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Volatility derivatives and default risk
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Transcript of Volatility derivatives and default risk
Volatility derivatives and default risk
ARTUR SEPP
Merrill Lynch
Quant Congress London
November 14-15, 2007
1
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
2
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
3
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
4
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)
5
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
6
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
7
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
8
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
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)
Nv(t) - Poisson process with intensity γ
Jv - the exponential jump with mean η
ε(t) - the vol-vol parameter
κ - the mean-reversion
10
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
11
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
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 forward-start date, its strike if defaulted between the forward-start date andmaturity
13
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
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 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
15
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
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) = 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
17
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
18
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
19
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
20
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)
21
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 ′,
22
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
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 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
24
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
25
Case Study: General Motors data I
GM volatility surface and the term structure of implied default prob-abilities observed in early September, 2007
26
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
27
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)
28
Model Fits. SV vs SVJD
SVJD model generates the deep skew for short-term options
SVJD model explains the skew across all maturities
29
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
30
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
31
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
32
Model Implied Delta and Gamma of Call Option
In SVJD model, as the spot price grows, the delta converges to onefaster
33
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
34
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)
35
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
36
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
37
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
38
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
39
THANK YOU FOR YOUR ATTENTION
40
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
41