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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
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