Equity Default Swaps under the Jump-to-Default Extended CEV Model

Transcription

1 Equity Default Swaps under the Jump-to-Default Extended CEV Model Rafael Mendoza-Arriaga Northwestern University Department of Industrial Engineering and Management Sciences Presentation of Paper to be Submitted to Finance and Stochastics in October 008 Joint work with Vadim Linetsky

2 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: Reference Entity Defaults Reference Stock Price drops significantly (L = 30%S 0 ) Similar to CDS Protection Buyer makes periodic Premium Payments on exchange of protection in case of a Credit Event. Protection Seller pays a recovery amount (1 r) for each dollar of principal at default time, if default occurs prior to Maturity. 1

3 Is an EDS a truly Credit Instrument? Against: A variation of a deep out of the money American Digital option for which the premium is paid over time conditional on the trigger event not having occur by that time. During periods of severe stock market movements the stock price may suffer a drop of 70% of the initial price which will trigger the EDS payoff but without having an actual bankruptcy. In Favor: Empirical evidence supports an inverse relationship between the stock price and the probability of default of the reference firm. EDS closes the gap between equity and credit instruments since it is structurally similar to the credit default swap. EDS spreads are higher than those corresponding to CDS

4 Empirical Evidence (Jobst & Servigny, 005a & 005b) Credit and equity factor analysis of EDS at different triggering levels Credit ratings and historical volatilities have the highest explanatory power. (Particularly true for L < 50%S 0, while for L > 50%S 0 equity type factors are more relevant) For low barrier levels (L 10%S 0 ) it is quite common (> 90% freq) that a default will occur in the same period of time (see Kendal s tau test) when the EDS payoff is triggered. Event Correlation (among firms) increases for higher barrier levels. (Adverse market conditions, common to different firms, will trigger EDS payoffs corresponding to these firms, if the barriers are high enough) Bottom-Line EDS behaves like a Credit instrument for lower triggering levels (L) EDS behaves like an Equity instrument for higher triggering levels (L) 3

5 Advantages of EDS over CDS Transparency on which an EDS payoff is triggered. It is easy to know whether a firm stock price has crossed a lower threshold (L) Using the Stock Price as the state variable to determine a credit event allows investors to have a Exposure to Firms for which CDS are not usually traded. (as in the case of firms with high yield debt) 4

6 EF D: 7 ; A >7 L 7? 9 EDS Mechanics OPQRST UVWXQYS ZX\ ^_`a = > > <? 9 :;< =? '()*+,*-). /+00+( B 7C ; < A@;4 54 ; < K I 7 G 8J IH ;G?H '()*+,*-). 13+( ; <7?7 ; N4I M > >! "!!! S(t) t #$%& Time (yrs) T t 5

7 Balance Equation ρ = {ρ PV(Protection Value)=PV(Premium Payments + Accrued Interest)} PV(Protection Payments) PV(Premium Payments) (1 r) E e r T L 1 {T L T } ρ t N i=1 e r t ie 1 {T L t i } PV(Accrued Interests) ρ E e r T L ( T L t T L t )1 {T L T } t r T TL ρ r Time Interval Recovery Maturity Payoff Triggering Time EDS rate Risk Free Rate 6

8 Jump-to-Default Extended CEV model (Carr & Linetsky 006) One-dimensional process with associated SDE (µ = r q), dx t = (µ + h(x t ))X t dt + σ(x t )X t db t, X 0 = S > 0 with σ(x t ) = ax β t, h(x t ) = b + cσ (X t ) = b + ca X β t Where σ(x) is the volatility (power) function with the volatility scale and volatility elasticity parameters denoted by a and β < 0, respectively The default intensity (killing) is given by h(x) which is an affine function of variance. Where b is a constant default intensity and c is the default sensitivity to variance Leverage Effect. Since β < 0, then as X 0, the volatility increases σ(x), as well as the default intensity since h(x) 7

9 Jump-to-Default Extended CEV model (cont.) Defaultable Stock Price Model S t = { Xt, t < ζ T L, t ζ T L, T L = inf{t : X t = L}, ζ = inf { t 0, : t 0 h(x u)du e } The Stock Price S is equal to the JDCEV process X if No Default has occurred by time t. ζ is the lifetime of process X while T L = ζ T L is the EDS triggering time of S. At time ζ the process X is sent to cemetery state, while S is sent to the cemetery state at time T L EDS payoff event occurs on whichever happens first of: the first hitting time of level L the first jump time, ie. the first time the integrated default intensity is greater than an exponentially distributed r.v. e with unit mean independent of X. 8

10 EDS Illustration S(t) T i m e ( yr s ) L e h(s) e T i m e ( ye a r s ) S t = { Xt, t < ζ T L, t ζ T L, T L = inf{t : X t = L}, ζ = inf { t 0, : t 0 h(x u)du e } 9

11 Protection Payment under JDCEV Protection = (1 r)e e r T L 1{T L T } = (1 r) E e r T L T L 0 h(x )du u 1 {TL T } + T }{{} Diffusion Term 0 e r u E e u h(xv)dv 0 h(x u )1 {TL >u} du }{{} Jump Term Recall that the first hitting time to L is given by T L = inf{t : X t = L}, and that the first jump time to is given by ζ = inf { t 0, : t 0 h(x u)du e } The default intensity is the power function: h(x t ) = b + ca X β t Notice. Since e is an exponentially distributed r.v. with unit mean, then Pζ > t = e t h(xu)du 0 and Pζ < t = t 0 h(x v)e v h(xu)du 0 dv 10

12 Premium Payment under JDCEV Premium = ρ t N i=1 e r t i E 1{T L t i } = ρ t N i=1 e r t i E e t i h(xu)du 0 1 {TL t i } }{{} NO jump to default & NO hitting level The premium is paid at times t i conditional on No default and that the stock price did Not drop to level L by time t i The default intensity is the power function: h(x t ) = b + ca X β t 11

13 Accrued Interests under JDCEV Acc. Interest = ρ E e r T L ( ) TL t T L t 1 {T L T } = ρ N 1 i=0 E ( e r T L T L t i ) 1 {T L (t i,t i+1 )} Expressed in terms of Diffusion and Jump components: = ρ T 0 ue r u Ee u h(xv)dv 0 h(x u )1 {TL u} du }{{} Jump Term N 1 +E N 1 i=1 (i t) t i+1 t i i=1 (i t) E e r T L T L 0 h(x )du u T L 1 {TL T } }{{} e r u E e u 0 h(x v)dv h(x u )1 {TL u} Diffusion Term e r T L T L 0 h(x )du u 1 {TL t i+1 } } {{ } Diffusion Term } {{ } Jump Term E du e r T L T L 0 h(x )du u 1 {TL t i } } {{ } Diffusion Term 1

14 Expectations to Solve: Jump Term and Diffusion Term Jump Term. E e u h(xv)dv 0 h(x u )1 {TL >u} Since the default intensity is given by a power function, h(x t ) = b + ca Xt β, we can solve, more generally, for a given p the expectation which we name p-truncated Moment E e u h(xv)dv 0 (X u ) p 1 {TL >u} Diffusion Term. This term can be seen as the Expected Discount (given no default) up to the first hitting time to level L E e r T L T L 0 h(x )du u 1 {TL T } The term contaning TL can be easily calculated as: E e r T L T L 0 h(x u )du T L 1 {TL T } = ddr E e r T L T L 0 h(x u )du 1 {TL T } 13

15 Solving the Expectations: The Green s function and the Fundamental Solutions The JDCEV process has the associated infinitesimal generator in the domain D = (L, ) { } G f (x) = 1 a x β+ d f dx (x) + (µ + b + ca x β )x d f dx (x) (b + ca x β ) f (x) Let p(t;x,y) be the transition density of the JDCEV process, then the Green s function can be found as G s (x,y) = 0 e st p(t;x,y)dt = m(y) w s { ψs (x)φ s (y), x y ψ s (y)φ s (x), y x. φ s (x),ψ s (x) are the solutions to the Sturm-Liouville problem: satisfying the boundary conditions G f (x) = s f (x) ψ s (x) = 0, x L, (increasing and vanishing at L) φ s (x) = 0, x, (decreasing and vanishing at ) 14

16 Solving the Expectations: The Green s function and the Fundamental Solutions (cont.) The speed and scale measures are given by m(x) = a x c β e εax β, s(x) = x c e εax β, where A := µ+b a β, ε := sign(β(µ + b)). The fundamental solutions φ s (x),ψ s (x) are given in terms of the Whittaker functions ψ s (x) = x 1 c+β e ε A x β ( W ) ( κ(s), ν AL β M ) ( κ(s), ν Ax β M ) ( κ(s), ν AL β W ) κ(s), ν Ax β ( φ s (x) = x 1 c+β e ε A x β W ) κ(s), ν Ax β The Wronskian is given by, w s = The rest of the parameters are: µ+b Γ(1+ν) a Γ(ν/+1/ κ(s)) W ( ) κ(s), ν AL β ν = 1+c β, κ(s) = ε1 ν s+ξ ω, where ω = β(µ + b), ξ = c(µ + b) + b 15

17 Solving the Expectations: The Resolvent Operator The Resolvent operator is the Laplace transform of the Expectation operator R s f (x) := 0 e st E x 1{ζ>t} f (X t ) dt = L = φ s(x) w s x L f (y)ψ s(y)m(y)dy + ψ s(x) w s x f (y)g s(x,y)dy f (y)φ s (y)m(y)dy where we exchanged the integrals by applying Fubinis theorem subject to L f (y)g s(x,y) dy < We recover the Expectation Operator by using the Bromwich Inversion Integral E x 1{ζ>t} f (X t ) = 1 ε+i πi ε i est R s f (x)ds Im(s) + i Singularities at s=p(µ+b) - (ωn+b) n=0,1,, o o Re(s) Singularities at -β {s W{κ(s),ν/}(AL )=0} i 16

18 Solving the Expectations: the p-truncated Moment The p-truncated Moment for L > 0 and µ + b > 0 is given by E x e t h(xu)du 0 1 {TL >t}(x t ) p = n=0 M ν 1 +n ( c+p β ), ν ( + n=1 (A 1 c p 4 β 1 ( 1 p β ) Γ(1+ c+p n β ) n!γ(1+ν) ( ) M ν 1 (AL Ax β ( +n c+p β ) β ), ν W ν 1 +n ( c+p β ), ν x 1 c+β e A x β e (p(µ+b) (b+ωn))t (AL β ) Wν 1 +n ( c+p β ), ν e (ω(κ n ν 1 )+ξ)t x 1 c+β e A x β M κn, ν (AL β )W κ n, ν (Ax β ) Γ(1+ν) d dκ W κ, ν (AL β ) κ=κ n A 1 c p 1 Γ(1 1 p 4 β β )Γ(1+ c+p β )Γ( ν 1 κ n c+p Γ( 1 ν κ n) ( 1 1 p β A1 ν L β 1+p Γ(ν) ( β 1+p) F β A1+ν L c+p β Γ( ν)γ( 1+ν κ n) ( β +c+p)γ( 1 ν κ n) where κ n = { κ W κ, ν ( AL β ) = 0 } β, 1 ν κ n β ) 1 p β, 1 ν ;AL β ( Ax β )) ) ( 1 + c+p β F, 1+ν κ n + c+p β, 1 + ν ;AL β )) 17

19 Solving the Expectations: the p-truncated Moment The p-truncated Moment for L = 0 (CDS case) and µ + b > 0 is given by E x e t h(xu)du 0 1 {TL >t}(x t ) p = n=0 A 1 c p 4 β 1 ( 1 p β ) Γ(1+ c+p n β ) n!γ(1+ν) x 1 c+β e A x βe (p(µ+b) (b+ωn))t M ν 1 +n ( c+p β ), ν where κ n = { ( κ W ) κ, ν AL β = 0 } ( Ax β ) 18

20 Solving the Expectations: Diffusion Term Similar to the Resolvent Operator approach we follow the following steps: 1. Laplace Transform 0 e st E x e r T L T L 0 h(x )du u 1 {TL T } dt = 1 s E x e (r+s) T L T L 0 h(x u )du. Solve the Sturm-Liouville problem Gu(x) = (r + s)u(x), u(x) <, u(l) = 1 (since T L = 0) 3. Invert the Laplace Transform (Bromwich Inversion Formula) The Diffusion Term E x ( x L)1 c+β e ε A (x β L β) 1 γ+i πi γ i est W κ(s+r), ν (Ax β ) sw κ(s+r), ν (AL β ) ds e r T L T L 0 h(x )du u 1 {TL T } Wε 1 ν r+ξ (Ax ω, ν β ) W ε 1 ν r+ξ ω, ν (AL β ) + n=1 ωe (ω(κn ε 1 ν = ( ) 1 x c+β L e ε A (x β L β) )+r+ξ)t (ω(κ n ε 1 ν )+r+ξ) W κ n, ν (Ax β ) κ W κ, ν (AL β ) κ=κ n 19

21 Numerical Example 1: the effect of changing the sensitivity to volatility c Increasing c shifts the EDS curves up while the spreads between EDS with L = {0,15,5} become tighter. The default intensity (h(x t ) = b + ca Xt β ) is increasing with c. We choose a = σ/s β 0 = 10!" # $ % & '( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % &!" # $ % & '( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % &!" # $ % & '( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % & * +% & '!" # $ % & ' ( % ' ) % & t r L S 0 r q b c β σ {0, 15, 5} {0, 0.0} {0, 1, }

22 Numerical Example : the effect of volatility σ Increasing σ the spreads between EDS with L = {0,15,5} become more ample, since it is more likely to hit the barrier level L. The default intensity (h(x t ) = b + ca X β t ) is increasing with c. We choose a = σ/s β 0 = 0! "#$ % & '$ & ( $ % ) * $ % &! "#$ % & '$ & ( $ % ) * $ % &! "#$ % & '$ & ( $ %! "#$ % & '$ & ( $ % ) * $ % &! "#$ % & '$ & ( $ % ) * $ % &! "#$ % & '$ & ( $ % t r L S 0 r q b c β σ {0, 15, 5} {0, 0.0} {0, 1}

23 Conclusions We develop a closed form solutions for pricing EDS in the intensity framework EDS is based on observable variables, ie. the stock process We capture the high event correlation between EDS payoff events for lower triggering levels and CDS events EDS with higher triggering levels (L 50%S 0 ) behaves like an equity instruments, while for lower triggering levels (L < 50%S 0 ) its behavior is more like a credit instrument. By using, the JDCEV specification for the stock price we are able to achieve both. On one hand, the JDCEV process is local volatility process which allows us to fit the volatility skews. On the other hand, the default intensity of the JDCEV process is an increasing function of variance, while variance is negative proportional to the stock price. Therefore, as the stock price drops the default intensity increases.