An exact formula for default swaptions pricing in the SSRJD stochastic intensity model
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1 An exact formula for default swaptions pricing in the SSRJD stochastic intensity model Naoufel El-Bachir (joint work with D. Brigo, Banca IMI) Radon Institute, Linz May 31, 2007 ICMA Centre, University of Reading p. 1/3
2 Outline Motivation Basics of default intensity modeling The SSRJD stochastic intensity model CDS and CDS options pricing An exact formula for CDS options Numerical implementation issues Proof of the formula ICMA Centre, University of Reading p. 2/3
3 Motivation Default swaptions are quoted by inputting a volatility in a market model formula How do we value default swaptions of different maturities, strikes, or more complex instruments? Strong evidence of jumps in default swap rates Relatively few calibration instruments The mark-to-market problem ICMA Centre, University of Reading p. 3/3
4 Intensity-based default model Market filtration: G = (G t ) (t 0) = F H H = (H t ) (t 0) where H t = σ ({τ < u},u t) F = (F t ) (t 0) is the usual filtration generated by the stochastic market variables except default τ is a totally G inaccessible stopping time: 1 {τ t} = M t + t τ 0 λ s ds where M is a uniformly integrable G martingale. The F adapted process λ is the G marginal default intensity. Constant loss given default: L (fraction of notional) ICMA Centre, University of Reading p. 4/3
5 The SSRJD intensity model λ t = y t + ψ(t;β), t 0 ψ a deterministic function, β = (κ, µ, ν, α, γ) the vector of parameters driving y: dy t = κ(µ y t )dt + ν y t dz t + dj t, with the condition 2κµ > ν 2. Z a brownian and J a pure jump process J t = M t Y i where M is a poisson process with intensity α, the Y s are exponentially distributed with mean γ i=1 ICMA Centre, University of Reading p. 5/3
6 The SSRJD intensity model The survival probability ( is S(t,T) = 1 {τ>t} S(t,T), where S(t,T) := E Q [exp ) ] T λ t u du F t is given by: Z T «S(t, T) = A(t, T) exp ψ(s, β)ds B(t, T)y t t A(t, T) = ξ(t, T)ζ(t, T) 0 2h exp h+κ+2γ 1 (T t) 2 ζ(t, T) A 2h + (κ + h + 2γ)(exp h(t t) 1) ξ(t, T) = 2h exp h+κ 2 (T t) 1 A 2h + (κ + h)(exp h(t t) 1) 2κµ ν 2 2αγ ν 2 2κγ 2γ 2 B(t, T) = 2(exp h(t t) 1) 2h + (κ + h)(exp h(t t) 1) h = p κ 2 + 2ν 2 ICMA Centre, University of Reading p. 6/3
7 The SSRJD intensity model From the expression of ζ(t,t), we have an additional condition on the model parameters: that is: γ h κ 2 Otherwise, ν 2 2κγ 2γ 2 0 theoretically ζ(t,t) = 1 and the jumps have no effect in numerical computations we would have divisions by zero ICMA Centre, University of Reading p. 7/3
8 The SSRJD intensity model We also make use of the following transform: satisfying E Q»exp Z T t «λ s ds λ T F t = T S(t, T) T S(t, T) =» 1 S(t, T) ξ(t, T) T ξ(t, T) y t T B(t, T) + 1 ζ(t, T) T ζ(t, T) ψ(t) T ξ(t, T) = 2κµ `e h(t t) 1 2h + (κ + h) `e h(t t) T) 1 ξ(t, T B(t, T) = T ζ(t, T) = 4h 2 eh(t t) ˆ2h `eh(t + (κ + h) t) 1 2 2αγ `e h(t t) 1 2h + (κ + h + 2γ) `e h(t t) T) 1 ζ(t, ICMA Centre, University of Reading p. 8/3
9 CDS pricing Brigo and Alfonsi (2005) find default swaps to be relatively insensitive to the correlation between brownians driving short rate and intensity Brigo and Cousot (2006) also confirm that it is relatively insignificant for default swaptions Brigo and Cousot (2006) find that the short rate volatility has relatively little impact on default swaptions, thus randomness of interest rates add little value to stochastic default intensity models. We assume interest rates to be deterministic. ICMA Centre, University of Reading p. 9/3
10 CDS pricing The value of a CDS paying a regular fee R is: CDS(t,Υ,R,L) = 1 {τ>t} [ RC a,b (t) + L GD Tb ] D(t,u) u S(t,u)du where [ b i=a+1 α i D(t,T i )S(t,T i ) Tb (u T (β(u) 1) )D(t,u) u S(t,u)du ] and T β(t) is the first date in the payment schedule Υ := {+1,...,T b } that follows t and α i = T i T i 1 is the year fraction between T i 1 and T i. ICMA Centre, University of Reading p. 10/3
![S(t,u)du ] and T β(t) is the first date in the payment schedule Υ := {+1,.](/docs-images/48/19279099/images/page_10.jpg "..,T b } that follows t and α i = T i T i 1 is the year fraction between T i 1 and T i.")
11 Default swaptions The payoff of a payer default swaption that is knocked out in case of default before the swaption maturity is: PSO(T, T, Υ, K) = 1 {τ>t } Ca,b (T) (R a,b (T) K) + = [CDS(T, Υ, K, L)] + The default swaption payoff can be valued as: PSO(t, T, Υ, K) = D(t, T)E Q [ (CDS(T, Υ, K, L)) + G t ] We derive a semi-analytical formula along the lines of Jamshidian (1989) s decomposition. ICMA Centre, University of Reading p. 11/3
![swaption payoff can be valued as: PSO(t, T, Υ, K) = D(t, T)E Q [ (CDS(T, Υ, K, L)) + G t ] We derive a](/docs-images/48/19279099/images/page_11.jpg "semi-analytical formula along the lines of Jamshidian (1989) s decomposition.")
12 An exact formula for CDS options If Z Tb 2 3 4L GD D(, u) u S(, u;0) + KS(, u;0)d(, u) `1 (u T β(u) 1 )r u 5du > 0 then PSO(t, T,Υ, K) = 1 {τ>t} D(t, )exp R Tb Z Ta t «ψ(s)ds h(u)a(, u)e R u ψ(s)ds Ψ(t,, y t, y, B(, u))du where y 0 satisfies: Z Tb h(u)s(, u; y )du = L GD Otherwise, PSO(t, T,Υ, K) = CDS(t,Υ, K, L GD ) ICMA Centre, University of Reading p. 12/3
13 An exact formula for CDS options h(u) = D(, u) i hl GD r u + δ Tb (u) + K 1 (u T β(u) 1 )r u Ψ(t, T, y t, ς, ) = e ς Π(T t, y t, ς,0) Π(T t, y t, ς, ) Π(T, y 0, ς, ) = 1 2 α ψ(t)e β ψ(t)y 0 1 π Z 0 e Uy 0[S cos(wy 0 + vς) + R sin(wy 0 + vς)] dv v α ψ (T) = 2 2hexp κ+h 2 4 T 3 5 2h + (h + κ + ν 2 )(e ht 1) κµ ν 2 «(h 2h(1 + γ) exp 2 (κ+2γ) 2 )(1 2 (h+κ)) 2(h κ 2γ+ (γ(h+κ) ν 2 )) T 2h(1 + γ) + [h + κ + ν 2 + γ(2 + (h κ))](e ht 1) αγ ν 2 2κγ 2γ 2 ICMA Centre, University of Reading p. 13/3
14 An exact formula for CDS options β ψ (T) = 2 h+(2+ (h κ))(eht 1) 2h+(h+κ+ ν 2 )(e ht 1) R = (J 2 + K 2 ) D 2 e G [E cos(h + D arctan(k/j)) F sin(h + D arctan(k/j))] S = (J 2 + K 2 ) D 2 e G [F cos(h + D arctan(k/j)) + E sin(h + D arctan(k/j))] U = δ + εeht + φe 2hT N W = 4vh2 e ht N E = (ex 2 + ey 2 ) κµ 2κµ ν 2 cos F = (ex 2 + ey 2 ) κµ ν 2 sin ««ey ν 2 arctan ex ««2κµ ey ν 2 arctan ex ICMA Centre, University of Reading p. 14/3
![arctan(k/j)) + E sin(h + D arctan(k/j))] U = δ + εeht + φe 2hT N W = 4vh2 e ht N E = (ex 2 + ey 2 ) κµ 2κµ](/docs-images/48/19279099/images/page_14.jpg "ν 2 cos F = (ex 2 + ey 2 ) κµ ν 2 sin ««ey ν 2 arctan ex ««2κµ ey ν 2 arctan ex ICMA Centre, University of")
15 An exact formula for CDS options ex = 2he(h+κ) T 2 [2h + (h + κ + ν 2 )(e ht 1)] N ey = 2he(h+κ) T 2 vν 2 [e ht 1] N D = 2γα ν 2 2γκ 2γ 2 G = αγt[(2 (h + κ))(h κ 2γ [ν2 γ(h + κ)]) + v 2 (h + κ)[ν 2 γ(h + κ)]] (h κ 2γ [ν 2 γ(h + κ)]) 2 + v 2 [ν 2 γ(h + κ)] 2 H = αγtv[(2 (h + κ))[ν2 γ(h + κ)] (h + κ)(h κ 2γ [ν 2 γ(h + κ)])] (h κ 2γ [ν 2 γ(h + κ)]) 2 + v 2 [ν 2 γ(h + κ)] 2 J = 1 + (eht 1)[(h + κ + 2γ)(1 + γ) + (ν 2 + γ(h κ))[ ( γ + 1) + v 2 γ]] 2h(1 + γ) 2 + 2hv 2 γ 2 ICMA Centre, University of Reading p. 15/3
![κ)] 2 H = αγtv[(2 (h + κ))[ν2 γ(h + κ)] (h + κ)(h κ 2γ [ν 2 γ(h + κ)])] (h κ 2γ [ν 2 γ(h + κ)]) 2 + v 2 [ν 2 γ(h + κ)] 2 J = 1](/docs-images/48/19279099/images/page_15.jpg "+ (eht 1)[(h + κ + 2γ)(1 + γ) + (ν 2 + γ(h κ))[ ( γ + 1) + v 2 γ]] 2h(1 + γ) 2 + 2hv 2 γ 2 ICMA Centre, University of Reading p.")
16 An exact formula for CDS options K = (eht 1)v[2γκ + 2γ 2 ν 2 ] 2h(1 + γ) 2 + 2hv 2 γ 2 N = (2h + (h + κ + ν 2 )[e ht 1]) 2 + v 2 ν 4 [e ht 1] 2 δ = 2(h κ) 4ν 2 + 2ν 2 (h + κ) + v 2 ν 2 (h + κ) ε = 4κ 4κ 2 2κ 2ν 2 2v 2 ν 2 κ φ = 2(h + κ) 4ν 2 2ν 2 (h κ) v 2 ν 2 (h κ) Although cumbersome, the formula is analytic up to the double integration. ICMA Centre, University of Reading p. 16/3
17 Numerical implementation issues lim v e Uy 0 [S cos(wy 0 + vς) + R sin(wy 0 + vς)] v = Figure 1: Plot of euy 0[S cos(wy 0 +vς)+r sin(wy 0 +vς)] v when v [1, 250] ICMA Centre, University of Reading p. 17/3
18 Numerical implementation issues for v = 0, we have [S cos(wy 0 + vς) + R sin(wy 0 + vς)] = Figure 2: Plot of [S cos(wy 0 + vς) + R sin(wy 0 + vς)] when v [0, 100] ICMA Centre, University of Reading p. 18/3
19 Numerical implementation issues lim v 0 e Uy 0 [S cos(wy 0 + vς) + R sin(wy 0 + vς)] v = e U(0)y 0 ( ) S c 1 R(0) + lim v 0 v lim v 0 lim v 0 S v F v ] = J(0) D e [c G(0) F 2 lim v 0 v + c 3 lim E v 0 = c 4 lim v 0 E = c 5 Hence both bounds are integrable singularities. ICMA Centre, University of Reading p. 19/3
20 Example: CDS term structure Figure 3: CDS (in bp) term structure with L GD = 0.7 r = 3%, y 0 = 0.005, κ = 0.229, µ = , ν = 0.078, α = 1.5, γ = ICMA Centre, University of Reading p. 20/3
21 Example: volatility smile Figure 4: CDS options volatility smile for different strikes (in bp) with = 1, T b = 5, L GD = 0.7 r = 3%, y 0 = 0.005, κ = 0.229, µ = , ν = 0.078, α = 1.5, γ = ICMA Centre, University of Reading p. 21/3
22 Proving the formula Proposition 1 The default swaption price satisfies the following formula: 1 {τ>t} D(t, )E Q ( exp Ta t λ s ds )( L GD Tb h(u)s(, u)du ) + F t where h is defined as: h(u) = D(, u) [ L GD ( ru + δ Tb (u) ) + K ( 1 (u T β(u) 1 )r u )] with δ Tb (u) the Dirac delta function centered at T b. ICMA Centre, University of Reading p. 22/3
23 Proof for proposition 1 Integrating by parts: PSO(t, T, Υ, K) = D(t, T)E Q [ (CDS(T, Υ, K, L)) + G t ] Tb D(, u) u S(, u)du = D(, T b )S(, T b ) 1 Tb S(, u) u D(, u)du D(, T b )S(, T b ) = Tb D(, u)s(, u)δ Tb (u)du ICMA Centre, University of Reading p. 23/3
24 Proof for proposition 1 Tb D(, u)(u T β(u) 1 ) u S(, u)du = b i=a+1 T b D(, u)s(, u)du T b S(, u)(u T β(u) 1 ) u D(, u)du α i D(, T i )S(, T i ) We use the formula (Bielecki and Rutkowski (2001), corollary p.145): E Q [1 {τ>ta }Y Ta G t ] = 1 {τ>t} E Q [exp ( Ta t λ s ds )Y Ta F t ] ICMA Centre, University of Reading p. 24/3
25 Proving the formula Corollary 1 If Z Tb 2 3 4L GD D(, u) u S(, u;0) + KS(, u;0)d(, u) `1 (u T β(u) 1 )r u 5du > 0 then the default swaption price is the solution to the following formula: 1 {τ>t} D(t, ) Z Tb h(u)e» Z Ta exp t «+ Ft λ s ds S(, u; y ) S(, u; y ) Ta du where y 0 satisfies: Z Tb h(u)s(, u; y )du = L GD Otherwise, the default swaption price is simply given by the corresponding forward default swap value CDS(t, Υ, K, L GD ) ICMA Centre, University of Reading p. 25/3
26 Proof of corollary 1 h(u) > 0, for all u and independent of y 0 S(, u; y) is monotonically decreasing in y for all and u. Hence, Tb is a monotonically decreasing function of y h(u)s(, u; y)du Integration by parts: Tb lim y h(u)s(, u; y)du = 0 < L GD Tb r u D(, u)s(, u)du = 1 D(, T b )S(, T b )+ Tb D(, u) u S(, u)du ICMA Centre, University of Reading p. 26/3
27 Proof of corollary 1 Z Tb lim h(u)s(, u; y)du = L GD + y 0 + Z Tb 2 4L GD D(, u) u S(, u;0) 3 + KS(, u;0)d(, u) `1 (u T β(u) 1 )r u 5du If the integral is negative, Tb lim y 0 + h(u)s(, u; y)du < L GD The payoff of the option is Q a.s. strictly positive and simplifies to a CDS. Otherwise, if the integral is negative, IVT!y : Tb h(u)s(, u; y )du = L GD ICMA Centre, University of Reading p. 27/3
28 Proof of corollary 1 sgn(s(, T i ; y ) S(, T i ; y Ta )) = sgn(s(, T j ; y ) S(, T j ; y Ta )), i, j Therefore: ( Tb h(u) ( S(, u; y ) S(, u; y Ta ) ) du) + = Tb h(u) ( S(, u; y ) S(, u; y Ta ) ) + du By Fubini s theorem, we change the order of integrations. ICMA Centre, University of Reading p. 28/3
29 Proving the formula Thus, when y is found, the default swaption satisfies: ( 1 {τ>t} D(t, ) exp Ta t ψ(s)ds ) Tb h(u)a(, u)e R u Ta ψ(s)ds Ψ(t,, y t, y, B(, u))du where [ ( Ψ(t, T, y t, ς, ) := E exp T t y s ds ) (e ς e y T ) + /Ft ] Proposition 2 Ψ(t, T, y t, ς, ) = e ς Π(T t, y t, ς, 0) Π(T t, y t, ς, ) ICMA Centre, University of Reading p. 29/3
30 Proof of proposition 2 {e ς e y T } {y T ς} Hence: Ψ(t, T, y t, ς, ) = [ ( e ς E exp [ ( E exp y T T t T t y s ds )1 {yt ς}/f t ] ] y s ds )1 {yt ς}/f t [ ( Π(T, y 0, ς, ) := E exp y T T 0 ) y s ds 1 {yt ς} ] Christensen (2002) derived the formula for Π, and we use the fact that y t is a homogenous and markovian jump-diffusion. ICMA Centre, University of Reading p. 30/3
31 Conclusion The SSRJD model allows for: dynamic deformations of the CDS term structure including jumps and stochastic volatility It also fits the CDS term structure and generates a volatility smile Pricing default swaptions in the SSRJD model is relatively tractable with a semi-analytical formula The model can be calibrated to a term structure of CDS and very few default swaptions, to price other products ICMA Centre, University of Reading p. 31/3
32 References [Bielecki and Rutkowski (2001)] Bielecki, T., and Rutkowski, M.: Credit risk: Modeling, Valuation and Hedging. Springer (2001). [Brigo and Alfonsi (2005)] Brigo, D., and Alfonsi, A.: Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance and Stochastics, Vol 9, n. 1, pp (2005). [Brigo and Cousot (2006)] Brigo, D., and Cousot, L.: A Comparison between the SSRD Model and the market model for CDS options pricing. International Journal of Theoretical and Applied Finance, Vol 9, n. 3, pp (2006). [Brigo and El-Bachir (2006)] Brigo, D., and El-Bachir, N.: Credit derivatives pricing with a smile-extended jump SSRJD stochastic intensity model. ICMA centre Discussion Papers in Finance DP (2006). ICMA Centre, University of Reading p. 32/3
33 References [Brigo and El-Bachir (2007)] Brigo, D., and El-Bachir, N.: An exact formula for default swaptions pricing in the SSRD and SSRJD stochastic intensity models. working paper (2007). [Christensen (2002)] Christensen, J. H.: Kreditderivater og deres prisfastsættelse. Thesis, Institute of Economics, University of Copenhagen (2002). [Jamshidian (1989)] Jamshidian, F.: An Exact Bond Option Formula. Journal of Finance, Vol. 44, pp (1989). ICMA Centre, University of Reading p. 33/3
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