Stochastic Volatility Modeling

Jean-Pierre Fouque

University of California Santa Barbara

2008 Daiwa Lecture Series

July 29 - August 1, 2008

Kyoto University, Kyoto

1

References:

Derivatives in Financial Marketswith Stochastic VolatilityCambridge University Press, 2000

Stochastic Volatility AsymptoticsSIAM Journal on Multiscale Modeling and Simulation, 2(1), 2003

Collaborators:

G. Papanicolaou (Stanford), R. Sircar (Princeton), K. Sølna (UCI)

http://www.pstat.ucsb.edu/faculty/fouque

2

What is Volatility?

Several notions of volatilityModel dependent or not, Data dependent or not

• Realized Volatility (historical data)

• Model Volatility:

– Local Volatility

– Stochastic Volatility

• Implied Volatility (option data)

3

Realized Volatility

t0 < t1 < · · · < tN = t (present time)

1

t − t0

∫ t

t0

σ2s ds ∼ 1

N

N∑

i=1

(

log Sti− log Sti−1

)2

ti − ti−1

depends on the choice of t0 and on the number of increments N

(assuming ti − ti−1 constant).

More details:

Zhang, L., Mykland, P.A., and Ait-Sahalia, Y. (2005). A tale of

two time scales: Determining integrated volatility with noisy

high-frequency data, J. Amer. Statist. Assoc. 100, 1394-1411.

http://galton.uchicago.edu/˜mykland/publ.html

4

Volatility Models

dSt = St (µdt + σtdWt)

• Local Volatility:

σt = σ(t, St)

where σ(t, x) is a deterministic function.

• Stochastic Volatility:

σt = f(Yt)

where Yt contains an additional source of randomness.

5

Implied Volatility

I(t, T, K) = σimplied(t, T, K)

where σimplied(t, T, K) is uniquely defined by inverting

Black-Scholes formula:

Cobserved(t, T, K) = CBS (t, St; T, K; σimplied(t, T, K))

given the call-option data.

t is present time, T is the option maturity date, and K is the strike

price.

6

0.8 0.85 0.9 0.95 1 1.05 1.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Moneyness K/x

Impl

ied

Vol

atili

ty

Historical Volatility

9 Feb, 2000

Excess kurtosis

Skew

Figure 1: S&P 500 Implied Volatility Curve as a function of moneyness

from S&P 500 index options on February 9, 2000. The current index value

is x = 1411.71 and the options have over two months to maturity. This is

typically described as a downward sloping skew.

7

“Parametrization” of the

Implied Volatility Surface I(t ; T, K)

REQUIRED QUALITIES

• Universal Parsimonous Parameters: Model Independence

• Stability in Time: Predictive Power

• Easy Calibration: Practical Implementation

• Compatibility with Price Dynamics: Applicability to

Pricing other Derivatives and Hedging

8

At least three approaches:

• Local Volatility Models: σt = σ(t, St)

+’s: market is complete (no additional randomness), Dupire

formula

σ2(T, K) = 2∂C∂T

+ rK ∂C∂K

K2 ∂2C∂K2

-’s: stability of calibration

• Implied Volatility Surface Models: dIt(T, K) = · · ·+’s: predictive power

-’s: no-arbitrage conditions not easy. Which underlying?

• Stochastic Volatility Models: σt = f(Yt)

9

Stochastic Volatility Framework

WHY?

• Distributions of returns are not log-normal

• Smile (Skew) effect observed in implied volatilities

HOW?

dSt = µStdt + σtStdWt

with, for instance:

σt = f(Yt)

dYt = α(m − Yt)dt + ν√

2α dW(1)t

d〈W, W (1)〉t = ρdt

10

The Popular Heston Model

dSt = µStdt + σtStdW(1)t

σt =√

Yt

dYt = α(m − Yt)dt + ν√

2αYt dW(2)t

d〈W (1), W(2)t 〉t = ρdt

Yt is a CIR (Cox-Ingersoll-Ross) process.

The condition m ≥ ν2 ensures that the process Yt stays strictly

positive at all time.

11

Mean-Reverting Stochastic Volatility Models

dXt = Xt (µdt + σtdWt)

σt = f(Yt)

For instance: 0 < σ1 ≤ f(y) ≤ σ2 for every y

dYt = α(m − Yt)dt + β(· · ·)dZt

Brownian motion Z correlated to W :

Zt = ρWt +√

1 − ρ2Zt , |ρ| < 1

so that

d〈W, Z〉t = ρdt

12

Pricing under Stochastic Volatility

Risk-neutral probability chosen by the market: IP ⋆(γ)

dXt = rXtdt + f(Yt)XtdW ⋆t

dYt =

[

α(m − Yt) − β

(

ρ(µ − r)

f (Yt)+ γ√

1 − ρ2

)]

dt + βdZ⋆t

Z⋆t = ρW ⋆

t +√

1 − ρ2 Z⋆t

Market price of volatility risk: γ = γ(y)

Pt = IE⋆(γ){e−r(T−t)h(XT )|Ft}

Markovian case:

P (t, x, y) = IE⋆(γ){e−r(T−t)h(XT )|Xt = x, Yt = y}

but y (or f(y)) is not directly observable!

13

Stochastic Volatility Pricing PDE

∂P

∂t+

1

2f(y)2x2 ∂2P

∂x2+ ρβxf(y)

∂2P

∂x∂y+

1

2β2 ∂2P

∂y2

+r

(

x∂P

∂x− P

)

+ α(m − y)∂P

∂y− βΛ

∂P

∂y= 0

where

Λ = ρ(µ − r)

f (y)+ γ√

1 − ρ2

Terminal condition: P (T, x, y) = h(x)

No perfect hedge!

14

Summary of the stochastic volatility approach

Positive aspects:

• More realistic returns distributions (fat tails and asymmetry )

• Smile effect with skew contolled by ρ

Difficulties:

• Volatility not directly observed, parameter estimation difficult

• No canonical model. Relevance of explicit formulas?

• Incomplete markets, no perfect hedge

• Volatility risk premium to be estimated from option prices

• Numerical difficulties due to higher dimension

15

Driving process Yt and intrinsic time scale

Markovian case: infinitesimal generator L• Two-state Markov chain:

L = α

−1 1

1 −1

• Pure jump Markov process

Lg(y) = α

∫

(g(z) − g(y))p(z)dz , p(z) =1

21(−1,1)(z)

• Diffusion: Ornstein-Uhlenbeck Process

αLOU = α

(

(m − y)∂

∂y+ ν2 ∂2

∂2y

)

16

Invariant probability distribution

L⋆Φ = 0

• Two-state Markov chain: linear system

Φ = {1

2,1

2}

• Pure jump Markov process: integral equation

Φ(y) =1

21[−1,1](y)

• OU diffusion process: differential equation

Φ(y) =1√2π ν

exp

(

− (y − m)2

2ν2

)

, ν2 = β2/2α

17

Convergence to Equilibrium

• Equilibrium:

〈g〉 =

∫

g(y)Φ(y)dy

• Exponential convergence to equilibrium with rate α:

|IE{g(Yt)|Y0 = y} − 〈g〉| ≤ Ce−αt

• Exponential decorrelation with rate α:

|IEΦ{g(Ys)h(Yt)} − 〈g〉〈h〉| ≤ Ce−α|t−s|

• Intrinsic time scale: 1/α

18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.15

0.2

0.25

0.3

0.35

α = 1

α = 20

Time t

α = 100

Figure 2: Simulated paths of σt = f(Yt), with (Yt) a two-state Markov

chain, showing the relation between burstiness and the mean holding time

1/α.

19

Ergodic Theorem

limt→∞

1

t

∫ t

0

g(Ys)ds = 〈g〉 almost surely (a.s.)

or t > 0 fixed and α → +∞:

1

t

∫ t

0

g(Ys) ≈ 〈g〉

In particular for α large:

σ2 =1

T − t

∫ T

t

f(Ys)2ds ≈ 〈f2〉 = σ2

LHS random RHS deterministic

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

α = 1

α = 20

α = 100

Time t

Figure 3: Simulated paths of σt = f(Yt), with (Yt) a pure jump Markov

process taking values in (0.05, 0.4), for increasing mean-reversion rates α.

The mean level 〈f〉 = 0.225.

21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

α = 1

α = 20

α = 100

Time t

Figure 4: Simulated paths of σt = f(Yt), with (Yt) a mean-reverting

OU process and f(y) = 0.35(arctan y + π/2)/π + 0.05, chosen so that

σt ∈ (0.05, 0.4). Notice how the mean-reversion rates α correspond to the

duration of the bursts.

22

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−1.5

−1

−0.5

0

0.5

1

1.5

2

Time

Ret

urn

s

S&P 500 Returns Process, 1996

Figure 5: 1996 S&P 500 returnscomputed from half-hourly data.

23

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1

α = 1

Vol

atility

Ret

urn

s

α = 100

Time (yrs.)Time (yrs.)

Figure 6: Simulated volatility and corresponding returns paths for small

and large rates of mean-reversion for the jump volatility model.

24

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4−0.5

0

0.5

1

α = 1 α = 200

Figure 7: Simulated volatility and corresponding returns paths for small

and large rates of mean-reversion for the OU model with f(y) = ey.

25

Exponential decorrelation – Variogram

Variogram: IE[f(Yt+lag) − f(Yt)]

2 ≈ 2 σ2(1 − e−αlag)

• Ys : Ornstein-Uhlenbeck driving process.

• f(x) = exp(x)

0 1 2 3 4 5 6 7 8 90

2

4

6

8x 10

−4 VARIOGRAM VOLATILITY

TRADING DAY LAG

26

Exponential decorrelation for returns

Variograms for raw and median filtered returns:

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3VARIOGRAM LOG RETURNS

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5VARIOGRAM LOG RETURNS (MEDIAN FILT)

TRADING DAY LAG

27

Estimation

• Stochastic volatility model:

dXt = µXtdt + f(Yt)XtdWt

• Volatility hidden, only see returns:

dXt

Xt

− µdt = f(Yt)dWt.

• The de-meaned return process:

Dti=

1√∆t

(

∆Xti

Xti

− mean

)

∼ f(Yti)∆Wti

= f(Yti)ǫti

where ǫtiis an i.i.d “white noise” sequence.

28

Variogram of log-returns

• Log-returns:

Lti= log |Dti

| = log(f(Yti)) + log |ǫti

| = f(Yti) + ǫti

.

• Variogram is like an exponential:

1

N

N∑

i=1

[

Lti+lag − Lti

]2

≈ c1(1 − e−αlag) + c2.

29

Median filtered S&P 500 variogram

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8S&P500 variogram (dotted) and fitted variogram (solid)

days

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8Variogram for synthetics (dotted) and fitted variogram (solid)

days

Notice the day effect removed by the fitted exponential.

The curvature determines 1/α : 1.5 ± 0.4 trading days

annualized α ≈ 130 − 230: large

30

Spectrum of log-returns

Energy spectrum of log returns:

→∫

IE[Lt+tLt] cos(ωt) dt ≈ d1 + d2α

α2+ω2 + d3δ(ω − ω1) :

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

S&P500 and Lorentzian spectra

Simulated Lorentzian spectra

1/day

1/day

31

Volatility Time Scales

Rescale the time of a diffusion process Y 1t :

Yαt

= Y1

αt

α large −→ “speeding up” the process Y 1t

α small −→ “slowing down” the process Y 1t

1/α is the characteristic time scale of the process Y αt .

Averaged square volatility:

σ2(0, T ) =1

T

∫ T

0

f2(Y αt )dt

32

Slowing Down the Time

dY 1t = c(Y 1

t )dt + g(Y 1t )dWt , Y 1

0 = y

dY αt = c(Y α

t )d(αt) + g(Y αt )dWαt

D= αc(Y α

t )dt +√

α g(Y αt )dWt , Y α

0 = y

Assuming that f is continuous,

σ2(0, T ) =1

T

∫ T

0

f2(Y αt )dt −→ f2(y) as α → 0

Volatility is “frozen” at its starting level f(y)

33

Rate of Convergence in the Slow Scale Limit

σ2(0, T ) − f2(y)

=1

T

∫ T

0

[

f2

(

y + α

∫ t

0

c(Y αs )ds +

√α

∫ t

0

g(Y αs )dWs

)

− f2(y)

]

dt

= 2√

α f(y)f ′(y)g(y)

(

1

T

∫ T

0

Wtdt

)

+ O(α)

(smoothness of f and g is assumed)

Risk neutral −→ a market price of volatility risk

−→ term: −√α g(Y α

s )Λ(Y αs )ds in the drift of Y α

t

−→ additional term: −2√

α f(y)f ′(y)g(y)Λ(y) (T/2)

Correlation with the BM driving the underlying will also come

into play at the order√

α

34

Speeding Up the Time

σ2(0, T ) =1

T

∫ T

0

f2(Yαt)dt =

1

αT

∫ αT

0

f2(Y 1s )ds , α → +∞

=1

T ′

∫ T ′

0

f2(Y 1s )ds , T′ ≡ αT → +∞

Assuming that Y 1 is ergodic with invariant distribution Φ

then:

limT ′→+∞

1

T ′

∫ T ′

0

f2(Y 1s )ds =

∫

f2(y)Φ(dy) ≡ 〈f2〉Φ.

Effective volatility: σ ≡√

〈f2〉Φ

limα→∞

σ2(0,T) = σ2

35

The Averaging Principle

Fast “oscillating” integral:

σ2(0,T) − σ2 =1

T

∫ T

0

(

f2(Y αs ) − σ2

)

ds, α → +∞

Observe that f2(Y αs ) does not converge for fixed s.

Introduce the Poisson equation:

LY1φ(y) = f2(y) − σ2

so that

σ2(0,T) − σ2 =1

T

∫ T

0

LY 1φ(Y αs ) ds

36

The Averaging Principle (continued)

Using Ito’s formula:

dφ(Y αs ) = LY αφ(Y α

s )ds +√

α φ′(Y αs )g(Y α

s )dWs

= αLY 1φ(Y αs )ds +

√α φ′(Y α

s )g(Y αs )dWs

Therefore

σ2(0,T) − σ2 =1

T

∫ T

0

LY 1φ(Y αs ) ds

=1

αT

∫ T

0

dφ(Y αs ) − 1√

αT

∫ T

0

φ′(Y αs )g(Y α

s )dWs

= − 1√α

(

1

T

∫ T

0

φ′(Y αs )g(Y α

s )dWs

)

+ O(

1

α

)

Risk neutral −→ −√α g(Y α

s )Λ(Y αs )ds in the drift of Y α

t

−→ additional term: 1√α

(

1T

∫ T

0g(Y α

s )Λ(Y αs )ds

)

37

Multiscale Stochastic Volatility Models

The spot volatility is a function of two factors:

σt = 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

(y, z) will denote the initial point for (Y, Z)

f is continuous with respect to z

Local Effective Volatility:

σ2(z) ≡⟨

f2(·, z)⟩

ΦY

38

ε << T << 1/δ

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

dXt = rXtdt + f(Yt, Zt)XtdW(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

39

Pricing Equation

P ε,δ(t, x, y, z) = IE⋆{

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

(

1

εL0 +

1√εL1 + L2 +

√δM1 + δM2 +

√

δ

εM3

)

P ε,δ = 0

P ε,δ(T, x, y, z) = h(x)

L0 = α∂

∂y+

1

2β2 ∂2

∂y2M1 = g

(

ρ2fx∂2

∂x∂z− Γ

∂

∂z

)

L1 = β

(

ρ1fx∂2

∂x∂y− Λ

∂

∂y

)

M2 = c∂

∂z+

g2

2

∂2

∂z2

L2 =∂

∂t+

1

2f2x2 ∂2

∂x2+ r

(

x∂

∂x− ·)

M3 = ρ12βg∂2

∂y∂z

40