Calibration to Implied Volatility Data

Transcription

1 Calibration to Implied Volatility Data Jean-Pierre Fouque University of California Santa Barbara 2008 Daiwa Lecture Series July 29 - August 1, 2008 Kyoto University, Kyoto 1

2 Calibration Formulas The implied volatility is an affine function of the LMMR: log-moneyness-to-maturity-ratio = logk/x)/t t) with I = a [LMMR] + b + O1/α) a = V 3 σ 3 b = σ + V 2 σ V 3 σ 3 r σ2 2 or for calibration purpose: ) V 2 = σ b σ) + ar σ2 2 ) V 3 = a σ 3 ) 2

3 In sample fit of implied volatilities Implied Vol LMMR I a [LMMR] + b Maturities less than 6 months) 3

4 A slow volatility factor is needed 0.5 Pure LMMR Fit Implied Volatility LMMR Implied volatility as a function of LMMR. The circles are from S&P 500 data, and the line almmr) + b shows the fit using maturities up to two years. 4

5 Two-Scale Stochastic Volatility Models ε << T << 1/δ dx t = rx t dt + fy t, Z t )X t dw 0) t 1 dy t = ε m Y t) ν ) 2 ΛY t, Z t ) ε dz t = δ cz t ) ) δ gz t )ΓY t, Z t ) dt + ν 2 ε dw 1) t dt + δ gz t )dw 2) t d < W 0), W 1) > t = ρ 1 dt d < W 0), W 2) > t = ρ 2 dt 5

6 Pricing Equation { } P ε,δ t, x, y, z) = IE e rt t) hx T ) X t = x, Y t = y, Z t = z 1 ε L ε L 1 + L 2 + δm 1 + δm 2 + P ε,δ T, x, y, z) = hx) ) δ ε M 3 P ε,δ = 0 L 0 = m y) y y 2 L 1 = ν 2 ρ 1 fx 2 x y Λ ) y L 2 = t f2 x 2 2 x + r x ) 2 x + ν2 2 M 1 = gγ z + ρ 2gfx 2 x z M 2 = c z + g2 2 2 z 2 M 3 = ν 2 ρ 12 g 2 y z 6

7 Double Expansion P ε,δ = P 0 + εp 1,0 + δp 0,1 + = P 0 + P 1 + Q 1 + Leading order term: P 0 t, x, z) = P BS t, x; σz)) Correction: P1 = εp 1,0 with V ε 2, V ε L BS σ) P 1 + P 1 T, x, z) = 0 P 1 t, x, z) = T t) V ε 2 x 2 2 P BS x 2 V ε 2 x 2 2 P BS x 2 3 z-dependent): + V ε 3 x x + V ε 3 x x x 2 2 P BS x 2 )) = 0 )) x 2 2 P BS x 2 7

8 Price Approximation: P ε,δ t, x, y, z) P BS t, x; T, σ) +T t) V2 ε x 2 2 P BS x 2 +T t) V0 δ P BS σ + V ε 3 x x + V δ 1 x 2 P BS x σ x 2 2 P BS x 2 ) )) L BS σ) Q Q 1 T, x) = 0 V0 δ P BS σ ) + V 1 δ x 2 P BS x σ = 0 KEY: P BS σ = T t)σx 2 2 P BS x 2 8

9 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)] logk/x) T t where the parameters σ, a ε, a δ, b ε, b δ ) depend on z and are related to the group parameters V0 δ, V1 δ, V2 ε, V3 ε ) by a ε = V 3 ε σ 3, b ε = V 2 ε σ V 3 ε σ2 r σ 3 2 ) a δ = V 1 δ σ 2, b δ = V δ 1 δ σ2 r σ 2 2 ), 9

10 α=a ε +a δ τ β=σ+b ε +b δ τ τ Term-structures fits 10

11 0.38 LMMR Fit to Residual δ adjusted Implied Volatility 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

12 A slow volatility factor is needed 0.5 Pure LMMR Fit Implied Volatility LMMR Implied volatility as a function of LMMR. The circles are from S&P 500 data, and the line almmr) + b shows the fit using maturities up to two years. 12

13 A fast volatility factor is needed 0.4 LM Fit to Residual 0.35 τ adjusted Implied Volatility 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

14 0.5 τ=43 days days days Implied Volatility LMMR LMMR LMMR τ=197 days 288 days days Implied Volatility 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

15 m 0 + m 1 τ τyrs) b 0 + b 1 τ τyrs) 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

16 Higher order terms in ε, δ and εδ I 4 a j τ) LM) j + 1 τ Φ t, j=0 where τ denotes the time-to maturity T t, LM denotes the moneyness logk/s), and Φ t is a rapidly changing component that varies with the fast volatility factor 16

17 0.5 5 June, 2003: S&P 500 Options, 15 days to maturity June, 2003: S&P 500 Options, 71 days to maturity Implied Volatility Implied Volatility Log Moneyness + 1 Log Moneyness June, 2003: S&P 500 Options, 197 days to maturity 3 5 June, 2003: S&P 500 Options, 379 days to maturity Implied Volatility 2 Implied Volatility Log Moneyness 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

18 a 4 2 a τ yrs) τyrs) a a τ yrs) Figure 4: S&P 500 Term-Structure Fit using second order approximation. Data from June 5,

19 a 4 4 a τ yrs.) τ a τ a τ Figure 5: S&P 500 Term-Structure Fit. Data from every trading day in May

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

21 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 PBS 2 V 0 σ + V 1x 2 PBS ) + V 3 x x 2 2 P ) BS x σ x x 2 Get the correction by solving the SAME PROBLEM with zero boundary conditions and the source. 21

22 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 P 2 V 0 σ + V 1x 2 P ) + V 3 x x 2 2 P ) x σ x x 2 Get the correction by solving the corresponding problem with fixed boundary x t), zero boundary conditions and the source. 22

23 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