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

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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

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

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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

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

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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

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