PART II:

Stochastic Volatility Modeling

Jean-Pierre Fouque

University of California Santa Barbara

Special Semester on Stochastics with Emphasis on Finance

Tutorial

September 5, 2008

RICAM, Linz, Austria

1

References:

Derivatives in Financial Markets

with Stochastic Volatility

Cambridge University Press, 2000

Stochastic Volatility Asymptotics

SIAM 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 volatility

Model 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

Fast Mean-Reverting Stochastic Volatility

Asymptotic Analysis

16

Model in the risk-neutral world IP ⋆(γ)in terms of ε = 1/α

Set β = ν√

2√ε

so that ν2 = β2/2α

dXεt = rXε

t dt + f(Y εt )Xε

t dW ⋆t

dY εt =

[1

ε(m − Y ε

t ) − ν√

2√ε

Λ(Y εt )

]dt +

ν√

2√ε

dZ⋆t

Market price of risks:

Λ(y) = ρ(µ − r)

f (y)+ γ(y)

√1 − ρ2

Skew:

Z⋆t = ρW ⋆

t +√

1 − ρ2Z⋆t , |ρ| < 1

17

Prices and Pricing PDE’s

P ε(t, x, y) = IE⋆(γ){

e−r(T−t)h(XεT )|Xε

t = x, Y εt = y

}

∂P ε

∂t+

1

2f(y)2x2 ∂2P ε

∂x2+

ρν√

2√ε

xf(y)∂2P ε

∂x∂y+

ν2

ε

∂2P ε

∂y2

+r

(x

∂P ε

∂x− P ε

)+

1

ε(m − y)

∂P ε

∂y− ν

√2√ε

Λ(y)∂P ε

∂y= 0

to be solved for t < T with the terminal condition

P ε(T, x, y) = h(x)

18

Operator Notation

(1

εL0 +

1√εL1 + L2

)P ε = 0

with

L0 = ν2 ∂2

∂y2+ (m − y)

∂

∂y= LOU

L1 =√

2ρνxf(y)∂2

∂x∂y−

√2νΛ(y)

∂

∂y

L2 =∂

∂t+

1

2f(y)2x2 ∂2

∂x2+ r

(x

∂

∂x− ·

)= LBS(f(y))

19

Formal Expansion

Expand:

P ε = P0 +√

εP1 + εP2 + ε√

εP3 + · · ·Compute:

(1

εL0 +

1√εL1 + L2

)(P0 +

√εP1 + εP2 + ε

√εP3 + · · ·

)= 0

Group the terms by powers of ε:

1

εL0P0 +

1√ε

(L0P1 + L1P0)

+ (L0P2 + L1P1 + L2P0)

+√

ε (L0P3 + L1P2 + L2P1)

+ · · · = 0

20

Diverging terms

• Order 1/ε:

L0P0 = 0

L0 = LOU , acting on y =⇒ P0 = P0(t, x)

with P0(T, x) = h(x)

• Order 1/√

ε:

L0P1 + L1P0 = 0

L1 takes derivatives w.r.t. y =⇒ L1P0 = 0

=⇒ L0P1 = 0

As for P0 : P1 = P1(t, x) with P1(T, x) = 0

• Important observation:

P0 +√

εP1 does not depend on y

21

Zero Order Term

L0P2 + (L1P1 = 0) + L2P0 = 0

Poisson equation in P2 with respect to L0 and the variable y.

Solution:

P2 = (−L0)−1(L2P0)

Only if L2P0 is centered

with respect to the

invariant distribution of Y .

22

Poisson Equations

L0χ + g = 0

Expectations w.r.t. the invariant distribution of the OU process:

〈g〉 = −〈L0χ〉 = −∫

(L0χ(y))Φ(y)dy =

∫χ(y)(L⋆

0Φ(y))dy = 0

limt→+∞

IE {g(Yt)|Y0 = y} = 〈g〉 = 0 (exponentially fast)

χ(y) =

∫ +∞

0

IE {g(Yt)|Y0 = y} dt

checked by applying L0

23

Leading Order Term

Centering:

〈L2P0〉 = 〈L2〉P0 = 0

〈L2〉 =

⟨∂

∂t+

1

2f(y)2x2 ∂2

∂x2+ r

(x

∂

∂x− ·

)⟩

=∂

∂t+

1

2〈f2〉x2 ∂2

∂x2+ r

(x

∂

∂x− ·

)

Effective volatility: σ2 =⟨f2

⟩

The zero order term P0(t, x) is the solution of the

Black-Scholes equation

LBS(σ)P0 = 0

with the terminal condition P0(T, x) = h(x)

24

Back to P2(t, x, y)

The centering condition 〈L2P0〉 = 0 being satisfied:

L2P0 = L2P0 − 〈L2P0〉 =1

2

(f(y)2 − σ2

)x2 ∂2P0

∂x2

=1

2L0φ(y)x2 ∂2P0

∂x2

for φ a solution of the Poisson equation:

L0φ = f(y)2 − 〈f2〉

Then

P2(t, x, y) = −L−10 (L2P0) = −1

2(φ(y) + c(t, x))x2 ∂2P0

∂x2

25

Terms of order√

ε

Poisson equation in P3:

L0P3 + L1P2 + L2P1 = 0

Centering condition:

〈L1P2 + L2P1〉 = 0

Equation for P1:

〈L2P1〉 = −〈L1P2〉 =1

2

⟨L1

[(φ(y) + c(t, x))x2 ∂2P0

∂x2

]⟩

P1 independent of y and L1 takes derivatives w.r.t. y

=⇒ LBS(σ)P1 =1

2〈L1φ(y)〉

[x2 ∂2P0

∂x2

]

with P1(T, x) = 0

26

The correction P ε

1(t, x) =

√εP1(t, x)

LBS(σ)P ε1 − ν

√2ε

2

⟨(ρxf(y)

∂2

∂x∂y− Λ(y)

∂

∂y

)φ(y)

⟩ [x2 ∂2P0

∂x2

]= 0

LBS(σ)P ε1 +

(V ε

2 x2 ∂2PBS

∂x2+ V ε

3 x∂

∂x

(x2 ∂2PBS

∂x2

))= 0

BS equation with source and zero terminal condition

with the two small parameters V ε2 and V ε

3 given by:

V ε2 =

ν√2α

〈Λφ′〉

V ε3 =

−ρν√2α

〈fφ′〉

Recall that α = 1/ε

27

Explicit Formula for the Corrected Price

P ε1 = (T − t)

(V ε

2 x2 ∂2PBS

∂x2+ V ε

3 x∂

∂x

(x2 ∂2PBS

∂x2

))

where V ε2 and V ε

3 are small numbers of order√

ε.

The corrected price is given explicitly by

P0 + (T − t)

(V ε

2 x2 ∂2PBS

∂x2+ V ε

3 x∂

∂x

(x2 ∂2PBS

∂x2

))

where P0 is the Black-Scholes price with constant volatility σ

28

Comments

• The small constants V ε2 and V ε

3 are complex functions of the

original model parameters (µ, m, ν, ρ, α; f) and γ

• Only (σ, V ε2 , V ε

3 ) are needed to compute the corrected price

• Probabilistic representation of (P0 + P ε1 )(t, x):

IE

{e−r(T−t)h(XT ) +

∫ T

t

e−r(s−t)H(s, Xs)ds|Xt = x

}

• Put-Call Parity is preserved at the order O(√

ε)

• The V ε2 term is a volatility level correction

σ⋆ =√

σ2 + 2V ε2

• The V ε3 term is the skew effect

ρ = 0 =⇒ V ε3 = 0

29

Accuracy of Approximation

Define

Zε = P0 +√

εP1 + εP2 + ε√

εP3 − P

so that

Zε(T, x, y) = ε(P2(T, x, y) +

√εP3(T, x, y)

)

Using how (P0, P1, P2, P3) have been chosen to cancel 1/ε, 1/√

ε,

O(1) and√

ε terms deduce

LεZε = ε(L1P3 + L2P2 +

√εL2P3

)

and conclude that source and terminal condition of order ε

=⇒ Zε = O(ε)

=⇒ P (t, x, y) = (P0(t, x) + P ε1 (t, x)) + O(ε)

30

Corrected Call Option Prices

h(x) = (x − K)+

and

P0(t, x) = CBS(t, x; K, T ; σ)

Compute the Delta, the Gamma and the Delta-Gamma

= ∂3P0/∂x3

Deduce the source

H =

(V ε

2 x2 ∂2PBS

∂x2+ V ε

3 x∂

∂x

(x2 ∂2PBS

∂x2

))

and the correction

P ε1 (t, x) = (T − t)H(t, x) =

xe−d2

1/2

σ√

2π

(−V3

d1

σ+ V2

√T − t

)

31

Expansion of Implied Volatilities

Recall

CBS(t, x; K, T ; I) = Cobserved

Expand

I = σ +√

εI1 + · · ·Deduce for given (K, T ):

CBS(t, x; σ) +√

εI1∂CBS

∂σ(t, x; σ) + · · · = P0(t, x) + P ε

1 (t, x) + · · ·

=⇒√

εI1 = P ε1 (t, x) [Vega(σ)]−1

Compute the Vega = ∂CBS/∂σ = xe−d2

1/2√

T − t/√

2π

and deduce

32

Calibration Formulas

The implied volatility is an affine function of the LMMR:

log-moneyness-to-maturity-ratio = log(K/x)/(T − t)

I = a [LMMR] + b + O(1/α)

with

a =V ε

3

σ3

b = σ − V ε3

σ3

(r − 1

2σ2

)+

V ε2

σ

or for calibration purpose:

V ε2 = σ

((b − σ) + a(r − 1

2σ2)

)

V ε3 = aσ3

33

400420

440460

480500

520

0

0.2

0.4

0.6

0.8

1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Strike Price KTime to Expiration

Implied

Vol

atility

Figure 2: A typical implied volatility surface predicted by the asymptotic

analysis. It is linear in the composite variable LMMR with slope a =

−0.154 and intercept b = 0.149 estimated from S&P 500 options data. We

take t = 0 and current asset price x = 460.

34

0.950.96

0.970.98

0.991

1.011.02

1.031.04

0.10.15

0.2

0.250.3

0.350.4

0.45

0.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Moneyness K/xTime to Maturity

Rat

ioP

1/(

P0

+P

1)

Figure 3: Ratio of correction P1 to corrected price P for a European

call option using parameter values calibrated from the observed S&P 500

implied volatility surface: a = −0.154, b = 0.149 and σ = 0.1, r = 0.02.

These give V2 = −0.0044 and V3 = 0.000154.

35

0 10 20 30 40 50 60

−0.25

−0.2

−0.15

−0.1

−0.05

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

Liquid Slope Estimates: Mean= −0.154, Std= 0.032

Liquid Intercept Estimates: Mean= 0.149, Std= 0.007

Trading Day Number: 9/20/94 - 12/19/94

Figure 4: Daily fits of S&P 500 European call option implied volatilties to

a straight line in LMMR, excluding days when there is insufficient liquidity

(16 days out of 60).

36

Exotic Derivatives (Binary, Barrier,Asian,...)

• Solve the corresponding problem with constant volatility σ

=⇒ P0

• Use V2 and V3 calibrated on the smile to compute the

source

V2x2 ∂2P0

∂x2+ V3x

∂

∂x

(x2 ∂2P0

∂x2

)

• Get the correction P1 by solving the SAME PROBLEM

with zero boundary conditions and the source.

37

American Options

• Solve the corresponding problem with constant volatility σ

=⇒ P0 and the free boundary x0(t)

• Use V2 and V3 calibrated on the smile to compute the

source

V2x2 ∂2P0

∂x2+ V3x

∂

∂x

(x2 ∂2P0

∂x2

)

• Get the correction P1 by solving the corresponding problem

with fixed boundary x0(t), zero boundary conditions and

the source.

38