Calibration to Implied Volatility Data Jean-Pierre Fouque - - PowerPoint PPT Presentation

PART III:

Calibration to Implied Volatility Data

Jean-Pierre Fouque University of California Santa Barbara Special Semester on Stochastics with Emphasis on Finance Tutorial September 5, 2008 RICAM, Linz, Austria

1

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 = V3 ¯ σ3 b = ¯ σ + V2 ¯ σ − V3 ¯ σ3

• r − ¯

σ2 2

• r for calibration purpose:

V2 = ¯ σ

• (b − ¯

σ) + a(r − ¯ σ2 2 )

• V3

= a¯ σ3

2

In sample fit of implied volatilities

−0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 LMMR Implied Vol.

I ≈ a [LMMR] + b (Maturities less than 6 months)

3

A slow volatility factor is needed

−2.5 −2 −1.5 −1 −0.5 0.5 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 LMMR Implied Volatility Pure LMMR Fit

Implied volatility as a function of LMMR. The circles are from S&P 500 data, and the line a(LMMR) + b shows the fit using maturities up to two years.

4

Two-Scale Stochastic Volatility Models ε << T << 1/δ dXt = rXtdt + f(Yt, Zt)XtdW (0)⋆

t

dYt =

• 1

ε(m − Yt) − ν √ 2 √ε Λ(Yt, Zt)

• dt + ν

√ 2 √ε 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

5

Pricing Equation P ε,δ(t, x, y, z) = I E⋆ 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 = (m − y) ∂ ∂y + ν2 ∂2 ∂y2 M1 = −gΓ ∂ ∂z + ρ2gfx ∂2 ∂x∂z L1 = ν √ 2

• ρ1fx ∂2

∂x∂y − Λ ∂ ∂y

• M2 = c ∂

∂z + g2 2 ∂2 ∂z2 L2 = ∂ ∂t + 1 2f 2x2 ∂2 ∂x2 + r

• x ∂

∂x − ·

• M3 = ν

√ 2 ˜ ρ12g ∂2 ∂y∂z

6

Double Expansion

P ε,δ = P0 + √εP1,0 + √ δP0,1 + · · · = P0 + ˜ P1 + ˜ Q1 + · · · Leading order term: P0(t, x, z) = PBS(t, x; ¯ σ(z)) Correction: ˜ P1 = √εP1,0 with V ε

2 , V ε 3 (z-dependent):

LBS(¯ σ) ˜ P1 +

• V ε

2 x2 ∂2PBS

∂x2 + V ε

3 x ∂

∂x

• x2 ∂2PBS

∂x2

• = 0

˜ P1(T, x, z) = 0 ˜ P1(t, x, z) = (T − t)

• V ε

2 x2 ∂2PBS

∂x2 + V ε

3 x ∂

∂x

• x2 ∂2PBS

∂x2

• 7

Price Approximation: P ε,δ(t, x, y, z) ≈ PBS(t, x; T, ¯ σ) +(T − t)

• V ε

2 x2 ∂2PBS

∂x2 + V ε

3 x ∂

∂x

• x2 ∂2PBS

∂x2

• +(T − t)
• V δ

∂PBS ∂σ + V δ

1 x∂2PBS

∂x∂σ

• LBS(¯

σ) ˜ Q1 + 2

• V δ

∂PBS ∂σ + V δ

1 x∂2PBS

∂x∂σ

• = 0

˜ Q1(T, x) = 0 KEY: ∂PBS ∂σ = (T − t)σx2 ∂2PBS ∂x2

8

Term Structure of Implied Volatility

I0 + Iε

1 + Iδ 1 =

¯ σ + [bε + bδ(T − t)] + [aε + aδ(T − t)]log(K/x) T − t , where the parameters (¯ σ, aε, aδ, bε, bδ) depend on z and are related to the group parameters (V δ

0 , V δ 1 , V ε 2 , V ε 3 ) by

aε = V ε

3

¯ σ3 , bε = V ε

2

¯ σ − V ε

3

¯ σ3 (r − ¯ σ2 2 ) aδ = V δ

1

¯ σ2 , bδ = V δ

0 − V δ 1

¯ σ2 (r − ¯ σ2 2 )

9

0.2 0.4 0.6 0.8 1 1.2 1.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 α=aε+aδτ 0.2 0.4 0.6 0.8 1 1.2 1.4 0.22 0.24 0.26 τ β=σ+bε+bδτ

Term-structures fits

10

−2.5 −2 −1.5 −1 −0.5 0.5 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 LMMR δ−adjusted Implied Volatility LMMR Fit to Residual

δ-adjusted implied volatility I − bδτ − aδ(LM) as a function of

• LMMR. The circles are from S&P 500 data, and the line

R + aε(LMMR) shows the fit using the estimated parameters.

11

A slow volatility factor is needed

−2.5 −2 −1.5 −1 −0.5 0.5 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 LMMR Implied Volatility Pure LMMR Fit

Implied volatility as a function of LMMR. The circles are from S&P 500 data, and the line a(LMMR) + b shows the fit using maturities up to two years.

12

A fast volatility factor is needed

−0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.15 0.2 0.25 0.3 0.35 0.4 LM τ−adjusted Implied Volatility LM Fit to Residual

The circles are from S&P 500 data, and the line aδ(LM) + ¯ σ shows the fit using the estimated parameters from only a slow factor fit.

13

−5 5 −0.1 0.1 0.2 0.3 0.4 0.5 LMMR Implied Volatility τ=43 days −2 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 LMMR 71 days −1 1 0.1 0.15 0.2 0.25 0.3 0.35 LMMR 106 days −0.5 0.5 0.15 0.2 0.25 LMMR Implied Volatility τ=197 days −0.05 0.05 0.18 0.185 0.19 0.195 0.2 LMMR 288 days −0.2 0.2 0.16 0.18 0.2 0.22 0.24 LMMR 379 days

Figure 1: S&P 500 Implied Volatility data on June 5, 2003 and fits to the

affine LMMR approximation for six different maturities.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −0.25 −0.2 −0.15 −0.1 −0.05 τ(yrs) m0 + m1 τ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.188 0.189 0.19 0.191 0.192 0.193 0.194 τ(yrs) b0 + b1 τ

Figure 2: S&P 500 Implied Volatility data on June 5, 2003 and fits to

the two-scales asymptotic theory. The bottom (rep. top) figure shows the linear regression of b (resp. a) with respect to time to maturity τ = T − t.

15

Higher order terms in ε, δ and √ εδ I ≈

4

• j=0

aj(τ) (LM)j + 1 τ Φt, where τ denotes the time-to maturity T − t, LM denotes the moneyness log(K/S), and Φt is a rapidly changing component that varies with the fast volatility factor

16

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Log−Moneyness + 1 Implied Volatility 5 June, 2003: S&P 500 Options, 15 days to maturity 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Log−Moneyness + 1 Implied Volatility 5 June, 2003: S&P 500 Options, 71 days to maturity 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 5 June, 2003: S&P 500 Options, 197 days to maturity Log−Moneyness + 1 Implied Volatility 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 5 June, 2003: S&P 500 Options, 379 days to maturity Log−Moneyness + 1 Implied Volatility

Figure 3: S&P 500 Implied Volatility data on June 5, 2003 and quartic

fits to the asymptotic theory for four maturities.

17

0.5 1 1.5 2 1 2 3 4 τ (yrs) a4 0.5 1 1.5 2 2 4 6 8 τ(yrs) a3 0.5 1 1.5 2 −1 1 2 3 4 5 a2 0.5 1 1.5 2 −0.5 −0.4 −0.3 −0.2 −0.1 τ (yrs) a1

Figure 4: S&P 500 Term-Structure Fit using second order approximation.

Data from June 5, 2003.

18

0.5 1 1.5 2 4 6 8 10 τ (yrs.) a4 0.5 1 1.5 5 10 15 20 25 τ a3 0.5 1 1.5 2 4 6 8 10 12 τ a2 0.5 1 1.5 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 τ a1

Figure 5: S&P 500 Term-Structure Fit. Data from every trading day in

May 2003.

19

Parameter Reduction and Direct Calibration LBS(¯ σ)

• ˜

P1 + ˜ Q1

• +
• V2x2 ∂2PBS

∂x2 + V3x ∂ ∂x

• x2 ∂2PBS

∂x2

• +

2

• V0

∂PBS ∂σ + V1x∂2PBS ∂x∂σ

• = 0

Set σ⋆ = √ ¯ σ2 + 2V2. At the same order, the correction is: (T − t)

• V0

∂P ⋆

BS

∂σ + V1x∂2P ⋆

BS

∂x∂σ + V3x ∂ ∂x

• x2 ∂2P ⋆

BS

∂x2

• I ≈ b⋆ + τbδ +
• aε + τaδ

LMMR b⋆ = σ⋆ + V3 2σ⋆

• 1 − 2r

σ⋆2

• ,

aε = V3 σ⋆3 bδ = V0 + V1 2

• 1 − 2r

σ⋆2

• ,

aδ = V1 σ⋆2

20

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

• Calibrate σ⋆, V0, V1 and V3 on the implied volatility surface
• Solve the corresponding problem with constant volatility σ⋆

= ⇒ P0 = PBS(σ⋆)

• Use V0, V1 and V3 to compute the source

2

• V0

∂P ⋆

BS

∂σ + V1x∂2P ⋆

BS

∂x∂σ

• + V3x ∂

∂x

• x2 ∂2P ⋆

BS

∂x2

• Get the correction by solving the SAME PROBLEM

with zero boundary conditions and the source.

21

American Options

• Calibrate σ⋆, V0, V1 and V3 on the implied volatility surface
• Solve the corresponding problem with constant volatility σ⋆

= ⇒ P ⋆ and the free boundary x⋆(t)

• Use V0, V1 and V3 to compute the source

2

• V0

∂P ⋆ ∂σ + V1x ∂2P ⋆ ∂x∂σ

• + V3x ∂

∂x

• x2 ∂2P ⋆

∂x2

• Get the correction by solving the corresponding problem with

fixed boundary x⋆(t), zero boundary conditions and the source.

22

Conclusions

• A short time-scale of order few days is present in

volatility dynamics

• It cannot be ignored in option pricing and hedging
• It can be dealt with by using singular perturbation

methods

• It is efficient as a parametrization tool for the term

structure of implied volatilities when combined with a regular perturbation

23