Applications to Fixed Income and Credit Markets

Applications to Fixed Income and Credit Markets Jean-Pierre Fouque University of California Santa Barbara 28 Daiwa Lecture Series July 29 - August 1, 28 Kyoto University, Kyoto 1

Fixed Income Perturbations around Vasicek (for instance) to account for: Volatility Time Scales Fit to Yield Curves Reference: Stochastic Volatility Corrections for Interest Rate Derivatives Mathematical Finance 14(2), April 24 2

Constant Volatility Vasicek Model Under the physical probability IP: d r t = a( r r t )dt + σd W t Under the risk-neutral pricing probability IP : d r t = a(r r t )dt + σd W t with a constant market price of interest rate risk λ: r = r λ σ a 3

Bonds Prices { Λ(t, T) = IE e T t r s ds F t } { = IE e T t r s ds r t } = P(t, r t ; T) Vasicek PDE: P t + 1 2 σ2 2 P x 2 + a(r x) P x x P = with the terminal condition P(t, x; T) = 1. Introduce the time-to-maturity τ = T t and seek a solution of the form: P(T τ, x; T) = A(τ)e B(τ)x by solving linear ODE s with A() = 1 and B() =. 4

Affine Yields B(τ) = 1 e aτ a { 1 e A(τ) = exp [R aτ τ R a + σ2 ( 1 e aτ ) ]} 2 4a 3 with Yield Curve: R = r σ2 2a 2 = r λ σ a σ2 2a 2 R(t, τ) = 1 τ log (Λ(t, t + τ)) = B(τ) r t + log A(τ) = R (R r t ) 1 e aτ aτ + σ2 4a 3 τ ( 1 e aτ ) 2 5

1.8 BOND PRICES.6.4.2 5 1 15 2 25 3 MATURITY.95.9 YIELD.85.8.75.7 5 1 15 2 25 3 MATURITY Figure 1: Bond prices (top) and cblue Yield curve (bottom) in the Vasicek model with a = 1, r =.1 and σ =.1. Maturity τ runs from to 3 years. R =.95 and the initial rate is x =.7. 6

Bond Options Prices Example: a Call Option with strike K and maturity T written on a zero-coupon bond with maturity T > T. The payoff h(λ(t, T)) = (Λ(T, T) K) + is a function of r T since Λ(T, T) = P(T, r T ; T) Call Option Price: { C(t, x; T, T ) = IE e T t } r s ds h (Λ(T, T)) r t = x solution of Vasicek PDE with terminal condition at t = T : C(T, x; T, T ) = ( P(T, x; T) K) + C(t, x; T, T ) = P(t, x; T)N(h 1 ) K P(t, x; T )N(h 2 ) 7

Stochastic Volatility Vasicek Models Under the physical measure: dr t = a(r r t )dt + f(y t )dw t where f is a positive function of a mean-reverting volatility driving process Y t. Example: Y t is an OU process: dy t = α(m Y t )dt + ν 2αdẐ t where Ẑt is a Brownian motion possibly correlated to the Brownian motion W t driving the short rate: Ẑ t = ρw t + 1 ρ 2 Z t (W t, Z t ) independent Brownian motions. 8

Stochastic Volatility Vasicek Pricing Models Under the risk-neutral pricing probability IP (λ,γ) : dr t = (a(r r t ) λ(y t )f(y t )) dt + f(y t )dwt ( dy t = α(m Y t ) ν 2α [ρλ(y t ) + γ(y t ) ]) 1 ρ 2 dt +ν ( 2α ρdwt + ) 1 ρ 2 dzt for bounded market prices of risk λ(y) and γ(y). Under fast mean-reversion: α is large 9

Bond Pricing P(t, x, y; T) = IE (λ,γ) {e T t } r s ds r t = x, Y t = y P t + 1 2 f(y)2 2 P x + (a(r 2 x) λ(y)f(y)) P x xp ( ) + α ν 2 2 P + (m y) P y2 y + ν ( 2α ρf(y) 2 P [ x y ρλ(y) + γ(y) ] ) P 1 ρ 2 y = with the terminal condition P(T, x, y; T) = 1 for every x and y. Expand : P ε = P + εp 1 + εp 2 + ε εp 3 + ε = 1/α 1

Leading Order Term P t + 1 2 σ2 x 2 2 P x 2 + a (r x) P x xp = Effective volatility σ 2 = f 2 and r = r λf /a The zero order term P (t, x) is the Vasicek bond price P (T τ, x; T) = P(T τ, x; T) = A(τ)e B(τ)x computed with the constant parameters (a, r, σ). 11

The Correction P 1 = εp 1 The correction P 1 solves the source problem: ( L V asicek (a, r, σ) P 1 = V 1 x + V 2 with the zero terminal condition P 1 (T, x) =. 2 x 2 + V 3 It involves the constant quantities, small of order 1/ α V 3 = ν 2α ρ fφ V 2 = ν (ρ λφ + ) 1 ρ 2 γφ 2α V 1 = ν 2 α ( ρ λψ + ) 1 ρ 2 γψ 3 ) P x 3 2 νρ α fψ 12

The Correction P 1 : explicit computation Using the variable τ = T t and the explicit form P = Ae Bx : P 1 τ = 1 2 σ2 2 P 1 x 2 + â(r x) P 1 x x P 1 +A(τ)e B(τ)x ( V 3 B(τ) 3 V 2 B(τ) 2 + V 1 B(τ) ) We seek a solution of the form P 1 (T τ, x; T) = D(τ)A(τ)e B(τ)x with the condition D() = so that P 1 (T, x; T) = We get: and D(τ) = V 3 â 3 V 2 â 2 D = V 3 B 3 V 2 B 2 + V 1 B (τ B(τ) 12âB(τ)2 13â2 B(τ) 3 ) (τ B(τ) 12âB(τ)2 ) + V 1 â (τ B(τ)) 13

Summary The corrected bond price is given by P(T τ, x, y; T) P (T τ, x; T) + P 1 (T τ, x; T) = A(τ) (1 + D(τ))e B(τ)x where D is a small factor of order 1/ α. The error P ε (t, x, y; T) ( P (t, x : T) + P ) 1 (t, x; T) is of order 1/α. Corrections for bond options prices are also obtained. 14

1.8 BOND PRICES.6.4.2 5 1 15 2 25 3 MATURITY.95.9.85 YIELD.8.75.7.65 5 1 15 2 25 3 MATURITY Figure 2: Top: bond prices and corrected bond prices (dotted curve). Bottom: yield curve and corrected yield curve (dotted curve) in the simulated Vasicek model (constant and stochastic volatility) with: a = 1, r =.1 and σ =.1 as in Figure 3. Correction: V 3 = 1/ α (ρ ), α = 1 3 and λ = γ = implying V 1 = and V 2 =. Maturity τ runs from to 3 years and the initial rate is x =.7. 15

Model Parameters Rate of mean-reversion of short-rate: a Long-run mean under IP: r Specific volatility distribution: f( ) Correction Parameters a r Mean volatility σ Rate of mean-reversion of volatility : α Group parameter V 1 Mean-level of (Y t ): m Group parameter V 2 V-vol : β Group parameter V 3 Correlation: ρ Interest-rate risk premium: λ( ) Volatility risk premium: γ( ) 16

.62 Vasicek with stochastic volatility correction.6 bond yield.58.56.54 1 2 3 4 5 6 7 years to maturity.62 CIR with jumps.6 bond yield.58.56.54 1 2 3 4 5 6 7 years to maturity Figure 3: Snapshot of the yield curve fit with the stochastic volatility corrected Vasicek model (top) and with the single factor CIR model and down jumps (bottom) for September 6, 1998. 17

Credit Perturbations around Merton/Black-Cox (in the context of the structural approach for instance) to account for: Volatility Time Scales in Default Times Fit to Yield Spreads References: Stochastic Volatility Effects on Defaultable Bonds Applied Mathematical Finance 26 with R. Sircar and K. Solna Modeling Correlated Defaults: First Passage Model under Stochastic Volatility Journal of Computational Finance 28 with B. Wignall and X. Zhou 18

Defaultable Bonds In the first passage structural approach, the payoff of a defaultable zero-coupon bond written on a risky asset X is h(x) = 1 {inf s T X s >B}. By no-arbitrage, the value of the bond is P B (t, T) = IE { e r(t t) 1 {inf s T X s >B} F t } = 1 {inf s t X s >B}e r(t t) IE { 1 {inft s T X s >B} F t }, Using the predictable stopping time τ t = inf{s t, X s B}: IE { 1 {inft s T X s >B} F t } = IP {τ t > T F t }. This defaultable zero-coupon bond is in fact a binary down-an-out barrier option where the barrier level and the strike price coincide. 19

Constant Volatility: Merton s Approach dx t = rx t dt + σx t dwt ( X t = X exp (r 1 2 σ2 )t + σwt ). In the Merton s approach, default occurs if X T < B: Defaultable bond = European digital option u d (t, x) = IE { e rτ 1 {XT >B} X t = x } = e rτ IP {X T > B X t = x} = e rτ N(d 2 (τ)) with the usual notation τ = T t and the distance to default: log ( ) ) x B + (r σ2 2 τ d 2 (τ) = σ τ 2

Constant Volatility: Black-Cox Approach IE { } 1 {inft s T X s >B} F t { ) = IP inf ((r σ2 t s T 2 )(s t) + σ(w s Wt ) > log ( B x ) } X t = x computed using distribution of minimum, or using PDE s: IE { e r(t t) 1 {inft s T X s >B} F t } = u(t, X t ) where u(t, x) is the solution of the following problem which is to be solved for x > B. L BS (σ)u = on x > B, t < T u(t, B) = for any t T u(t, x) = 1 for x > B, 21

Constant Volatility: Barrier Options Using the European digital pricing function u d (t, x) L BS (σ)u d = on x >, t < T u d (T, x) = 1 for x > B, and otherwise By the method of images one has: u(t, x) = u d (t, x) where we denote ( x B ) 1 2r σ 2 u d ( t, B2 x = e r(t t) ( N(d + 2 (T t)) ( x B d ± 2 (τ) = ± log ( x B ) ) + (r σ2 2 σ τ ) 1 2r σ 2 N(d 2 (T t)) ) ) τ 22

Yield Spreads Curve The yield spread Y (, T) at time zero is defined by e Y (,T)T = P B (, T) P(, T), where P(, T) is the default free zero-coupon bond price given here, in the case of constant interest rate r, by P(, T) = e rt, and P B (, T) = u(, x), leading to the formula Y (, T) = 1 T log (N (d 2 (T)) ( x B ) 1 2r σ 2 N ( d 2 (T))) 23

45 4 35 Yield spread in basis points 3 25 2 15 1 5 1 1 1 1 1 1 2 Time to maturity in years Figure 4: The figure shows the sensitivity of the yield spread curve to the volatility level. The ratio of the initial value to the default level x/b is set to 1.3, the interest rate r is 6% and the curves increase with the values of σ: 1%, 11%, 12% and 13% (time to maturity in unit of years, plotted on the log scale; the yield spread is quoted in basis points) 24

45 4 35 Yield spread in basis points 3 25 2 15 1 5 1 1 1 1 1 1 2 Time to maturity in years Figure 5: This figure shows the sensitivity of the yield spread to the leverage level. The volatility level is set to 1%, the interest rate is 6%. The curves increases with the decreasing ratios x/b: (1.3, 1.275, 1.25, 1.225, 1.2). 25

Challenge: Yields at Short Maturities As stated by Eom et.al. (empirical analysis 21), the challenge for theoretical pricing models is to raise the average predicted spread relative to crude models such as the constant volatility model, without overstating the risks associated with volatility or leverage. Several approaches (within structural models) have been proposed that aims at the modeling in this regard. These include Introduction of jumps (Zhou,...) Stochastic interest rate (Longstaff-Schwartz,...) Imperfect information (on X t ) (Duffie-Lando,...) Imperfect information (on B) (Giesecke) 26

Stochastic Volatility Models where we assume that dx t = µx t dt + f(y t )X t dw () t dy t = α(m Y t )dt + ν 2α dw (1) t f non-decreasing, < c 1 f c 2 Invariant distribution of Y : N(m, ν 2 ) independent of α α > is the rate of mean reversion of Y The standard Brownian motions W () and W (1) are correlated d W (), W (1) = ρ 1 dt t 27

Stochastic Volatility Models under IP In order to price defaultable bonds under this model for the underlying we rewrite it under a risk neutral measure IP, chosen by the market through the market price of volatility risk Λ 1, as follows dx t = rx t dt + f(y t )X t dw () t, ( dy t = α(m Y t ) ν ) 2αΛ 1 (Y t ) dt + ν 2α dw (1) t. Here W () and W (1) are standard Brownian motions under IP correlated as W () and W (1). We assume that the market price of volatility risk Λ 1 is bounded and a function of y only. 28

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path.25.15 1 1 1 1 1 1 2.2.1.5 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 6: Uncorrelated slowly mean-reverting stochastic volatility: α =.5 and ρ 1 =. 29

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path.25.15 1 1 1 1 1 1 2.2.1.5 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 7: Correlated slowly mean-reverting stochastic volatility: α =.5 and ρ 1 =.5. 3

Yield spread in basis points 4 3 2 1 2 4 6 8 1 12 14 16 18 2 SV path 4 3 2 1.15.5 1 1 1 1 1 1 2.2.1 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 8: Uncorrelated stochastic volatility: α =.5 and ρ 1 =. 31

Yield spread in basis points 4 3 2 1 2 4 6 8 1 12 14 16 18 2 SV path 4 3 2 1 1 1 1 1 1 1 2.4.3.2.1 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 9: Correlated stochastic volatility: α =.5 and ρ 1 =.5. 32

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 1: Uncorrelated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =. 33

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 11: Correlated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =.5. 34

Yield spread in basis points 6 4 2 5 1 15 2 6 4 2 1 1 1 1 1 1 2.4 SV path.2 5 1 15 2 Time to maturity in years Figure 12: Highly correlated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =.5. 35

Yield spread in basis points 8 6 4 2 2 4 6 8 1 12 14 16 18 2 SV path 8 6 4 2 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 13: High leverage correlated fast mean-reverting stochastic volatility: x/b = 1.2, α = 1 and ρ 1 =.5. 36

Barrier Options under Stochastic Volatility u(t, x, y) = e r(t t) IE { h(x T )1 {inft s T X s >B} X t = x, Y t = y }, P B (t, T) = 1 {inf s t X s >B}u(t, X t, Y t ). The function u(t, x, y) satisfies for x B the problem ( t + L X,Y r ) u = on x > B, t < T u(t, B) = for any t T u(t, x) = h(x) for x > B where L X,Y is the infinitestimal generator of the process (X, Y ) under IP. 37

Leading Order Term under Stochastic Volatility In the regime α large, as in the European case, u(t, x, y) is approximated by u (t, x) which solves the constant volatility problem L BS (σ )u = u (t, B) = on x > B, t < T for any t T u (T, x) = h(x) for x > B where σ is the corrected effective volatility. 38

Stochastic Volatility Correction Define the correction u 1(t, x) by L BS (σ )u 1 = V 3 x x u 1(t, B) = ( ) x 2 2 u x 2 on x > B, t < T for any t T u 1(T, x) = for x > B. Remarkably, the small parameter V 3 is the same as in the European case (calibrated to implied volatilities). 39

Define Computation of the Correction v 1(t, x) = u 1(t, x) (T t)v 3 x x so that v 1(t, x) solves the simpler problem L BS (σ )v 1 = v 1(t, B) = g(t) ( x 2 2 u ), x 2 on x > B, t < T for any t T v1(t, x) = for x > B ( )) g(t) = V 3 (T t) lim x B (x x x 2 2 u x 2 To summarize we have u(t, x, y) u (t, x) + (T t)v 3 x x with explicit computation in the case h(x) = 1. ( x 2 2 u ) x 2 + v 1(t, x) 4

25 Term structure of yield 2 15 1 5 2 4 6 8 1 12 14 16 18 2 25 2 15 1 5 1 1 1 1 1 Time to maturity in years Figure 14: The price approximation for σ =.12,r =.,V 3 =.3, x/b = 1.2. 41

Slow Factor Correction The first correction u (z) 1 (t, x) solves the problem L BS ( σ(z))u (z) 1 = 2 ( V (z) u BS σ + V 1(z)x x ( ubs σ )) on x > B, t < T, u (z) 1 (t, B) = for t T, u (z) 1 (T, x) = for x > B, where u BS is evaluated at (t, x, σ(z)), and V (z) and V 1 (z) are small parameters of order δ, functions of the model parameters, and depending on the current level z of the slow factor. 42

7 Fits to Ford Yields Spreads, 12/9/4 6 Yield Spreads (%) 5 4 3 2 Black Cox Stochastic Volatility Data 1 2 4 6 8 1 Time to maturity Figure 15: Black-Cox and two-factor stochastic volatility fits to Ford yield spread data. The short rate is fixed at r =.25. The fitted Black-Cox parameters are σ =.35 and x/b = 2.875. The fitted stochastic volatility parameters are σ =.385, corresponding to R 2 =.129, R 3 =.12, R 1 =.16 and R =.8. 43

1 Add R 3 1 Add R Yield Spreads (%) 5 5 1 5 5 1 1 Add R 1 1 Add R 2 σ * Yield Spreads (%) 5 5 1 Time to maturity 5 5 1 Time to maturity 44

6 Fit to IBM Yield Spreads 12/1/4 5 Yield Spread (%) 4 3 2 Black Cox Stochastic Volatility Data 1 5 1 15 2 25 Time to maturity (years) Figure 16: Black-Cox and two-factor stochastic volatility fits to IBM yield spread data. The short rate is fixed at r =.25. The fitted Black-Cox parameters are σ =.35 and x/b = 3. The fitted stochastic volatility parameters are σ =.36, corresponding to R 2 =.355, R 3 =.112, R 1 =.13 and R =.45. 45

Multiname Model Setup Under risk neutral pricing probability: dx (1) t = rx (1) t dt + f 1 (Y t, Z t )X (1) t dw (1) t, dx (2) t = rx (2) t dt + f 2 (Y t, Z t )X (2) t dw (2) t, dx (n) dw (n) t = rx (n) t dt + f n (Y t, Z t )X (n) t t, [ 1 dy t = ε (m Y Y t ) ν ] Y 2 Λ 1 (Y t, Z t ) dt + ν Y 2 dw (Y ) ε ε t, [ ] (Z) dz t = δ(m Z Z t ) ν Z 2δΛ2 (Y t, Z t ) dt + ν Z 2δdW t, where the W (i) t s are independent standard Brownian motions and d W (Y ), W (i) t = ρ iy dt, d W (Z), W (i) t = ρ iz dt, d W (Y ), W (Z) t = ρ Y Z dt. with n i=1 ρ2 iy 1 and n i=1 ρ2 iz 1. 46

Objective Find the joint (risk-neutral) survival probabilities u ε,δ u ε,δ (t,x, y, z) IP { τ (1) t > T,...,τ (n) t > T } X t = x, Y t = y, Z t = z, where t < T, X t (X (1) t the default time of firm i: τ (i) t = inf,...,x (n) t ), x (x 1,...,x n ), and τ (i) t { s t X (i) s } B i (s), is where B i (t) is the exogenously pre-specified default threshold at time t for firm i. Following Black and Cox (1976) we assume B i (t) = K i e η it, with constant parameters K i > and η i. 47

PDE Formulation L ε,δ u ε,δ (t,x, y, z) =, x i > B i (t), for all i, t < T L ε,δ = 1 ε L + 1 ε L 1 + L 2 + δm 1 + δm 2 + δ ε M 3 Boundary conditions: u ǫ,δ (t, x 1, x 2,...,x n, y, z) =, i {1,,n}, x i = B i (t), t T, Terminal condition: u ε,δ (T, x 1, x 2,...,x n, y, z) = 1, x i > B i (t), for all i 48

Expansion and Approximation u ε,δ = u + εu 1, + δu,1 }{{} + εu 2, + εδu 1,1 + δu,2 + Leading Order Term u : L 2 u =, x i > B i (t), for all i, t < T u (t, x 1, x 2,...,x n ) =, i {1,,n}, x i = B i (t), t T, u (T, x 1, x 2,...,x n ) = 1, x i > B i (t), for all i L 2 = t + n σ i (z) = i=1 ( 1 2 σ i(z) 2 x 2 i 2 x 2 i ) + rx i x i f 2 i (, z), : average w.r.t. N(m Y, ν 2 Y ) 49

A Formula for u u = n Q i i=1 n i=1 [ ( ) N d + 2(i) ( ) pi xi ( N d B i (t) 2(i)) ], where N( ) is the standard normal distribution function, ( ) ± ln x i d ± B i (t) + r η i σ2 i (z) 2 (T t) 2(i) = σ i (z), T t σ i (z) = fi 2(,z), p i = 1 2(r η i) σ 2 i (z). 5

Correction Term εu 1, L 2 u 1, = Au, x i > B i (t), for all i, t < T u 1, (t, x 1, x 2,...,x n ) =, i {1,,n}, x i = B i (t), t T, u 1, (T, x 1, x 2,...,x n ) =, x i > B i (t), for all i ν Y 2 n i=1 n j=1 ρ iy f i (, z) φ j x i y x i ( x 2 j 2 x 2 j ) A = L 1 L 1 (L 2 L 2 ) = n Λ 1 (, z) φ j x 2 2 j y j=1 x 2 j where the φ i s are given by the Poisson equations w.r.t. y: L φ i (y, z) = f 2 i (y, z) f 2 i (, z). Then use u (t, x 1,,x n ) = n i=1 Q i(t, x i ). 51

Correction Term δ u,1 L 2 u,1 = M 1 u, x i > B i (t), for all i, t < T u,1 (t, x 1, x 2,...,x n ) =, i {1,,n}, x i = B i (t), t T, u,1 (T, x 1, x 2,...,x n ) =, x i > B i (t), for all i. n ν Z 2 i=1 n j=1 [ n 2 M 1 = ν Z 2 ρ iz f(, z) x i i=1 ρ iz f(, z) σ j(z)x i x i ( σ j ) x i z Λ 2(, z) z Λ 2 (, z) n i=1 ] σ i(z) σ i = Then use u (t, x 1,,x n ) = n i=1 Q i(t, x i ). 52

Homogeneous Portfolio Case u (t, x,,x) = n Q i (t, x) = Q(t, x) n q n i=1 εu1, = n (R (2) 1 w(2) 1 +n(n 1)R (3) δ u,1 = n (R () 1 w() 1 +n(n 1)R (1) ) (t, x) + R(3) 1 w(3) 1 (t, x) q n 1 12 w(3) 12 (t, x, x)qn 2 ) (t, x) + R(1) 1 w(1) 1 (t, x) q n 1 12 w(1) 12 (t, x, x)qn 2 Joint survival probabilities: S n ũ u + ǫ u 1, + δ u,1 = q n + Anq n 1 + Bn(n 1)q n 2 A = R () B = R (1) 1 w() 1 12 w(1) 12 (t, x) + R(1) 1 w(1) 1 (t, x, x) + R(3) 12 w(3) 12 (t, x) + R(2) (t, x, x) 1 w(2) 1 (t, x) + R(3) 1 w(3) 1 (t, x) 53

Loss Distribution For N names perfectly symmetric, if L is the number of defaults by time T, then IP (L = k) = ( ) k N k j= ) k ( N k i= ( ) k N +A k +B ( N k ( ) k ( 1) j S N+j k j ( ) k ( 1) k i q N i i i= ) k i= I + AI 1 + BI 2 ( ) k ( 1) k i (N i)q N i 1 i ( ) k ( 1) k i (N i)(n i 1)q N i 2 i 54

Loss Distribution Formulas IP (L = k) I + AI 1 + BI 2 with I = I 1 = I 2 = ( ) N (1 q) k q N k k [ N k k ] I q 1 q [ (N k)(n k 1) q 2 2k(N k) q(1 q) + k(k 1) (1 q) 2 ] I 55

.14 binomial perturbed.12.1 probability.8.6.4.2 1 2 3 4 5 6 7 8 9 1 number of defaults 1.9 binomial perturbed.8.7 cumulative probability.6.5.4.3.2.1 1 2 3 4 5 6 7 8 9 1 number of defaults N = 1, q =.9, A =., B =.6 56

Models with Name-Name Correlation d W (i), W (j) t = ρ ij dt, ρ ij < 1 for i j L ǫ,δ,ρ = L ǫ,δ + with n i<j ρ ij L (ij) ρ L (ij) ρ = f i (y, z)f j (y, z)x i x j 2 x i x j Expand u ǫ,δ,ρ = u ǫ,δ + n i<j ρ ij ( u (ij),,1 + ǫu (ij) 1,,1 + δu (ij),1,1 + ) + and retain the first corrections, ũ u + ǫ u 1, + δ u,1 + n i<j ρ ij u (ij),,1 57

Correction Terms ρ ij u (ij),,1 L 2 u (ij),,1 = L (ij) ρ u, x l > B l (t), for all l, t < T u (ij),,1 (t, x 1, x 2,...,x n ) =, l {1,,n}, x l = B l (t), t T u (ij),,1 (T, x 1, x 2,...,x n ) =, x l > B l (t), for all l where L (ij) ρ = f i (, z)f j (, z) x i x j 2 x i x j Then use u (t, x 1,,x n ) = n i=1 Q i(t, x i ) to deduce n ρ ij u (ij),,1 = R (4) ij w(4) ij Q k k=1 k i,j R (4) ij = ρ ij f i (, z)f j (, z), i j 58

Homogeneous Portfolio Case ρ ij = ρ u (t, x,,x) = n Q i (t, x) = Q(t, x) n q n i=1 Joint survival probabilities: S n ũ u + ǫu 1, + δ u,1 + n i<j ρ ij u (ij),,1 = q n + Anq n 1 + (B + B ρ )n(n 1)q n 2 A = R () B = R (1) 1 w() 1 12 w(1) 12 B ρ = 1 2 R(4) 12 w(4) 12 (t, x) + R(1) 1 w(1) 1 (t, x, x) + R(3) 12 w(3) 12 (t, x) + R(2) (t, x, x) (t, x, x), R(4) 12 = ρσ2 (z) 1 w(2) 1 (t, x) + R(3) 1 w(3) 1 (t, x) 59

Comparison of the Two Sources of Correlation For a single maturity T: the correlations generated by stochastic volatility and name-name correlation are of the same form to leading order. Term structure of correlation across several maturities: the shape of the function w (4) 12 is different from the shapes of w(1) 12 and w (3) 12 and therefore the nature of the correlation plays a role. 6