PART II: Stochastic Volatility Modeling...SIAM Journal on Multiscale Modeling and Simulation, 2(1),...
Transcript of PART II: Stochastic Volatility Modeling...SIAM Journal on Multiscale Modeling and Simulation, 2(1),...
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