Stochastic Volatility Modeling Jean-Pierre Fouque University of - - PowerPoint PPT Presentation

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

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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

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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)

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Realized Volatility

t0 < t1 < · · · < tN = t (present time) 1 t − t0 t

t0

σ2

s 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

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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.

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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.

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0.8 0.85 0.9 0.95 1 1.05 1.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Moneyness K/x Implied Volatility 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.

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“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

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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)

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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

dW, W (1)t = ρdt

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The Popular Heston Model

dSt = µStdt + σtStdW (1)

t

σt =

• Yt

dYt = α(m − Yt)dt + ν

• 2αYt dW (2)

t

dW (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.

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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(· · ·)d ˆ Zt Brownian motion ˆ Z correlated to W: ˆ Zt = ρWt +

• 1 − ρ2Zt ,

|ρ| < 1 so that dW, ˆ Zt = ρdt

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Pricing under Stochastic Volatility

Risk-neutral probability chosen by the market: I P ⋆(γ) dXt = rXtdt + f(Yt)XtdW ⋆

t

dYt =

• α(m − Yt) − β
• ρ(µ − r)

f (Yt) + γ

• 1 − ρ2
• dt + βd ˆ

Z⋆

t

ˆ Z⋆

t

= ρW ⋆

t +

• 1 − ρ2 Z⋆

t

Market price of volatility risk: γ = γ(y) Pt = I E⋆(γ){e−r(T −t)h(XT )|Ft} Markovian case: P(t, x, y) = I E⋆(γ){e−r(T −t)h(XT )|Xt = x, Yt = y} but y (or f(y)) is not directly observable!

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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 = where Λ = ρ(µ − r) f (y) + γ

• 1 − ρ2

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

No perfect hedge!

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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

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Fast Mean-Reverting Stochastic Volatility Asymptotic Analysis

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Model in the risk-neutral world I P ⋆(γ) in terms of ε = 1/α Set β = ν

√ 2 √ε so that ν2 = β2/2α

dXε

t

= rXε

t dt + f(Y ε t )Xε t dW ⋆ t

dY ε

t

=

• 1

ε(m − Y ε

t ) − ν

√ 2 √ε Λ(Y ε

t )

• dt + ν

√ 2 √ε d ˆ Z⋆

t

Market price of risks: Λ(y) = ρ(µ − r) f (y) + γ(y)

• 1 − ρ2

Skew: ˆ Z⋆

t = ρW ⋆ t +

• 1 − ρ2Z⋆

t

, |ρ| < 1

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Prices and Pricing PDE’s

P ε(t, x, y) = I E⋆(γ) e−r(T −t)h(Xε

T )|Xε t = x, Y ε t = 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 = to be solved for t < T with the terminal condition P ε(T, x, y) = h(x)

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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))

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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

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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

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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 .

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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

lim

t→+∞ I

E {g(Yt)|Y0 = y} = g = 0

(exponentially fast)

χ(y) = +∞ I E {g(Yt)|Y0 = y} dt checked by applying L0

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Leading Order Term

Centering: L2P0 = L2P0 = 0 L2 = ∂ ∂t + 1 2f(y)2x2 ∂2 ∂x2 + r

• x ∂

∂x − ·

• =

∂ ∂t + 1 2f 2x2 ∂2 ∂x2 + r

• x ∂

∂x − ·

• Effective volatility: ¯

σ2 =

• f 2

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)

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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 − f 2 Then P2(t, x, y) = −L−1 (L2P0) = −1 2 (φ(y) + c(t, x)) x2 ∂2P0 ∂x2

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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

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The correction P ε

1 (t, x) = √εP1(t, x)

LBS(¯ σ)P ε

1 − ν

√ 2ε 2

• ρxf(y) ∂2

∂x∂y − Λ(y) ∂ ∂y

• φ(y)

x2 ∂2P0 ∂x2

• =

LBS(¯ σ)P ε

1 +

• V ε

2 x2 ∂2PBS

∂x2 + V ε

3 x ∂

∂x

• x2 ∂2PBS

∂x2

• =

BS equation with source and zero terminal condition with the two small parameters V ε

2 and V ε 3 given by:

V ε

2

= ν √ 2αΛφ′ V ε

3

= −ρν √ 2α fφ′ Recall that α = 1/ε

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Explicit Formula for the Corrected Price

P ε

1

= (T − t)

• V ε

2 x2 ∂2PBS

∂x2 + V ε

3 x ∂

∂x

• x2 ∂2PBS

∂x2

• where V ε

2 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

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Comments

• The small constants V ε

2 and V ε 3 are complex functions of the

• riginal model parameters (µ, m, ν, ρ, α; f) and γ
• Only (¯

σ, V ε

2 , V ε 3 ) are needed to compute the corrected price

• Probabilistic representation of (P0 + P ε

1 )(t, x):

¯ I E

• 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 ε

2 term is a volatility level correction

σ⋆ =

• ¯

σ2 + 2V ε

2

• The V ε

3 term is the skew effect

ρ = 0 = ⇒ V ε

3 = 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 ε

1 (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 ε

1 (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 ε

1 (t, x) [Vega(¯

σ)]−1 Compute the Vega = ∂CBS/∂σ = xe−d2

1/2√

T − t/ √ 2π and deduce

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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 ε

3

¯ σ3

• r − 1

2 ¯ σ2

• + V ε

2

¯ σ

• r for calibration purpose:

V ε

2

= ¯ σ

• (b − ¯

σ) + a(r − 1 2 ¯ σ2)

• V ε

3

= a¯ σ3

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400 420 440 460 480 500 520 0.2 0.4 0.6 0.8 1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Strike Price K Time to Expiration Implied Volatility 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.

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0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Moneyness K/x Time to Maturity Ratio P1/(P0 + P1) 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.

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10 20 30 40 50 60 −0.25 −0.2 −0.15 −0.1 −0.05 10 20 30 40 50 60 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).

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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.

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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.

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