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 V 3 a = σ 3 ¯ σ 2 σ + V 2 σ − V 3 � r − ¯ � b = ¯ σ 3 ¯ ¯ 2 or for calibration purpose : σ 2 � σ ) + a ( r − ¯ � V 2 = σ ¯ ( b − ¯ 2 ) σ 3 V 3 = a ¯ 2

In sample fit of implied volatilities 0.16 0.14 0.12 0.1 Implied Vol. 0.08 0.06 0.04 0.02 0 −0.02 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 LMMR I ≈ a [ LMMR ] + b (Maturities less than 6 months) 3

A slow volatility factor is needed Pure LMMR Fit 0.5 0.45 0.4 Implied Volatility 0.35 0.3 0.25 0.2 0.15 −2.5 −2 −1.5 −1 −0.5 0 0.5 LMMR 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 rX t dt + f ( Y t , Z t ) X t dW (0) ⋆ dX t = t √ √ � � 1 ε ( m − Y t ) − ν 2 dt + ν 2 √ ε dW (1) ⋆ √ ε Λ( Y t , Z t ) dY t = t √ √ � � δ g ( Z t ) dW (2) ⋆ dZ t = δ c ( Z t ) − δ g ( Z t )Γ( Y t , Z t ) dt + t d < W (0) ⋆ , W (1) ⋆ > t = ρ 1 dt d < W (0) ⋆ , W (2) ⋆ > t = ρ 2 dt 5

Pricing Equation E ⋆ � � e − r ( T − t ) h ( X T ) | X t = x, Y t = y, Z t = z P ε,δ ( t, x, y, z ) = I � � √ � 1 ε L 0 + 1 δ P ε,δ = 0 √ ε L 1 + L 2 + δ M 1 + δ M 2 + ε M 3 P ε,δ ( T, x, y, z ) = h ( x ) ∂y + ν 2 ∂ 2 ∂z + ρ 2 gfx ∂ 2 L 0 = ( m − y ) ∂ M 1 = − g Γ ∂ ∂y 2 ∂x∂z √ ρ 1 fx ∂ 2 ∂z + g 2 ∂ 2 � � ∂x∂y − Λ ∂ M 2 = c ∂ L 1 = ν 2 ∂z 2 ∂y 2 √ 2 f 2 x 2 ∂ 2 ρ 12 g ∂ 2 L 2 = ∂ ∂t + 1 � x ∂ � ∂x 2 + r ∂x − · M 3 = ν 2 ˜ ∂y∂z 6

Double Expansion √ P 0 + √ εP 1 , 0 + P ε,δ = δP 0 , 1 + · · · ˜ ˜ = P 0 + P 1 + Q 1 + · · · Leading order term: P 0 ( t, x, z ) = P BS ( t, x ; ¯ σ ( z )) P 1 = √ εP 1 , 0 with V ε Correction: ˜ 2 , V ε 3 ( z -dependent): 2 x 2 ∂ 2 P BS x 2 ∂ 2 P BS � 3 x ∂ � �� σ ) ˜ V ε + V ε L BS (¯ P 1 + = 0 ∂x 2 ∂x 2 ∂x ˜ P 1 ( T, x, z ) = 0 2 x 2 ∂ 2 P BS x 2 ∂ 2 P BS � 3 x ∂ � �� ˜ V ε + V ε P 1 ( t, x, z ) = ( T − t ) ∂x 2 ∂x 2 ∂x 7

Price Approximation: P ε,δ ( t, x, y, z ) ≈ P BS ( t, x ; T, ¯ σ ) 2 x 2 ∂ 2 P BS x 2 ∂ 2 P BS � 3 x ∂ � �� V ε + V ε +( T − t ) ∂x 2 ∂x 2 ∂x 1 x∂ 2 P BS � ∂P BS � V δ + V δ +( T − t ) 0 ∂σ ∂x∂σ 1 x∂ 2 P BS � ∂P BS � σ ) ˜ V δ + V δ L BS (¯ Q 1 + 2 = 0 0 ∂σ ∂x∂σ ˜ Q 1 ( T, x ) = 0 KEY: ( T − t ) σx 2 ∂ 2 P BS ∂P BS = ∂x 2 ∂σ 8

Term Structure of Implied Volatility I 0 + I ε 1 + I δ 1 = σ + [ b ε + b δ ( T − t )] + [ a ε + a δ ( T − t )]log( K/x ) ¯ , T − t σ, a ε , a δ , b ε , b δ ) depend on z and are related where the parameters (¯ to the group parameters ( V δ 0 , V δ 1 , V ε 2 , V ε 3 ) by σ 2 a ε = V ε b ε = V ε σ − V ε σ 3 ( r − ¯ 3 2 3 , 2 ) σ 3 ¯ ¯ ¯ σ 2 a δ = V δ 0 − V δ σ 2 ( r − ¯ b δ = V δ 1 1 , 2 ) σ 2 ¯ ¯ 9

0 −0.05 −0.1 α =a ε +a δ τ −0.15 −0.2 −0.25 −0.3 −0.35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.26 β = σ +b ε +b δ τ 0.24 0.22 0 0.2 0.4 0.6 0.8 1 1.2 1.4 τ Term-structures fits 10

LMMR Fit to Residual 0.38 0.36 0.34 0.32 δ −adjusted Implied Volatility 0.3 0.28 0.26 0.24 0.22 0.2 0.18 −2.5 −2 −1.5 −1 −0.5 0 0.5 LMMR δ -adjusted implied volatility I − b δ τ − a δ ( LM ) as a function of LMMR. The circles are from S&P 500 data, and the line R + a ε ( LMMR ) shows the fit using the estimated parameters. 11

A slow volatility factor is needed Pure LMMR Fit 0.5 0.45 0.4 Implied Volatility 0.35 0.3 0.25 0.2 0.15 −2.5 −2 −1.5 −1 −0.5 0 0.5 LMMR 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 LM Fit to Residual 0.4 0.35 τ −adjusted Implied Volatility 0.3 0.25 0.2 0.15 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 LM The circles are from S&P 500 data, and the line a δ ( LM ) + ¯ σ shows the fit using the estimated parameters from only a slow factor fit. 13

τ =43 days 71 days 106 days 0.35 0.5 0.35 0.3 0.4 0.3 Implied Volatility 0.25 0.3 0.25 0.2 0.2 0.2 0.15 0.1 0.15 0.1 0 0.05 0.1 −0.1 −5 0 5 −2 0 2 −1 0 1 LMMR LMMR LMMR τ =197 days 288 days 379 days 0.24 0.2 0.25 0.22 0.195 Implied Volatility 0.2 0.2 0.19 0.18 0.185 0.15 0.16 0.18 −0.5 0 0.5 −0.05 0 0.05 −0.2 0 0.2 LMMR LMMR LMMR Figure 1: S&P 500 Implied Volatility data on June 5, 2003 and fits to the affine LMMR approximation for six different maturities. 14

−0.05 −0.1 m 0 + m 1 τ −0.15 −0.2 −0.25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 τ (yrs) 0.194 0.193 0.192 b 0 + b 1 τ 0.191 0.19 0.189 0.188 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 τ (yrs) 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 4 a j ( τ ) ( LM ) j + 1 � I ≈ τ Φ t , j =0 where τ denotes the time-to maturity T − t , LM denotes the moneyness log( K/S ), and Φ t is a rapidly changing component that varies with the fast volatility factor 16

5 June, 2003: S&P 500 Options, 15 days to maturity 5 June, 2003: S&P 500 Options, 71 days to maturity 0.5 0.5 0.45 0.45 0.4 0.4 Implied Volatility Implied Volatility 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Log−Moneyness + 1 Log−Moneyness + 1 5 June, 2003: S&P 500 Options, 197 days to maturity 5 June, 2003: S&P 500 Options, 379 days to maturity 0.28 0.23 0.22 0.26 0.21 0.24 0.2 Implied Volatility Implied Volatility 0.22 0.19 0.2 0.18 0.18 0.17 0.16 0.16 0.14 0.15 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Log−Moneyness + 1 Log−Moneyness + 1 Figure 3: S&P 500 Implied Volatility data on June 5, 2003 and quartic fits to the asymptotic theory for four maturities. 17

4 8 3 6 a 4 2 a 3 4 1 2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 τ (yrs) τ (yrs) 5 0 4 −0.1 3 −0.2 a 2 2 a 1 −0.3 1 −0.4 0 −1 −0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 τ (yrs) Figure 4: S&P 500 Term-Structure Fit using second order approximation. Data from June 5, 2003. 18

10 25 8 20 6 15 a 4 a 3 4 10 2 5 0 0 0 0.5 1 1.5 0 0.5 1 1.5 τ τ (yrs.) 12 0 −0.1 10 −0.2 8 −0.3 a 2 6 a 1 −0.4 4 −0.5 2 −0.6 0 −0.7 0 0.5 1 1.5 0 0.5 1 1.5 τ τ Figure 5: S&P 500 Term-Structure Fit. Data from every trading day in May 2003. 19

Parameter Reduction and Direct Calibration V 2 x 2 ∂ 2 P BS x 2 ∂ 2 P BS � + V 3 x ∂ � �� � � P 1 + ˜ ˜ L BS (¯ σ ) Q 1 + ∂x 2 ∂x 2 ∂x + V 1 x∂ 2 P BS � ∂P BS � + 2 V 0 = 0 ∂σ ∂x∂σ √ Set σ ⋆ = σ 2 + 2 V 2 . ¯ At the same order, the correction is: ∂P ⋆ + V 1 x∂ 2 P ⋆ x 2 ∂ 2 P ⋆ � ∂x∂σ + V 3 x ∂ � �� BS BS BS ( T − t ) V 0 ∂x 2 ∂σ ∂x I ≈ b ⋆ + τb δ + a ε + τa δ � � LMMR b ⋆ = σ ⋆ + V 3 � 1 − 2 r � a ε = V 3 , σ ⋆ 2 σ ⋆ 3 2 σ ⋆ � � b δ = V 0 + V 1 1 − 2 r a δ = V 1 , σ ⋆ 2 σ ⋆ 2 2 20

Exotic Derivatives (Binary, Barrier, Asian,...) • Calibrate σ ⋆ , V 0 , V 1 and V 3 on the implied volatility surface • Solve the corresponding problem with constant volatility σ ⋆ ⇒ P 0 = P BS ( σ ⋆ ) = • Use V 0 , V 1 and V 3 to compute the source ∂P ⋆ + V 1 x∂ 2 P ⋆ x 2 ∂ 2 P ⋆ � � � � + V 3 x ∂ BS BS BS 2 V 0 ∂x 2 ∂σ ∂x∂σ ∂x • Get the correction by solving the SAME PROBLEM with zero boundary conditions and the source . 21

American Options • Calibrate σ ⋆ , V 0 , V 1 and V 3 on the implied volatility surface • Solve the corresponding problem with constant volatility σ ⋆ ⇒ P ⋆ and the free boundary x ⋆ ( t ) = • Use V 0 , V 1 and V 3 to compute the source ∂σ + V 1 x ∂ 2 P ⋆ x 2 ∂ 2 P ⋆ ∂P ⋆ � � � � + V 3 x ∂ 2 V 0 ∂x 2 ∂x∂σ ∂x • Get the correction by solving the corresponding problem with fixed boundary x ⋆ ( t ), zero boundary conditions and the source . 22