Multiscale Stochastic Volatility Models

Jean-Pierre FouqueUniversity of California Santa Barbara

6th World Congress of the Bachelier Finance Society

Toronto, June 25, 2010

Multiscale Stochastic Volatility for Equity,Interest-Rate and Credit Derivatives

J.-P. Fouque, G. Papanicolaou, R. Sircar, K. Sølna

Cambridge University Press. To appear (soon...)

www.pstat.ucsb.edu/faculty/fouque

Price Expansion

P: price of a vanilla European option (to start with)

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

D1 = S∂

∂S(Delta), D2 = S2 ∂2

∂S2(Gamma) ∂σ =

∂

∂σ(V ega) · · ·

vi = vi(τ), payoff independent, τ = time-to-maturity

P0 is typically a constant volatility price → closed-form formula

Black-Scholes in Equity (Vasicek or CIR in Fixed Income, Black-Cox in

Credit, ...)

Where do we get such an expansion?

What do we expect from it?

Wish List

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

• Accuracy: the truncated expansion should be a good

approximation (vi → 0 fast enough)

• Stability: the coefficients v’s should be stable in time

“short-time tight-fit vs. long-time rough fit”

• Should be useful for hedging under physical measure (the

v’s are calibrated under risk-neutral)

• Should lead to practical consistent pricing of exotic

derivatives

Let’s look at calibration first →

Wish List

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

• Accuracy: the truncated expansion should be a good

approximation (vi → 0 fast enough)

• Stability: the coefficients v’s should be stable in time

“short-time tight-fit vs. long-time rough fit”

• Should be useful for hedging under physical measure (the

v’s are calibrated under risk-neutral)

• Should lead to practical consistent pricing of exotic

derivatives

Let’s look at calibration first →

Wish List

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

• Accuracy: the truncated expansion should be a good

approximation (vi → 0 fast enough)

• Stability: the coefficients v’s should be stable in time

“short-time tight-fit vs. long-time rough fit”

• Should lead to practical consistent pricing of

path-dependent derivatives

• Should be useful for hedging under physical measure (the

v’s are calibrated under risk-neutral)

Let’s look at calibration first →

Wish List

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

• Accuracy: the truncated expansion should be a good

approximation (vi → 0 fast enough)

• Stability: the coefficients v’s should be stable in time

“short-time tight-fit vs. long-time rough fit”

• Should lead to practical consistent pricing of

path-dependent derivatives

• Should be useful for hedging under physical measure (the

v’s are calibrated under risk-neutral)

Let’s look at calibration first →

Wish List

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0

+ v4∂2σσP0 + · · ·

• Accuracy: the truncated expansion should be a good

approximation (vi → 0 fast enough)

• Stability: the coefficients v’s should be stable in time

“short-time tight-fit vs. long-time rough fit”

• Should lead to practical consistent pricing of

path-dependent derivatives

• Should be useful for hedging under physical measure (the

v’s are calibrated under risk-neutral)

Let’s look at calibration first →

Calibration on Implied Volatilities

For vanilla European options we have: ∂σP0 = τ σ̄D2P0 so that

P = P0 + v0∂σP0 + v1D1∂σP0 +v2

σ̄τ∂σP0 +

v3

σ̄τD1∂σP0 + · · ·

For Calls, P0 = CBS and by direct computation

P = CBS +

{

v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)}

∂σCBS + · · ·

where d1 =−LM+(r+ 1

2σ̄2)τ

σ̄√

τ, and LM ≡ log(K/S)

Expanding the implied volatility I = σ̄ + I1 + · · · →

P ≡ CBS(σ̄ + I1 + · · ·) = CBS + I1∂σCBS + · · ·

=⇒ I1 = v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)

+ · · ·

Affine in LMMR: I = b + a LMτ + (quartic in LM) + · · ·

where the term structure of the v’s (τ dependence) is important.

Calibration on Implied Volatilities

For vanilla European options we have: ∂σP0 = τ σ̄D2P0 so that

P = P0 + v0∂σP0 + v1D1∂σP0 +v2

σ̄τ∂σP0 +

v3

σ̄τD1∂σP0 + · · ·

For Calls, P0 = CBS and by direct computation

P = CBS +

{

v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)}

∂σCBS + · · ·

where d1 =−LM+(r+ 1

2σ̄2)τ

σ̄√

τ, and LM ≡ log(K/S)

Expanding the implied volatility I = σ̄ + I1 + · · · →

P ≡ CBS(σ̄ + I1 + · · ·) = CBS + I1∂σCBS + · · ·

=⇒ I1 = v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)

+ · · ·

Affine in LMMR: I = b + a LMτ + (quartic in LM) + · · ·

where the term structure of the v’s (τ dependence) is important.

Calibration on Implied Volatilities

For vanilla European options we have: ∂σP0 = τ σ̄D2P0 so that

P = P0 + v0∂σP0 + v1D1∂σP0 +v2

σ̄τ∂σP0 +

v3

σ̄τD1∂σP0 + · · ·

For Calls, P0 = CBS and by direct computation

P = CBS +

{

v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)}

∂σCBS + · · ·

where d1 =−LM+(r+ 1

2σ̄2)τ

σ̄√

τ, and LM ≡ log(K/S)

Expanding the implied volatility I = σ̄ + I1 + · · · →

P ≡ CBS(σ̄ + I1 + · · ·) = CBS + I1∂σCBS + · · ·

=⇒ I1 = v0 +v2

σ̄τ+(

v1 +v3

σ̄τ

)(

1 − d1

σ̄√

τ

)

+ · · ·

Affine in LMMR: I = b + a LMτ + (quartic in LM) + · · ·

where the term structure of the v’s (τ dependence) is important.

Calibration Examples

Goal: fit

I = b + aLM

τ+ (quartic in LM) + · · ·

to the observed implied volatility surface.

We typically fit the parameters a, b, ... by regressing in LMMR

maturity-by-maturity, then we fit their dependence in τ .

We will see that our expansion leads to a, b which are affine in τ .

Some examples −→

−5 0 5

−0.1

0

0.1

0.2

0.3

0.4

0.5

LMMR

Impl

ied

Vol

atili

ty

τ=43 days

−2 0 2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

LMMR

71 days

−1 0 1

0.1

0.15

0.2

0.25

0.3

0.35

LMMR

106 days

−0.5 0 0.5

0.15

0.2

0.25

LMMR

Impl

ied

Vol

atili

ty

τ=197 days

−0.05 0 0.050.18

0.185

0.19

0.195

0.2

LMMR

288 days

−0.2 0 0.2

0.16

0.18

0.2

0.22

0.24

LMMR

379 days

S&P 500 Implied Volatility data on June 5, 2003 and fits to the affine

LMMR approximation for six different maturities.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−0.25

−0.2

−0.15

−0.1

−0.05

τ(yrs)

m0 +

m1 τ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.188

0.189

0.19

0.191

0.192

0.193

0.194

τ(yrs)

b 0 + b

1 τ

S&P 500 Implied Volatility data on June 5, 2003 and fits to the

two-scales asymptotic theory. The bottom (rep. top) figure shows the

linear regression of b (resp. a) with respect to time to maturity τ .

0 50 100 150 200 250 300 350

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

a

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

b

Trading Day Number

Higher Order Expansion

I ∼4∑

j=0

aj(τ) (LM)j+

1

τΦt,

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.150.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Log−Moneyness + 1

Impl

ied

Vol

atili

ty

5 June, 2003: S&P 500 Options, 15 days to maturity

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Log−Moneyness + 1

Impl

ied

Vol

atili

ty

5 June, 2003: S&P 500 Options, 71 days to maturity

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.14

0.16

0.18

0.2

0.22

0.24

0.26

0.285 June, 2003: S&P 500 Options, 197 days to maturity

Log−Moneyness + 1

Impl

ied

Vol

atili

ty

0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.235 June, 2003: S&P 500 Options, 379 days to maturity

Log−Moneyness + 1

Impl

ied

Vol

atili

ty

S&P 500 Implied Volatility data on June 5, 2003 and quartic fits to the

asymptotic theory for four maturities.

0 0.5 1 1.5 20

1

2

3

4

τ (yrs)

a 4

0 0.5 1 1.5 20

2

4

6

8

τ(yrs)

a 30 0.5 1 1.5 2

−1

0

1

2

3

4

5

a 2

0 0.5 1 1.5 2−0.5

−0.4

−0.3

−0.2

−0.1

0

τ (yrs)

a 1

S&P 500 Term-Structure Fit using second order approximation. Data

from June 5, 2003.

Stochastic Volatility Models

Equity for instance.

Under physical measure:

dSt

St= µdt + σtdW

(0)t

σt = f(Yt, Zt, · · ·)dYt = α(Yt)dt + β(Yt)dW

(1)t

dZt = c(Zt)dt + g(Zt)dW(2)t

· · ·

Volatility factors can be differentiated by their time scales

Multiscale Stochastic Volatility Modelsσt = f(Yt,Zt)

• Yt is fast mean-reverting (ergodic on a fast time scale):

dYt =1

εα(Yt)dt +

1√εβ(Yt)dW

(1)t , 0 < ε ≪ 1

• Zt is slowly varying:

dZt = δc(Zt)dt +√

δ g(Zt)dW(2)t , 0 < δ ≪ 1

Separation of time scales: ε << T << 1/δ

1

T

∫ T

0

σ2t dt =

1

T

∫ T

0

f2(Yt, Zt)dt −→⟨f2(·, z)

⟩

ΦY

Local Effective Volatility: σ̄2(z) ≡⟨f2(·, z)

⟩

ΦY

Multiscale Stochastic Volatility Modelsσt = f(Yt,Zt)

• Yt is fast mean-reverting (ergodic on a fast time scale):

dYt =1

εα(Yt)dt +

1√εβ(Yt)dW

(1)t , 0 < ε ≪ 1

• Zt is slowly varying:

dZt = δc(Zt)dt +√

δ g(Zt)dW(2)t , 0 < δ ≪ 1

Separation of time scales: ε << T << 1/δ

(assuming f continuous in z):

1

T

∫ T

0

σ2t dt =

1

T

∫ T

0

f2(Yt, Zt)dt −→⟨f2(·, z)

⟩

ΦY

Local Effective Volatility: σ̄2(z) ≡⟨f2(·, z)

⟩

ΦY

P0 = PBS(σ̄(z))

Multiscale Stochastic Volatility Modelsσt = f(Yt,Zt)

• Yt is fast mean-reverting (ergodic on a fast time scale):

dYt =1

εα(Yt)dt +

1√εβ(Yt)dW

(1)t , 0 < ε ≪ 1

• Zt is slowly varying:

dZt = δc(Zt)dt +√

δ g(Zt)dW(2)t , 0 < δ ≪ 1

Separation of time scales: ε << T << 1/δ

(assuming f continuous in z):

1

T

∫ T

0

σ2t dt =

1

T

∫ T

0

f2(Yt, Zt)dt −→⟨f2(·, z)

⟩

ΦY

Local Effective Volatility: σ̄2(z) ≡⟨f2(·, z)

⟩

ΦY

P0 = PBS(σ̄(z))

Market Prices of Volatility Risk

Under the risk neutral measure IP ⋆ chosen by the market:

dSt = rStdt + f(Yt, Zt)StdW(0)⋆t

dYt =

(1

εα(Yt) −

1√εβ(Yt)Λ(Yt, Zt)

)

dt +1√εβ(Yt)dW

(1)⋆t

dZt =(

δ c(Zt) −√

δ g(Zt)Γ(Yt, Zt))

dt +√

δ g(Zt)dW(2)⋆t

d < W (0)⋆, W (1)⋆ >t = ρ1dt

d < W (0)⋆, W (2)⋆ >t = ρ2dt

Λ and Γ: market prices of volatility risk

Pricing Equation

P ε,δ(t, x, y, z) = IE⋆{

e−r(T−t)h(ST )|St = x, Yt = y, Zt = z}

Feynman–Kac:(

1

εLY +

1√εLρ1,Λ + L +

√δLρ2,Γ + δLZ +

√

δ

εLρ12

)

P ε,δ = 0

P ε,δ(T, x, y, z) = h(x)

with

L = LBS(f(y, z)) =∂

∂t+

1

2f2(y, z)x2 ∂2

∂x2+ r

(

x∂

∂x− ·)

Regular-Singular Perturbations

P ε,δ =∑

i,j

εi/2 δj/2 Pi,j = P0 +√

ε P1,0 +√

δ P0,1 + · · ·

LBS(σ̄(z))P0 = 0, P0(T, x) = h(x) =⇒ P0 = PBS(σ̄(z))

P0 is independent of y and z is a parameter.

bfLBS(σ̄(z))(√

εP1,0

)+ V ε

2 D2PBS + V ε3 D1D2PBS = 0

bfLBS(σ̄(z))(√

δP0,1

)

+ 2(V δ

0 ∂σPBS + V δ1 D1∂σPBS

)= 0

P1,0(T,x) = P0,1(T,x) = 0

Vδ0 and Vε

2 are volatility level adjustments due to Γ and Λ resp.

Vδ1 and Vε

3 are skew parameters proportional to ρ2 and ρ1 resp.

Regular-Singular Perturbations

P ε,δ =∑

i,j

εi/2 δj/2 Pi,j = P0 +√

ε P1,0 +√

δ P0,1 + · · ·

LBS(σ̄(z))P0 = 0, P0(T, x) = h(x) =⇒ P0 = PBS(σ̄(z))

P0 is independent of y and z is a parameter.

LBS(σ̄(z))(√

εP1,0

)+ Vε

2D2PBS + Vε3D1D2PBS = 0

LBS(σ̄(z))(√

δP0,1

)

+ 2(Vδ

0∂σPBS + Vδ1D1∂σPBS

)= 0

P1,0(T,x) = P0,1(T,x) = 0

Vδ0 and Vε

2 are volatility level adjustments due to Γ and Λ resp.

Vδ1 and Vε

3 are skew parameters proportional to ρ2 and ρ1 resp.

Important: these Black-Scholes equations will hold for exotic

options with additional boundary conditions, but with the

same group parameters V ’s

Regular-Singular Perturbations

P ε,δ =∑

i,j

εi/2 δj/2 Pi,j = P0 +√

ε P1,0 +√

δ P0,1 + · · ·

LBS(σ̄(z))P0 = 0, P0(T, x) = h(x) =⇒ P0 = PBS(σ̄(z))

P0 is independent of y and z is a parameter.

LBS(σ̄(z))(√

εP1,0

)+ Vε

2D2PBS + Vε3D1D2PBS = 0

LBS(σ̄(z))(√

δP0,1

)

+ 2(Vδ

0∂σPBS + Vδ1D1∂σPBS

)= 0

P1,0(T,x) = P0,1(T,x) = 0

Vδ0 and Vε

2 are volatility level adjustments due to Γ and Λ resp.

Vδ1 and Vε

3 are skew parameters proportional to ρ2 and ρ1 resp.

Important: these Black-Scholes equations will hold for exotic

options with additional boundary conditions, but with the

same group parameters V ’s

Explicit formulas for Vanilla European Options

Notation: T − t = τ

√εP1,0 = τ (Vε

2D2PBS + Vε3D1D2PBS)

easily checked by using LBSDi = DiLBS

bf√

δP0,1 = τ(V δ

0 ∂σPBS + V δ1 D1∂σPBS

)

easily checked by using ∂PBS = τ σ̄D2PBS and then LBSDi = DiLBS .

• Back to our expansion −→

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0 + · · ·

v0 = τVδ0 , v1 = τVδ

1

v2 = τVε2 , v3 = τVε

3

In terms of calibration to implied volatilities:

Explicit formulas for Vanilla European Options

Notation: T − t = τ√

εP1,0 = τ (Vε2D2PBS + Vε

3D1D2PBS)

easily checked by using LBSDi = DiLBS

√δP0,1 = τ

(Vδ

0∂σPBS + Vδ1D1∂σPBS

)

easily checked by using first ∂PBS = τ σ̄D2PBS and then

LBSDi = DiLBS

• Back to our expansion −→

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0 + · · ·

v0 = τVδ0 , v1 = τVδ

1

v2 = τVε2 , v3 = τVε

3

In terms of calibration to implied volatilities:

Explicit formulas for Vanilla European Options

Notation: T − t = τ

√εP1,0 = τ (Vε

2D2PBS + Vε3D1D2PBS)

easily checked by using LBSDi = DiLBS

√δP0,1 = τ

(Vδ

0∂σPBS + Vδ1D1∂σPBS

)

easily checked by using ∂PBS = τ σ̄D2PBS and then LBSDi = DiLBS .

• Back to our expansion −→

P = P0 + v0∂σP0 + v1D1∂σP0 + v2D2P0 + v3D1D2P0 + · · ·

v0 = τVδ0 , v1 = τVδ

1

v2 = τVε2 , v3 = τVε

3

In terms of calibration to implied volatilities −→

Implied Volatility Calibration Formulas

σ̄ +V2

σ̄+

V3

2σ̄(1 − 2r

σ̄2) + τ

(

V0 +V1

2(1 − 2r

σ̄2))

︸ ︷︷ ︸

+(

V3

σ̄3+ τ

V1

σ̄2

)

︸ ︷︷ ︸

LMMR

intercept b slope a

Either

• one estimates σ̄ from historical data (preferred for hedging

where V0 and V2 do not appear), and then fitting

maturity-by-maturity and regressing in τ , one gets:

1. V1 and V3 from the slope a

2. V0 and V2 from the intercept b

• or one uses the adjusted effective volatility σ⋆ ≡√

σ̄2 + 2V2

calibrated from option data , along with V0, V1, and V3

σ⋆ +

V3

2σ⋆(1 − 2r

σ̄⋆2) + τ

(

V0 +V1

2(1 − 2r

σ⋆2))

+(

V3

σ⋆3+ τ

V1

σ⋆2

)

LMMR

Implied Volatility Calibration Formulas

σ̄ +V2

σ̄+

V3

2σ̄(1 − 2r

σ̄2) + τ

(

V0 +V1

2(1 − 2r

σ̄2))

︸ ︷︷ ︸

+(

V3

σ̄3+ τ

V1

σ̄2

)

︸ ︷︷ ︸

LMMR

intercept b slope a

Either

• one estimates σ̄ from historical data (preferred for hedging

where V0 and V2 do not appear), and then fitting

maturity-by-maturity and regressing in τ , one gets:

1. V1 and V3 from the slope a

2. V0 and V2 from the intercept b

• or one uses the adjusted effective volatility σ⋆ ≡√

σ̄2 + 2V2

calibrated from option data , along with V0, V1, and V3 (preferred

for pricing):

σ⋆ +

V3

2σ⋆(1 − 2r

σ̄⋆2) + τ

(

V0 +V1

2(1 − 2r

σ⋆2))

+(

V3

σ⋆3+ τ

V1

σ⋆2

)

LMMR

Back to the Wish List: Accuracy

If the payoff function h is smooth:

P ε,δ

=(

P0 +√

εP1,0 + εP2,0 + ε3/2P3,0

)

+√

δ(P0,1 +

√εP1,1 + εP2,1

)+ Rε,δ

=(

P0 +√

εP1,0 +√

δP0,1

)

+ O(ε + δ) + Rε,δ

then the residual Rε,δ satisfies

Lε,δRε,δ = O(ε + δ)

Rε,δ(T ) = O(ε + δ)

and therefore Rε,δ = O(ε + δ).

If h is non-smooth (call option in particular), then use a careful

regularization.

Path-Dependent Derivatives (Barrier, Asian,...)

• Calibrate σ⋆, V0, V1 and V3 on the implied volatility surface

• Solve the corresponding problem with constant volatility σ⋆

=⇒ P0 = PBS(σ⋆)

• Use V0, V1 and V3 to compute the source

2 (V0∂σP⋆BS + V1D1∂σP⋆

BS) + V3D1D2P⋆BS

• Get the correction by solving the SAME PROBLEM

with zero boundary conditions and the source.

American Options

• Calibrate σ⋆, V0, V1 and V3 on the implied volatility surface

• Solve the corresponding problem with constant volatility σ⋆

=⇒ P⋆ and the free boundary x⋆(t)

• Use V0, V1 and V3 to compute the source

2 (V0∂σP⋆BS + V1D1∂σP⋆

BS) + V3D1D2P⋆BS

• Get the correction by solving the corresponding problem with

fixed boundary x⋆(t), zero boundary conditions and the

source.

Cost of the Black-Scholes Hedging Strategy

PBS(T, ST ) = h(ST )

PBS(t, St) = atSt + btert , at = ∂xPBS

Infinitesimal cost:

dPBS(t, St)− (atdSt + rbtertdt)

︸ ︷︷ ︸= 1

2

(f2(Yt, Zt) − σ2

)D2PBS(t, St)dt

self-financing part

Cumulative financing cost:

EBS(t) =1

2

∫ t

0

e−rs(f2(Ys,Zs) − σ2

)D2PBS(s,Ss)ds

Choice of σ ?

Choice of σ ?

Since Yt is fast mean-reverting (ε << 1), integrals like∫ t

0

(f2(Ys,Zs) − σ2

)Ψsds will be small with ε if

σ2 = σ̄2(z) = 〈f2(·, z)〉Φ(Y)

Therefore two choices:

• σ2 = σ̄2(Zt) and PBS = PBS(t, St; σ̄(Zt)), in which case σ̄(Zt)

needs to be estimated continuously (and dPBS revisited)

• σ2 = σ̄2(Z0) and PBS = PBS(t, St; σ̄(Z0)) with

f2(Ys,Zs) − σ2 =(f2(Ys,Zs) − σ̄2(Zt)

)+(σ̄2(Zt) − σ̄2(Z0)

)

in which case parameters are frozen at time zero, an additional

cost of order√

δ comes from the second term (offset in practice

by re-calibration at√

δ-frequency).

Corrected Hedging Strategy

A careful analysis of the cost shows

E0(t) =1

2

∫ t

0

e−rs(f2(Ys,Zs) − σ̄2(Zt)

)D2PBS(s,Ss)ds

=√

ε (Bεt + Mε

t) + O(ε + δ),

where Mεt is a martingale, and

Bεt = −ρ1

2

∫ t

0

e−rsβ(Ys)∂φ

∂yf(Ys,Zs)D1D2PBS(s,Ss)ds

is a bounded variation bias which can be compensated by using

the corrected hedging ratio at given by

∂xPBS + (T − t)V3∂xD1D2PBS + (T − t)V1∂xD1∂σPBS

The last term compensates for the bias generated by σ̄2(Zt) − σ̄2(Z0)

Examples of other:

• Models

• Regimes

• Applications

A Model with Volatility Time-Scale of Order One

In the model σt = f(Yt,Zt), if one wants to:

• keep Y fast mean-reverting

• let Z be on a time scale comparable to maturity (or add one

such factor)

• keep the computational tractability

then, one needs to make sure that the SV model σ̄2(Zt) is tractable.

An interesting choice is the Heston model:

“A Fast Mean-Reverting Correction to Heston Stochastic Volatility

Model” with Matthew Lorig (PhD student, UCSB), where we

develop this idea.

An example of fit −→

A Model with Volatility Time-Scale of Order One

In the model σt = f(Yt,Zt), if one wants to:

• keep Y fast mean-reverting

• let Z be on a time scale comparable to maturity (or add one

such factor)

• keep the computational tractability

then, one needs to make sure that the SV model σ̄2(Zt) is tractable.

An interesting choice is the Heston model:

“A Fast Mean-Reverting Correction to Heston Stochastic Volatility

Model” with Matthew Lorig (PhD student, UCSB), where we

develop this idea.

An example of fit −→

−0.1 −0.05 0 0.05 0.1 0.150.1

0.11

0.12

0.13

0.14

0.15

0.16Days to Maturity = 65

log(K/x)

Impl

ied

Vol

atili

ty

Market DataHeston FitMultiscale Fit

SPX Implied Volatilities from May 17, 2006

Fast Mean-Reverting SV and Short Maturities

If the time scale of the fast mean-reverting factor Y is ε << 1, and

if the maturity of interest is small but still large compared with ε,

then, one can consider the regime

ε << T ∼√

ε << 1

It involves a non-trivial mixture of Large Deviation (short

maturity) and Homogenization (fast mean reverting coefficient):

“Short maturity asymptotics for a fast mean reverting Heston

stochastic volatility model” with Jin Feng and Martin Forde

(SIAM Journal on Financial Mathematics, Vol. 1, 2010).

Interestingly, in this regime and for this model, we derive explicit

formulas for the limiting implied volatility which looks like −→

Three parameters which control the implied volatility skew’s

level (θ), slope (ρ) and convexity (ν/κ).

A Cool Application to Forward-Looking Betas

Discrete time CAPM model:

Ra − Rf = βa(RM − Rf ) + ǫa

Christoffersen, Jacobs, and Vainberg (2008, McGill University):

βa =(

SKEWa

SKEWM

) 1

3(

V ARa

V ARM

) 1

2

where VAR and SKEW are variance and risk-neutral skewness

With Eli Kollman (PhD 2009, UCSB), we propose in

“Calibration of Stock Betas from Skews of Implied Volatilities”

(Applied Mathematical Finance, 2010):

β̂a =

(

Va,ǫ3

VM,ǫ3

)1/3

=

(aa,ǫ

aM,ǫ

)1/3(ba∗

bM∗

)

A Cool Application to Forward-Looking Betas

Discrete time CAPM model:

Ra − Rf = βa(RM − Rf ) + ǫa

Christoffersen, Jacobs, and Vainberg (2008, McGill University):

βa =(

SKEWa

SKEWM

) 1

3(

V ARa

V ARM

) 1

2

where VAR and SKEW are variance and risk-neutral skewness

With Eli Kollman (PhD 2009, UCSB), we propose in

“Calibration of Stock Betas from Skews of Implied Volatilities”

(Applied Mathematical Finance, 2010):

β̂a =

(

Va,ǫ3

VM,ǫ3

)1/3

=

(aa,ǫ

aM,ǫ

)1/3(ba∗

bM∗

)

LMMR fits (2/18/2009): S&P500 and Amgen, beta estimate is 1.03

−1 −0.5 0 0.5 10.3

0.35

0.4

0.45

0.5

0.55

LMMR

Impl

ied

Vol

S&P 500

−1 −0.5 0 0.5 10.3

0.35

0.4

0.45

0.5

0.55

LMMR

Impl

ied

Vol

AMGN

LMMR fits (2/19/2009): S&P500 and Goldman Sachs, beta estimate is 2.28

−1 −0.5 0 0.5 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

LMMR

Impl

ied

Vol

S&P 500

−1 −0.5 0 0.5 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

LMMR

Impl

ied

Vol

GS

THANKS FOR YOUR ATTENTION