Hedging against changes in the realized/implied volatility. â Speculation and directional trading. Credit Default Swaps. â Hedging against the default of the ...
Volatility derivatives and default risk ARTUR SEPP Merrill Lynch Quant Congress London November 14-15, 2007
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Plan of the presentation 1) Heston stochastic volatility model with the term-structure of ATM volatility and the jump-to-default: interaction between the realized variance and the default risk 2) Analytical and numerical solution methods for the pricing problem 3) Case study: application of the model to the General Motors data, implications
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References Theoretical and practical details for my presentation can be found in: 1) Sepp, A. (2008) Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility, Journal of Computational Finance, Vol. 11, No. 4, pp. 33-70 http://ssrn.com/abstract=1408005 2) Sepp, A. (2007) Affine Models in Mathematical Finance: an Analytical Approach, PhD thesis, University of Tartu http://math.ut.ee/~spartak/papers/seppthesis.pdf 3) Sepp, A. (2006) Extended CreditGrades Model with Stochastic Volatility and Jumps, Wilmott Magazine, September, 50-60 http://ssrn.com/abstract=1412327
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Financial Motivation Volatility Products ⊗ Hedging against changes in the realized/implied volatility ⊗ Speculation and directional trading Credit Default Swaps ⊗ Hedging against the default of the issuer ⊗ Speculation and directional trading Volatility and Credit Products ⊗ The degree of correlation ? ⊗ Relative value analysis
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Volatility Products I The asset realized variance: IN (t0, tN ) =
N AF X
ln
S(tn) S(tn−1)
!2
,
(1)
N n=1 S(tn) is the asset closing price observed at times t0 (inception), .., tN (maturity) N is the number of observations AF is annualization factor (typically, AF=252 - daily sampling) Realized variance swap with payoff function: U (T, I) = IN (0, T ) − Kf2air
Kf2air - the fair variance which equates the value of the var swap at the inception to zero Call on the realized variance swap with payoff function:
2 U (T, I) = max IN (0, T ) − Kf air , 0 5
Volatility Products II Forward-start call: S(T ) − K, 0 U (TF , T ) = max S(TF )
!
where TF - forward start time, T - maturity Forward-start variance swap: U (TF , T ) = IN (TF , T ) − Kf2air Option on the future implied volatility (VIX-type option): U (∆T, T ) = max
q
E[IN (T, T + ∆T )] − K, 0
The values of these products are sensitive to the evolution of the volatility surface 6
Credit Products Credit default swap (CDS) - the protection against the default of the reference name in exchange for quarterly coupon payments Deep out-of-the money put option - tiny value under the lognormal model unless a huge volatility parameter is used The value of a deep OTM put is almost proportional to its strike and the default probability up to its maturity Forward-start options - would typically lose their value if the default occurs up to the forward-start date The value of the forward-start option is sensitive to the evolution of the default probability curve
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Our Motivation Develop a model for the for pricing and risk-managing of volatility and credit products on single names For this purpose we need to describe the joint evolution of: the asset price S(t) its variance V (t), its realized variance I(t), the jump-to-default intensity λ(t) Design efficient semi-analytical and numerical solution methods Analyze model implications
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Heston model with volatility jumps and jump-to-default I We adopt the following joint dynamics under the pricing measure Q: q dS(t) = µ(t)dt + σ(t) V (t)dW s(t) − dN d(t), S(0) = S0 S(t−) q
dV (t) = κ(1 − V (t))dt + ε(t) V (t)dW v (t) + J v dN v (t), V (0) = 1, dI(t) = σ 2(t)V (t)dt, I(0) = I0, λ(t) = α(t) + β(t)V (t), (2) V (t) is ”normalized” variance σ(t) - is ”ATM-volatility” N d(t) - Poisson process with intensity λ(t) min{ι : N d(ι) = 1} is the default time 9
Heston model with volatility jumps and jump-to-default II µ(t) = r(t) − d(t) + λ(t) - the risk-neutral drift ρ(t) - the instantaneous correlation between W s(t) and W v (t) N v (t) - Poisson process with intensity γ J v - the exponential jump with mean η ε(t) - the vol-vol parameter κ - the mean-reversion
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Model Interpretation: Asset Realized Variance The expected variance: γη −T κ Q 1−e V (T ) := E [V (T )|V (0) = 1] = 1 +
(3)
κ
Assuming for moment no default risk, the asset realized variance in the continuous-time limit becomes: I(T ) = lim
N →∞
X tn ∈π N
S(tn) ln S(tn−1)
!2
=
Z T 0
σ 2(t0)V (t0)dt0
(4)
The expected realized variance: I(T ) := EQ[I(T )|V (0) = 1] =
Z T 0
σ 2(t0)V (t0)dt0
(5)
Given the values of mean-reversion parameters κ and jump parameters η and γ, we can extract the term structure of σ 2(t) from the fair variance curve observed from the market data 11
Model Interpretation: Jump-to-Default The probability of survival up to time T : R − tT λ(t0 )dt0 Q Q ] Q(t, T ) = E [ι > T |ι > t] = E [e
(6)
The probability of defaulting up to time T is connected to the integrated expected variance: Qc(t, T ) = EQ[ι ≤ T |ι > t] = 1 − Q(t, T ) ≈
Z T t
(α(t0) + β(t0)V (t0))dt0 (7)
Variation of the default intensity: < λ(t) >= β 2 < V (t) >
(8)
Parameter β can be extracted form the time series or from non-linear CDS contracts The term structure of parameter α(t), is backed-out from the survival probabilities implied CDS quotes 12
Recovery Assumption I Should be specified by the contract terms Can be simplified by the modeling purposes Asset price: zero Call option payoff: zero Put option payoff: its strike Forward-start call option payoff: zero Forward-start put option: zero if defaulted before the forwardstart date, its strike if defaulted between the forward-start date and maturity 13
Recovery Assumption II Realized Variance: Iˆ(T ) - the cap level on the realized variance Typically, Iˆ(T ) = 3K V (T ) where K V (T ) is the fair variance observed today for swap with maturity T Now the model implied expected realized variance at time T becomes: EQ[I(T )] ≈ Q(0, T )
Z T 0
σ 2(t0)V (t0)dt0 + Qc(0, T )Iˆ(T ),
(9)
”≈” since we ignore the cap on the realized pre-default variance and dependence between V (t) and Q(t, T ) In general, we compute: EQ[I(T )] = EQ
"Z
T
0
#
σ 2(t0)V (t0)dt0 | ι > T + Qc(0, T )Iˆ(T ),
(10)
Given the jump-to-default probabilities we use (9) or (10) to fit σ 2(t) to the term structure of the fair variance 14
Model Interpretation: Volatility Jumps Introduce the fat right tail to the density of the variance Explain the positive skew observed in the VIX options At the same time: Decrease the (terminal) correlation between the spot and both the implied variance and realized variance Increase the variance of the realized variance while give little impact on the asset (terminal) variance As a result, calibrating the variance jumps to the deep skews is not reasonable - we need to calibrate them to the volatility products 15
Convergence of Discretely Sampled Realized Variance to Continuous Time Limit, T = 1y, S0 = 1, V0 = 1, µ = 0.05, σ = 0.2, κ = 2, ε = 1, ρ = −0.8, γ = 0.5, η = 1
As the number of fixings decreases, the mean of the discrete sample decreases while its variance increases 16
General Pricing Problem under Model (2) I For calibration and pricing we need to model the joint evolution of (X(t), V (t), I(t)) with X(t) = ln S(t) Kolmogoroff forward equation for the joint transition density function G(t, T, V, V 0, X, X 0, I, I 0): 1 1 2 GT − (µ(T ) − σ 2(T )V 0)G + σ (T )V 0G 2 2 X0 X0 X 0 1 + ρ(T )ε(T )σ 2(T )V 0G 0 0 + κ(1 − V 0)G 0 + ε2(T )V 0G XV V 2 V 0V 0 Z ∞ 1 − 1η J v v 0 v − σ(T )V G 0 − γ(T ) dJ (G(V − J ) − G) e I η 0 − (α(T ) + β(T )V 0)G = 0, G(t, t, V, V 0X, X 0, I, I 0) = δ(X 0 − X)δ(V 0 − V )δ(I 0 − I), (11)
Here, (X 0, V 0, I 0) are variables (future states of the world), (X, V, I) are initial data 17
General Pricing Problem under Model (2) II Kolmogoroff backward equation for the value function U (t, T, V, V 0X, X 0, I, I 0): 1 2 1 2 Ut + (µ(t) − σ (t)V )UX + σ (t)V UXX 2 2 1 2 2 + ρ(t)ε(t)σ (t)V UXV + κ(1 − V )UV + ε (t)V UV V + σ 2(t)V UI 2 Z ∞ (12) 1 − 1η J v v v dJ − (α(t) + β(t)V )U + γ(t) (U (V + J ) − G) e η −∞ = (α(t) + β(t)V )R(t, V, V 0X, X 0, I, I 0) + U2(t, V, V 0X, X 0, I, I 0) U (T, T, V, V 0X, X 0, I, I 0) = U1(V, V 0X, X 0, I, I 0) U1(V, V 0X, X 0, I, I 0) - terminal pay-off function U2(t, V, V 0X, X 0, I, I 0) - instantaneous reward function R(t, V, V 0X, X 0, I, I 0) - the recovery value paid upon the default event Here, (X, V, I) are variables, (X 0, V 0, I 0) are parameters 18
Analytical Solution using the Fourier Transform We apply 3-dimensional generalized Fourier transform to forward PDE (11): b G(t, T, V, Θ, X, Φ, I, Ψ) =
Z ∞ Z ∞ Z ∞ −∞ −∞ −∞
0 0 0 e−X Φ−V Θ−I ΨGdX 0dV 0dI 0,
√(13) where Θ = ΘR + iΘI , Φ = ΦR + iΦI , Ψ = ΨR + iΨI i = −1, ΘR , ΘI , ΦR , ΦI , ΨR , ΨI ∈ R We obtain: b G(t, T, V, Θ, X, Φ, I, Ψ) = e−Φ(X+
RT
(r(t0 )−d(t0 ))dt0 )−ΨI+A(t,T )+B(t,T )V
, (14) where functions A(t, T ) and B(t, T ) are computed in closed-form by recursion t
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Marginal Transition Densities and Convergence Asymptotic convergence rate is important to set-up the bounds for quadrature and FFT inversion methods We first recall that for the Black-Scholes model with constant V : 2 2 −1 X b G (t, T, V, Φ, X) ∼ e 2 σ V0ΦI , |ΦI | → ±∞
For our model we obtain: b X (t, T, V, Φ, X) = G(t, b G T, V, 0, X, Φ, I, 0) ∼ b I (t, T, V, Ψ, I) = G(t, b G T, V, 0, X, 0, I, Ψ) ∼ e
((T −t)κ+σ 2 V0 )(1−ρ2 ) |ΦI | ε e− ,
−
2(T −t)κ+σ 2 V0 √ |ΨI | ε2 ,
|ΦI | → ±∞
|ΨI | → ±∞,
− 2κ ln |ΘI | V b b G (t, T, V, Θ) = G(t, T, V, Θ, X, 0, I, 0) ∼ e ε2 , |ΘI | → ±∞ R x = 0T x(t0)dt0 and ∼ stands for the leading term of the real part
b X , moderate for G bI, In relative terms, the convergence is fast for G bV and slow for G 20
Moments All moments are can be computed numerically by approximating the partial derivatives: h i Q k j l E X (T )V (T )I (T )
= (−1)k+j+l
∂ k+j+l j ∂ΦkR ∂ΘR ∂ΨlR
b G(t, T, V, Θ, X, Φ, I, Ψ) |Φ=0,Θ=0,Ψ=0
The survival probability is computed by: b I (t, T, V, 1, I) Q(t, T ) = G
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Option Pricing I The general pricing problem includes computing the expectation of the pay-off and reward functions: R − tT (r(t0 )+λ(t0 ))dt0 Q u1(X(T ), V (T ), I(T )) U (t, X, I, V ) = E e
+
Z T t
R t0
#
− t (r(t00 )+λ(t00 ))dt00 u2(t0, X(t0), V (t0), I(t0))dt0 , e
= U1(t, X, I, V ) + U2(t, X, I, V ) (15) We compute the Fourier-transformed pay-off and reward functions: b 1(Φ, Θ, Ψ) = u b 2(t, Φ, Θ, Ψ) = u
Z ∞ Z ∞ Z ∞ −∞ −∞ −∞ Z ∞ Z ∞ Z ∞ −∞ −∞ −∞
0 +ΘV 0 +ΨI 0 ΦX e u1(X 0, V 0, I 0)dX 0dV 0dI 0, 0 0 0 eΦX +ΘV +ΨI u2(t, X 0, V 0, I 0)dX 0dV 0dI 0,
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Option Pricing II The value of the option is then computed by inversion: ∞ ∞ ∞ 1 U1(t, X, I, V ) = 3 h 8π −∞ −∞ −∞ i b b 1(Φ, Θ, Ψ) dΦI dΘI dΨI , < G(t, T, V, Θ, X, Φ, I, Ψ)u
Z
Z
Z
∞ ∞ T ∞ 1 U2(t, X, I, V ) = 3 t 8π −∞ −∞ −∞ i h 0 0 b b 2(t , Φ, Θ, Ψ) dΦI dΘI dΨI dt0 < G(t, t , V, Θ, X, Φ, I, Ψ)u
Z
Z
Z
Z
In one (two) dimensional case these formulas reduce to one (two) dimensional integrals For example, for call option on the asset price with strike K we have: U (t, X, I, V ) = −
e−
RT t
r(t0 )dt0
π
Z ∞ 0
(Φ+1) ln K e X b (t, T, V, Φ, X) dΦ , < G I
Φ(Φ + 1)
where −1 < ΦR < 0 23
Numerical Solution using Craig-Sneyd ADI method I ⊗ Allows to solve the pricing problem in its most general form ⊗ Can be applied for both forward and backward equations in a consistent way Introduce the following discretesized operators: LI - the explicit convection vector operator in I direction LX - the implicit convection-diffusion operator in X direction LV - the implicit convection-diffusion operator in V direction CXV - the explicit correlation operator JV - the explicit jump operator in V direction For the forward equation the transition from solution Gn at time tn to Gn+1 at time tn+1 is computed by: G∗ = (I + LI )Gn (I + LX )G∗∗ = (I − LX − 2LV + CXV )G∗ (I + LV )Gn+1 = (I + LV + JV )G∗∗
(16)
Steps 2 and 3 lead to a system of tridiagonal equations Jump operator is handled by a fast recursive algorithm 24
Numerical Solution using Craig-Sneyd ADI method II Allows to analyze volatility products with general accrual variable: I(t, T ) =
Z T t
f (t0, V, X, I)dt0
(17)
For example, for conditional up and down variance swap with upper level U (t) and lower level L(t) (in continuous time limit): f up(t, V, X) = 1{eX(t)≥U (t)}σ 2(t)V (t), f down(t, V, X) = 1{eX(t)
