volatility derivatives and default risk
DESCRIPTION
Volatility, Credit, Affine Models, Jump-to-Default, Variance SwapTRANSCRIPT
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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 ATMvolatility and the jump-to-default: interaction between the realizedvariance 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 He-ston Model with Jumps in Returns and Volatility, Journal of Compu-tational Finance, Vol. 11, No. 4, pp. 33-70http://ssrn.com/abstract=1408005
2) Sepp, A. (2007) Affine Models in Mathematical Finance: an An-alytical Approach, PhD thesis, University of Tartuhttp://math.ut.ee/~spartak/papers/seppthesis.pdf
3) Sepp, A. (2006) Extended CreditGrades Model with StochasticVolatility and Jumps, Wilmott Magazine, September, 50-60http://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) =AF
N
N∑n=1
(ln
S(tn)
S(tn−1)
)2
, (1)
S(tn) is the asset closing price observed at times t0 (inception), .., tN(maturity)N is the number of observationsAF is annualization factor (typically, AF=252 - daily sampling)
Realized variance swap with payoff function:
U(T, I) = IN(0, T )−K2fair
K2fair - 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:
U(T, I) = max(IN(0, T )−K2
fair,0)
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Volatility Products II
Forward-start call:
U(TF , T ) = max
(S(T )
S(TF )−K,0
)where TF - forward start time, T - maturity
Forward-start variance swap:
U(TF , T ) = IN(TF , T )−K2fair
Option on the future implied volatility (VIX-type option):
U(∆T, T ) = max(√
E[IN(T, T + ∆T )]−K,0)
The values of these products are sensitive to the evolution of thevolatility surface
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Credit Products
Credit default swap (CDS) - the protection against the default ofthe reference name in exchange for quarterly coupon payments
Deep out-of-the money put option - tiny value under the log-normal model unless a huge volatility parameter is used
The value of a deep OTM put is almost proportional to its strike andthe default probability up to its maturity
Forward-start options - would typically lose their value if the defaultoccurs up to the forward-start date
The value of the forward-start option is sensitive to the evolution ofthe default probability curve
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Our Motivation
Develop a model for the for pricing and risk-managing of volatilityand 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:
dS(t)
S(t−)= µ(t)dt+ σ(t)
√V (t)dW s(t)− dNd(t), S(0) = S0
dV (t) = κ(1− V (t))dt+ ε(t)√V (t)dW v(t) + JvdNv(t), V (0) = 1,
dI(t) = σ2(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”
Nd(t) - Poisson process with intensity λ(t)
min{ι : Nd(ι) = 1} is the default time
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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)
Nv(t) - Poisson process with intensity γ
Jv - 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:
V (T ) := EQ[V (T )|V (0) = 1] = 1 +γη
κ
(1− e−Tκ
)(3)
Assuming for moment no default risk, the asset realized variance inthe continuous-time limit becomes:
I(T ) = limN→∞
∑tn∈πN
(ln
S(tn)
S(tn−1)
)2
=∫ T
0σ2(t′)V (t′)dt′ (4)
The expected realized variance:
I(T ) := EQ[I(T )|V (0) = 1] =∫ T
0σ2(t′)V (t′)dt′ (5)
Given the values of mean-reversion parameters κ and jump parametersη and γ, we can extract the term structure of σ2(t) from the fairvariance curve observed from the market data
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Model Interpretation: Jump-to-Default
The probability of survival up to time T :
Q(t, T ) = EQ[ι > T |ι > t] = EQ[e−∫ Tt λ(t′)dt′] (6)
The probability of defaulting up to time T is connected to the inte-grated expected variance:
Qc(t, T ) = EQ[ι ≤ T |ι > t] = 1−Q(t, T ) ≈∫ Tt
(α(t′)+β(t′)V (t′))dt′ (7)
Variation of the default intensity:
< λ(t) >= β2 < V (t) > (8)
Parameter β can be extracted form the time series or from non-linearCDS contracts
The term structure of parameter α(t), is backed-out from the survivalprobabilities implied CDS quotes
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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 forward-start date, its strike if defaulted between the forward-start date andmaturity
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Recovery Assumption II
Realized Variance: I(T ) - the cap level on the realized variance
Typically, I(T ) = 3KV (T ) where KV (T ) is the fair variance observedtoday for swap with maturity T
Now the model implied expected realized variance at time T becomes:
EQ[I(T )] ≈ Q(0, T )∫ T
0σ2(t′)V (t′)dt′+Qc(0, T )I(T ), (9)
”≈” since we ignore the cap on the realized pre-default variance anddependence between V (t) and Q(t, T )
In general, we compute:
EQ[I(T )] = EQ[∫ T
0σ2(t′)V (t′)dt′ | ι > T
]+Qc(0, T )I(T ), (10)
Given the jump-to-default probabilities we use (9) or (10) to fit σ2(t)to the term structure of the fair variance
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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 theimplied variance and realized variance
Increase the variance of the realized variance while give little impacton the asset (terminal) variance
As a result, calibrating the variance jumps to the deep skews is notreasonable - we need to calibrate them to the volatility products
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Convergence of Discretely Sampled Realized Variance to Con-tinuous 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 sampledecreases while its variance increases
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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) = lnS(t)
Kolmogoroff forward equation for the joint transition density functionG(t, T, V, V ′, X,X ′, I, I ′):
GT −(
(µ(T )−1
2σ2(T )V ′)G
)X ′
+(
1
2σ2(T )V ′G
)X ′X ′
+(ρ(T )ε(T )σ2(T )V ′G
)X ′V ′
+(κ(1− V ′)G
)V ′
+(
1
2ε2(T )V ′G
)V ′V ′
−(σ(T )V ′G
)I ′− γ(T )
∫ ∞0
(G(V − Jv)−G)1
ηe−1ηJ
vdJv
− (α(T ) + β(T )V ′)G = 0,
G(t, t, V, V ′X,X ′, I, I ′) = δ(X ′ −X)δ(V ′ − V )δ(I ′ − I),(11)
Here, (X ′, V ′, I ′) are variables (future states of the world), (X,V, I)are initial data
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General Pricing Problem under Model (2) II
Kolmogoroff backward equation for the value function U(t, T, V, V ′X,X ′, I, I ′):
Ut + (µ(t)−1
2σ2(t)V )UX +
1
2σ2(t)V UXX
+ ρ(t)ε(t)σ2(t)V UXV + κ(1− V )UV +1
2ε2(t)V UV V + σ2(t)V UI
+ γ(t)∫ ∞−∞
(U(V + Jv)−G)1
ηe−1ηJ
vdJv − (α(t) + β(t)V )U
= (α(t) + β(t)V )R(t, V, V ′X,X ′, I, I ′) + U2(t, V, V ′X,X ′, I, I ′)
U(T, T, V, V ′X,X ′, I, I ′) = U1(V, V ′X,X ′, I, I ′)
(12)
U1(V, V ′X,X ′, I, I ′) - terminal pay-off function
U2(t, V, V ′X,X ′, I, I ′) - instantaneous reward function
R(t, V, V ′X,X ′, I, I ′) - the recovery value paid upon the default event
Here, (X,V, I) are variables, (X ′, V ′, I ′) are parameters
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Analytical Solution using the Fourier Transform
We apply 3-dimensional generalized Fourier transform to forward PDE(11):
G(t, T, V,Θ, X,Φ, I,Ψ) =∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
e−X′Φ−V ′Θ−I ′ΨGdX ′dV ′dI ′,
(13)where Θ = ΘR + iΘI , Φ = ΦR + iΦI , Ψ = ΨR + iΨI i =
√−1,
ΘR,ΘI ,ΦR,ΦI ,ΨR,ΨI ∈ R
We obtain:
G(t, T, V,Θ, X,Φ, I,Ψ) = e−Φ(X+∫ Tt (r(t′)−d(t′))dt′)−ΨI+A(t,T )+B(t,T )V ,
(14)where functions A(t, T ) and B(t, T ) are computed in closed-form byrecursion
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Marginal Transition Densities and Convergence
Asymptotic convergence rate is important to set-up the bounds forquadrature and FFT inversion methods
We first recall that for the Black-Scholes model with constant V :
GX(t, T, V,Φ, X) ∼ e−12σ
2V0Φ2I , |ΦI | → ±∞
For our model we obtain:
GX(t, T, V,Φ, X) = G(t, T, V,0, X,Φ, I,0) ∼ e−((T−t)κ+σ2V0)(1−ρ2)
ε |ΦI |, |ΦI | → ±∞
GI(t, T, V,Ψ, I) = G(t, T, V,0, X,0, I,Ψ) ∼ e−2(T−t)κ+σ2V0
ε2
√|ΨI |, |ΨI | → ±∞,
GV (t, T, V,Θ) = G(t, T, V,Θ, X,0, I,0) ∼ e−2κε2
ln |ΘI |, |ΘI | → ±∞
x =∫ T0 x(t′)dt′ and ∼ stands for the leading term of the real part
In relative terms, the convergence is fast for GX, moderate for GI,and slow for GV
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Moments
All moments are can be computed numerically by approximating thepartial derivatives:
EQ[Xk(T )V j(T )Il(T )
]= (−1)k+j+l ∂k+j+l
∂ΦkR∂Θj
R∂ΨlR
G(t, T, V,Θ, X,Φ, I,Ψ) |Φ=0,Θ=0,Ψ=0
The survival probability is computed by:
Q(t, T ) = GI(t, T, V,1, I)
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Option Pricing I
The general pricing problem includes computing the expectation ofthe pay-off and reward functions:
U(t,X, I, V ) = EQ[e−∫ Tt (r(t′)+λ(t′))dt′u1(X(T ), V (T ), I(T ))
+∫ Tte−∫ t′t (r(t′′)+λ(t′′))dt′′u2(t′, X(t′), V (t′), I(t′))dt′
],
= U1(t,X, I, V ) + U2(t,X, I, V )(15)
We compute the Fourier-transformed pay-off and reward functions:
u1(Φ,Θ,Ψ) =∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
eΦX ′+ΘV ′+ΨI ′u1(X ′, V ′, I ′)dX ′dV ′dI ′,
u2(t,Φ,Θ,Ψ) =∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
eΦX ′+ΘV ′+ΨI ′u2(t,X ′, V ′, I ′)dX ′dV ′dI ′,
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Option Pricing II
The value of the option is then computed by inversion:
U1(t,X, I, V ) =1
8π3
∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
<[G(t, T, V,Θ, X,Φ, I,Ψ)u1(Φ,Θ,Ψ)
]dΦIdΘIdΨI ,
U2(t,X, I, V ) =1
8π3
∫ Tt
∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
<[G(t, t′, V,Θ, X,Φ, I,Ψ)u2(t′,Φ,Θ,Ψ)
]dΦIdΘIdΨIdt
′
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−∫ Tt r(t′)dt′
π
∫ ∞0<
GX(t, T, V,Φ, X)e(Φ+1) lnK
Φ(Φ + 1)
dΦI ,
where −1 < ΦR < 0
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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 con-sistent way
Introduce the following discretesized operators:LI - the explicit convection vector operator in I directionLX - the implicit convection-diffusion operator in X directionLV - the implicit convection-diffusion operator in V directionCXV - the explicit correlation operatorJV - 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 equationsJump operator is handled by a fast recursive algorithm
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Numerical Solution using Craig-Sneyd ADI method IIAllows to analyze volatility products with general accrual variable:
I(t, T ) =∫ Ttf(t′, V,X, I)dt′ (17)
For example, for conditional up and down variance swap with upperlevel U(t) and lower level L(t) (in continuous time limit):
fup(t, V,X) = 1{eX(t)≥U(t)}σ2(t)V (t), fdown(t, V,X) = 1{eX(t)<L(t)}σ
2(t)V (t)
The implied density for up-variance with U = 1 and down-variancewith L = 1 using the above given model parameters
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Case Study: General Motors data I
GM volatility surface and the term structure of implied default prob-abilities observed in early September, 2007
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Case Study: General Motors data II
For illustration we calibrate two models:
1) SV - the dynamics (2) without jump-to-default
2) SVJD - the dynamics (2) with jump-to-default
The term structure of σ(t) is backed-out from the ATM volatilities,other parameters are kept constant, no volatility jumps
Jump-to-default intensity parameter α is inferred from the term struc-ture of implied probabilities for GM CDS (which is pretty flat), β = 0
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The term structure of σ(t) and model parameters
SV SVJDκ 3.4804 0.0739ε 2.6254 0.3665ρ -0.7330 -0.7874α 0.1035
SVJD model implies:Less variable variance process (some part of the skew is explain bythe jump-to-default)
The decreasing term structure of ATM vols (in the long-term, theimpact of the jump-to-default increases)
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Model Fits. SV vs SVJD
SVJD model generates the deep skew for short-term options
SVJD model explains the skew across all maturities
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Variance Density
In SV model, since the volatility of the variance process is high, themodel implies sizable likelihood of observing small values of the vari-ance
This presents challenges for numerical methods
SVJD model dynamics looks more reasonable
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Annualized Realized Variance Density
In SV model, the realized variance have very heavy right tail
In SVJD model, the peak of the annualized realized variance movesto the left
As a result, in SVJD model a bigger part of the realized variance isexplained by the jump-to-default
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Asset Price Density
In SV model, the asset price density becomes convoluted for long-term maturities - the SV model virtually implies the default event
In SVJD model, the asset price density is stable across maturities
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Model Implied Delta and Gamma of Call Option
In SVJD model, as the spot price grows, the delta converges to onefaster
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Sensitivity to Jump-to-Default Intensity
The sensitivity to the jump-to-default intensity is positive and almostlinear in maturity time
The forward-start call starting at TF = 0.5 has extra exposure to thedefault risk because of the possibility of defaulting up to the optionstart date
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Vega sensitivity for SVJD. Change in the implied volatility sur-face following the shift in V (t) (dV) and the parallel shift in σ(t)(dSigma)
Vega risk can be defined as change in V (t) and as the parallel shiftin the term structure of the ATM volatility σ(t)
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Implied Volatility of the Forward Start Call
IN SVJD model, the sort-term forward implied volatility is high be-cause it reflects the risk of defaulting before the forward start date
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Products on the Realized Variance I
In the SVJD model, the fair variance explained by the diffusive vari-ance decreases in maturity time and a growing part becomes explainedby the jump-to-default risk
Here we use recovery cap equal to one - in SVJD models it is impor-tant to describe the recovery value for variance swaps
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Products on the Realized Variance II
The SVJD model introduces the positive volatility skew for the vari-ance options - the out-of-the-money calls have higher vols
In pure SV model the skew is minimal, so that we need to include thejumps in the variance to model the variance skew
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Conclusions
We have presented a unified approach to price and hedge the volatilityproducts
We have shown that it is important to account for the default risk bymodeling single name equities
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THANK YOU FOR YOUR ATTENTION
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References
Sepp, A. (2008) Pricing Options on Realized Variance in the HestonModel with Jumps in Returns and Volatility, Journal of ComputationalFinance, Vol. 11, No. 4, pp. 33-70http://ssrn.com/abstract=1408005
Sepp, A. (2007) Affine Models in Mathematical Finance: an Analyt-ical Approach, PhD thesis, University of Tartuhttp://math.ut.ee/~spartak/papers/seppthesis.pdf
Sepp, A. (2006) Extended CreditGrades Model with Stochastic Volatil-ity and Jumps, Wilmott Magazine, September, 50-60http://ssrn.com/abstract=1412327
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