5 NUMERICAL EXAMPLE 2/ 29

3 OUTLINE Inroducion 1 INTRODUCTION / 29

4 INTRODUCTION Inroducion There exis wo main approaches in he credi risk modelling: he srucural approach and he reduced-form approach. The wo approaches are relaed by he accessibiliy of informaion on he underlying asse value process (Duffie-Lando 2001, Jeanblanc-Valchev 2005, Coculescu-Geman-Jeanblanc 2006, Guo-Jarrow-Zeng 2008), delayed or noisy observaion of he underlying process, on he defaul hreshold (El Karoui 1999, Giesecke-Goldberg 2008), consan or random defaul barrier. AIM sudy he impac of he informaion concerning he defaul hreshold in he credi analysis, in addiion o he parial observaion of he underlying asse process. 4/ 29

5 INTRODUCTION Inroducion (Ω, A, P) probabiliy space. (X ) 0 posiive coninous-ime process : asse value of he firm. F = (F = σ(x s, s )) 0 saisfying he usual condiions Defaul hreshold L, random variable in A. Defaul ime τ τ = inf{ : X L} where X 0 > L wih he convenion ha inf = + We inroduce he decreasing process X defined as X = inf{x s, s }. We assume ha he filraion F is generaed by a Brownian moion B. 5/ 29

6 OUR FRAMEWORK The manager of he firm has full informaion concerning he underlying asse of he firm and he chooses he defaul barrier. L is a random variable se by he manager of he firm. The invesors on he marke have differen levels of informaion on he fundamenal process (X ) 0 and on he defaul hreshold L. Full informaion knowledge of (X ) 0 and L (manager of he firm) Noisy full informaion knowledge of (X ) 0 + noisy signal on L + defaul ime observable Progressive informaion knowledge of (X ) 0 + defaul ime observable Delayed informaion delayed informaion on (X ) 0 + defaul ime observable 6/ 29

7 PRICING FRAMEWORK differen level of informaion { differen filraion (H ) differen pricing measure Q Evaluae a credi-sensiive derivaive claim of mauriy T : he value process a ime < τ T is given by [ T V = R E Q CR 1 T 1 {τ>t} + 1 {τ>u} R 1 u ] dg u + Z τ 1 {τ T} R 1 H τ (1) where C (F T -measurable) represens he paymen a he mauriy T (if τ T) G (F-adaped) represens he dividend paymen Z (F-predicable) represens he recovery paymen a he defaul ime τ R is he discoun facor process. 7/ 29

8 OUTLINE Inroducion 1 Inroducion 2 THE INFORMATIONAL STRUCTURE / 29

9 FULL INFORMATION The manager knows he hreshold L ω-wise from he beginning. Thus his informaion is given as he iniial enlargemen of he filraion F wih respec o L : ASSUMPTION ( H S ) G M = (G M ) 0 wih G M := F σ(l). We assume ha P(L F )(ω) P(L ) for all for P almos all ω Ω. EXAMPLES Assumpion (H S ) is saisfied if L is independen of F. Anoher example in a finie ime horizon T < T : L = l s 1 ]0,a[ (X T ) + l i 1 ]a,+ [ (X T ), l i < l s 9/ 29

10 RISK NEUTRAL PRICING MEASURE FOR THE FULL INFORMATION PROPOSITION There exiss a G M adaped process (ρ Ṃ (L)) (called informaion drif) such ha Y M (L) = E(. 0 ρm s (L)(dB s ρ M s (L)ds)) is a probabiliy densiy Any (F, Q)-local maringale is an (G M, Y M (L)Q)-local maringale. 10/ 29

11 THE PROGRESSIVE INFORMATION This is he sandard informaion srucure in he credi risk analysis. Invesors know a each ime wheher or no defaul has occured. Thus heir informaion is given as he progressive enlargemen of filraion of F wih respec o τ: ASSUMPTION ( H N ) G = (G ) 0 wih G = F σ({τ s}, s ). REMARK If L is independen of F, he sandard (H)-hypohesis is saisfied: every (F, P) local maringale is also a (G, P) local maringale. 11/ 29

12 THE DELAYED INFORMATION F D := { F 0 F δ() if δ(), if > δ(), consan delay ime model : δ() = δ discree observaion model : δ() = (m) i, (m) i < (m) i+1 where 0 = (m) 0 < (m) 1 < < m (m) = T are he only discree daes on which he informaion is renewed. The delayed informaion is he progressive enlargemen of filraion of F D wih respec o τ: ASSUMPTION ( H D ) G D = (G D ) 0 wih G D = F D σ({τ s}, s ). 12/ 29

13 NOISY FULL INFORMATION The invesor observes he value of he firm (X ) and receives a noisy signal (L s = f (L, ɛ s )) s 0 on he hreshold L. ASSUMPTION ( H N ) G I = (F I σ({τ s}, s )) 0 wih F I = u> (F u σ(f (L, ɛ s ), s u)) f : R 2 R is a given measurable funcion. ɛ = {ɛ, 0} is independen of F σ(l). P(L F )(ω) P(L ) for all for P almos all ω Ω. EXAMPLE Example in a finie ime horizon T < T : L s = L + W g(t s) wih W an independen Brownian moion and g : [0, T ] [0, [ a sricly increasing bounded funcion wih g(0) = 0 13/ 29

14 RISK NEUTRAL PRICING MEASURE FOR THE NOISY INFORMATION PROPOSITION There exiss a F I adaped process (ρ Ị ) such ha Y I = E(. 0 ρi s(db s ρ I sds)) is a probabiliy densiy Any (F, Q)-local maringale is an (F I, Y I Q)-local maringale. 14/ 29

15 OUTLINE Inroducion 1 Inroducion 2 3 PRICING UNDER THE HISTORICAL PROBABILITY / 29

16 AIM Compue he price of he coningen claim (C, G, Z) wih mauriy T given differen sources of informaion : [ T V = R E P CR 1 T 1 {τ>t} + 1 {τ>u} R 1 u P is he hisorical probabiliy measure ] dg u + Z τ 1 {τ T} R 1 H (H ) 0 describes he accessible informaion for he invesors. (H ) 0 = (G M ) 0 for he full informaion, (H ) 0 = (G I ) 0 for a noisy full informaion. (H ) 0 = (G ) 0 for he progressive informaion. (H ) 0 = (G D ) 0 for he delayed informaion. τ 16/ 29

17 PRICING FOR THE FULL INFORMATION PROPOSITION We define F M (x) := p (x)1 {X >x} where p (x)(ω) = dpl dp L (ω, x), P L (ω, dx) = a regular version of he condiional law of L given F, P L = he law of L. The value process of he coningen claim (C, G, Z) given he full informaion (G M ) 0 is V M Ṽ M (L) = 1 {τ>} p (L) where Ṽ M (L) = R E P [ CR 1 T FM T (x)+ T T ] Fs M (x)r 1 s dg s Z s R 1 s dfs M (x) F. x=l 17/ 29

18 PRICING FOR THE PROGRESSIVE AND DELAYED INFORMATION PROPOSITION We define S := P(τ > F ). The value process given he progressive informaion flow G is [ R T V = 1 {τ>} E P R 1 T S S TC + T R 1 u S udg u R 1 u Z uds u F ]. The value process for a delay-informed invesor is V D = 1 {τ>} E[S F D ] E [ R T R T R P S T C + S u dg u Z u ds u R T R u R u F D ]. 18/ 29

19 PRICING FOR THE NOISY INFORMATION PROPOSITION We assume (H N ) wih L = L + ɛ, ɛ being a coninuous process wih backwardly independen incremens and whose marginal has densiy q. The value process for he noisy full informaion flow G I is given by V I 1 {τ>} = Ṽ R FM (l)q (L l)p L (dl) M (l)q (L l)p L (dl) where ṼM and F M are defined for he full informaion. F M (x) := p (x)1 {X >x}. Ṽ M (L) = R E P [ CR 1 T FM T (x)+ T T ] Fs M (x)r 1 s dg s Z s R 1 s dfs M (x) F. x=l 19/ 29

20 OUTLINE Inroducion 1 Inroducion RISK NEUTRAL PRICING 5 20/ 29

21 RISK NEUTRAL PROBABILITIES We assume ha a pricing probabiliy Q is given wih respec o he filraion F of he fundamenal process X (for example, Q such ha X is an (F, Q) local maringale). We wan o focus on he change of probabiliy measures due o he differen sources of informaions and on is impac on he pricing of credi derivaives, wihou loss of generaliy, we ake he hisorical probabiliy P o be he benchmark pricing probabiliy Q on F and G. The pricing probabiliy for he manager is Q M where dqm dq = YM (L) wih Y M (L) = E(. 0 ρm s (L)(dB s ρ M s (L)ds)) The pricing probabiliy for he noisy full informaion is Q I where dq I dq = YI wih Y I = E(. 0 ρi s (db s ρ I s ds)) We also ake Q as he pricing probabiliy for he delayed informaion 21/ 29

22 RISK NEUTRAL PRICING FOR THE FULL INFORMATION PROPOSITION 1) Define F QM (l) = 1 {X >l}. Then he value process of a credi sensiive claim (C, G, Z) for he manager s full informaion under he risk neural probabiliy measure Q M is given by V QM = 1 {τ>} R E P [ CR 1 + T T F QM s FQM T (x) (x)r 1 s dg s T Z s R 1 s dfs QM (x) F ]x=l. 22/ 29

23 2) Le ɛ be a coninuous process wih backwardly independen incremens such ha he probabiliy law of ɛ has a densiy q ( ) w.r.. he Lebesgue measure. Le µ,θ be he probabiliy law of ɛ θ ɛ. Then he value process for he insider s noisy full informaion under Q I is given by where V QI = 1 {τ>} R FM (l)q (L l)p L (dl) [ Ṽ QI (l) = R E P CR 1 T FI,T(u, l) + T Ṽ QI (l)q (L l)p L (dl) T F,θ(u, I l)r 1 θ dg θ R 1 θ Z θdf,θ I (u, l) F ( θ ) 1F F,θ(u, I l) = E ρ I M s(u + y)µ,θ (dy)db s θ (l). ] u=l, 23/ 29

24 OUTLINE Inroducion 1 Inroducion NUMERICAL EXAMPLE 24/ 29

25 BINOMIAL MODEL Inroducion Le L be a (F ) 0 -independen random variable aking wo values l i, l s R, l i l s such ha P(L = l i ) = α, P(L = l s ) = 1 α (0 < α < 1). We suppose ha he asse values process X saisfies he Black Scholes model : dx X = µd + σdb, 0. We compue he value process of a defaulable bond. 25/ 29

26 VALUE PROCESS OF A DEFAULTABLE BOND Numerical values in he following simulaions : l i = 1, l s = 3, α = firm value manager progressive delayed noisy dynamic price of he defaulable bond firm value FIGURE: L = l i 26/ 29

27 manager progressive delayed noisy 9 8 firm value dynamic price of he defaulable bond firm value FIGURE: L = l s 27/ 29

28 We compare he dynamic price of he defaulable bond, for he same scenario of he firm value bu depending on he level of he hreshold fixed by he manager manager inf manager sup progressive bruie inf bruie sup 8 valeur de firm dynamic price of he defaulable bond firm value FIGURE: L = l s or L = l i 28/ 29

29 REFERENCES Inroducion Coculescu, D., Geman, H., Jeanblanc, M., Valuaion of Defaul sensiive claims under imperfec Informaion, Finance and Sochasics. Corcuera, J.M., Imkeller, P., Kohasu-Higa, A., Nualar, D., Addiional uiliy of insiders wih imperfec dynamical informaion, Finance and Sochasics, 8, Duffie, D., Lando, D Term Srucures of Credi Spreads wih Incomplee Accouning Informaion, Economerica, 69, Giesecke K, Goldberg L. R., The Marke Price of credi risk : he impac of Asymmeric Informaion. Grorud, A., Ponier, M., Insider Trading in a Coninuous Time Marke Model, Inernaional Journal of Theorical and Applied Finance Guo, X., Jarrow, R., Zeng, Y., Credi risk wih incomplee informaion, forhcoming Mahemaics of Operaions Research. 1, / 29