Stochastic Volatility Modeling

Transcription

1 Stochastic Volatility Modeling Jean-Pierre Fouque University of California Santa Barbara 28 Daiwa Lecture Series July 29 - August 1, 28 Kyoto University, Kyoto 1

2 References: Derivatives in Financial Markets with Stochastic Volatility Cambridge University Press, 2 Stochastic Volatility Asymptotics SIAM Journal on Multiscale Modeling and Simulation, 2(1), 23 Collaborators: G. Papanicolaou (Stanford), R. Sircar (Princeton), K. Sølna (UCI) 2

3 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

4 Realized Volatility t < t 1 < < t N = t (present time) 1 t t t t σ 2 s ds 1 N N i=1 ( log Sti log S ti 1) 2 t i t i 1 depends on the choice of t and on the number of increments N (assuming t i t i 1 constant). More details: Zhang, L., Mykland, P.A., and Ait-Sahalia, Y. (25). A tale of two time scales: Determining integrated volatility with noisy high-frequency data, J. Amer. Statist. Assoc. 1, mykland/publ.html 4

5 Volatility Models ds t = S t (µdt + σ t dw t ) Local Volatility: σ t = σ(t, S t ) where σ(t, x) is a deterministic function. Stochastic Volatility: σ t = f(y t ) where Y t contains an additional source of randomness. 5

6 Implied Volatility I(t, T, K) = σ implied (t, T, K) where σ implied (t, T, K) is uniquely defined by inverting Black-Scholes formula: C observed (t, T, K) = C BS (t, S t ; T, K; σ implied (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

7 Feb, 2.3 Skew Implied Volatility Excess kurtosis.1 Historical Volatility Moneyness K/x Figure 1: S&P 5 Implied Volatility Curve as a function of moneyness from S&P 5 index options on February 9, 2. The current index value is x = and the options have over two months to maturity. This is typically described as a downward sloping skew. 7

8 REQUIRED QUALITIES Parametrization of the Implied Volatility Surface I(t ; T, K) 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

9 At least three approaches: Local Volatility Models: σ t = σ(t, S t ) + s: market is complete (no additional randomness), Dupire formula σ 2 (T, K) = 2 - s: stability of calibration C T + rk C K K 2 2 C K 2 Implied Volatility Surface Models: di t (T, K) = + s: predictive power - s: no-arbitrage conditions not easy. Which underlying? Stochastic Volatility Models: σ t = f(y t ) 9

10 WHY? Stochastic Volatility Framework Distributions of returns are not log-normal Smile (Skew) effect observed in implied volatilities HOW? with, for instance: ds t = µs t dt + σ t S t dw t σ t = f(y t ) dy t = α(m Y t )dt + ν 2α dw (1) t d W, W (1) t = ρdt 1

11 The Popular Heston Model ds t = µs t dt + σ t S t dw (1) t σ t = Y t dy t = α(m Y t )dt + ν 2αY t dw (2) t d W (1), W (2) t t = ρdt Y t is a CIR (Cox-Ingersoll-Ross) process. The condition m ν 2 ensures that the process Y t stays strictly positive at all time. 11

12 Mean-Reverting Stochastic Volatility Models dx t = X t (µdt + σ t dw t ) σ t = f(y t ) For instance: < σ 1 f(y) σ 2 for every y dy t = α(m Y t )dt + β( )dẑt Brownian motion Ẑ correlated to W: Ẑ t = ρw t + 1 ρ 2 Z t, ρ < 1 so that d W, Ẑ t = ρdt 12

13 Pricing under Stochastic Volatility Risk-neutral probability chosen by the market: IP (γ) dx t = rx t dt + f(y t )X t dwt [ ( )] (µ r) dy t = α(m Y t ) β ρ f (Y t ) + γ 1 ρ 2 Ẑ t = ρw t + 1 ρ 2 Z t Market price of volatility risk: γ = γ(y) Markovian case: P t = IE (γ) {e r(t t) h(x T ) F t } P(t, x, y) = IE (γ) {e r(t t) h(x T ) X t = x, Y t = y} but y (or f(y)) is not directly observable! dt + βdẑ t 13

14 Stochastic Volatility Pricing PDE P t f(y)2 x 2 2 P x + ρβxf(y) 2 P 2 x y β2 2 P y ( 2 +r x P ) x P + α(m y) P y βλ P y = where Λ = ρ (µ r) f (y) Terminal condition: P(T, x, y) = h(x) + γ 1 ρ 2 No perfect hedge! 14

15 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

16 Driving process Y t and intrinsic time scale Markovian case: infinitesimal generator L Two-state Markov chain: L = α Pure jump Markov process Lg(y) = α (g(z) g(y))p(z)dz, p(z) = ( 1,1)(z) Diffusion: Ornstein-Uhlenbeck Process ( αl OU = α (m y) ) 2 + ν2 y 2 y 16

17 Invariant probability distribution L Φ = Two-state Markov chain: linear system Φ = { 1 2, 1 2 } Pure jump Markov process: integral equation Φ(y) = [ 1,1](y) OU diffusion process: differential equation Φ(y) = 1 ) (y m)2 exp ( 2π ν 2ν 2, ν 2 = β 2 /2α 17

18 Convergence to Equilibrium Equilibrium: g = g(y)φ(y)dy Exponential convergence to equilibrium with rate α: IE{g(Y t ) Y = y} g Ce αt Exponential decorrelation with rate α: IE Φ {g(y s )h(y t )} g h Ce α t s Intrinsic time scale: 1/α 18

19 .35 α = α = 2 α = Time t Figure 2: Simulated paths of σ t = f(y t ), with (Y t ) a two-state Markov chain, showing the relation between burstiness and the mean holding time 1/α. 19

20 Ergodic Theorem lim t 1 t t g(y s )ds = g or t > fixed and α + : almost surely (a.s.) In particular for α large: 1 t t g(y s ) g σ 2 = 1 T t LHS random T t f(y s ) 2 ds f 2 = σ 2 RHS deterministic 2

21 α = α = 2 α = Time t Figure 3: Simulated paths of σ t = f(y t ), with (Y t ) a pure jump Markov process taking values in (.5,.4), for increasing mean-reversion rates α. The mean level f =

22 .25 α = α = 2 α = Time t Figure 4: Simulated paths of σ t = f(y t ), with (Y t ) a mean-reverting OU process and f(y) =.35(arctany + π/2)/π +.5, chosen so that σ t (.5,.4). Notice how the mean-reversion rates α correspond to the duration of the bursts. 22

23 2 S&P 5 Returns Process, Returns Time Figure 5: 1996 S&P 5 returnscomputed from half-hourly data. 23

24 .5 α = 1.5 α = 1 Volatility Returns Time (yrs.) Time (yrs.) Figure 6: Simulated volatility and corresponding returns paths for small and large rates of mean-reversion for the jump volatility model. 24

25 .5 α = 1 α = Figure 7: Simulated volatility and corresponding returns paths for small and large rates of mean-reversion for the OU model with f(y) = e y. 25

26 Exponential decorrelation Variogram Variogram: IE[f(Y t+lag ) f(y t)] 2 2 σ 2 (1 e αlag ) Y s : Ornstein-Uhlenbeck driving process. f(x) = exp(x) 8 x 1 4 VARIOGRAM VOLATILITY TRADING DAY LAG 26

27 Exponential decorrelation for returns Variograms for raw and median filtered returns: 3 VARIOGRAM LOG RETURNS VARIOGRAM LOG RETURNS (MEDIAN FILT) TRADING DAY LAG 27

28 Stochastic volatility model: Estimation dx t = µx t dt + f(y t )X t dw t Volatility hidden, only see returns: dx t X t µdt = f(y t )dw t. The de-meaned return process: D ti = 1 ( ) Xti mean t X ti f(y ti ) W ti = f(y ti )ǫ ti where ǫ ti is an i.i.d white noise sequence. 28

29 Log-returns: Variogram of log-returns L ti = log D ti = log(f(y ti )) + log ǫ ti = f(y ti ) + ǫ ti. Variogram is like an exponential: 1 N N i=1 [ L ti +lag L t i ] 2 c1 (1 e αlag ) + c 2. 29

30 Median filtered S&P 5 variogram.8 S&P5 variogram (dotted) and fitted variogram (solid) days.8 Variogram for synthetics (dotted) and fitted variogram (solid) days Notice the day effect removed by the fitted exponential. The curvature determines 1/α : 1.5 ±.4 trading days annualized α 13 23: large 3

31 Spectrum of log-returns Energy spectrum of log returns: α IE[L t+tl t] cos(ωt) dt d 1 + d 2 α 2 +ω + d 2 3 δ(ω ω 1 ) : 14 S&P5 and Lorentzian spectra /day Simulated Lorentzian spectra 1 5 1/day

32 Volatility Time Scales Rescale the time of a diffusion process Y 1 t : Y α t = Y 1 αt α large speeding up the process Y 1 t α small slowing down the process Y 1 t 1/α is the characteristic time scale of the process Y α t. Averaged square volatility: σ 2 (, T) = 1 T T f 2 (Y α t )dt 32

33 Slowing Down the Time dy 1 t = c(y 1 t )dt + g(y 1 t )dw t, Y 1 = y dy α t = c(y α t )d(αt) + g(y α t )dw αt D = αc(y α t )dt + α g(y α t )dw t, Y α = y Assuming that f is continuous, σ 2 (, T) = 1 T T f 2 (Y α t )dt f 2 (y) as α Volatility is frozen at its starting level f(y) 33

34 Rate of Convergence in the Slow Scale Limit σ 2 (, T) f 2 (y) = 1 T ( t [f 2 y + α c(ys α )ds + t ) ] α g(ys α )dw s f 2 (y) dt T = 2 ( ) α f(y)f 1 T (y)g(y) W t dt + O(α) T (smoothness of f and g is assumed) Risk neutral a market price of volatility risk term: α g(ys α )Λ(Ys α )ds in the drift of Yt α additional term: 2 α f(y)f (y)g(y)λ(y) (T/2) Correlation with the BM driving the underlying will also come into play at the order α 34

35 Speeding Up the Time σ 2 (, T) = 1 T T f 2 (Y α t )dt = 1 αt αt = 1 T T f 2 (Y 1 s )ds, α + f 2 (Y 1 s )ds, T αt + Assuming that Y 1 is ergodic with invariant distribution Φ then: 1 T lim f 2 (Y 1 T + T s )ds = f 2 (y)φ(dy) f 2 Φ. Effective volatility: σ f 2 Φ lim α σ2 (,T) = σ 2 35

36 Fast oscillating integral: The Averaging Principle σ 2 (,T) σ 2 = 1 T T ( f 2 (Y α s ) σ 2) ds, α + Observe that f 2 (Y α s ) does not converge for fixed s. Introduce the Poisson equation: so that L Y 1φ(y) = f 2 (y) σ 2 σ 2 (,T) σ 2 = 1 T T L Y 1φ(Y α s ) ds 36

37 The Averaging Principle (continued) Using Ito s formula: Therefore dφ(y α s ) = L Y αφ(y α s )ds + α φ (Y α s )g(y α s )dw s σ 2 (,T) σ 2 = 1 T = αl Y 1φ(Y α s )ds + α φ (Y α s )g(y α s )dw s = 1 αt T T ( = 1 1 α T L Y 1φ(Y α s ) ds dφ(ys α ) 1 T αt T φ (Y α s )g(y α s )dw s φ (Y α s )g(y α s )dw s ) + O Risk neutral α g(ys α )Λ(Ys α )ds in the drift of Yt α ( ) 1 additional term: α 1 T T g(y s α )Λ(Ys α )ds ( ) 1 α 37

38 Multiscale Stochastic Volatility Models The spot volatility is a function of two factors: σ t = f(y t,z t ) Y t is fast mean-reverting (ergodic on a fast time scale): dy t = 1 ε α(y t)dt + 1 ε β(y t )dw (1) t, < ε 1 Z t is slowly varying: dz t = δc(z t )dt + δ g(z t )dw (2) t, < δ 1 (y, z) will denote the initial point for (Y, Z) f is continuous with respect to z Local Effective Volatility: σ 2 (z) f 2 (,z) Φ Y 38

39 ε << T << 1/δ Under the risk neutral measure IP chosen by the market: dx t = rx t dt + f(y t, Z t )X t dw () t ( 1 dy t = ε α(y t) 1 ) β(y t )Λ(Y t, Z t ) ε ( dz t = δ c(z t ) ) δ g(z t )Γ(Y t, Z t ) dt + 1 ε β(y t )dw (1) t dt + δ g(z t )dw (2) t d < W (), W (1) > t = ρ 1 dt d < W (), W (2) > t = ρ 2 dt Λ and Γ: market prices of volatility risk 39

40 Pricing Equation { } P ε,δ (t, x, y, z) = IE e r(t t) h(x T ) X t = x, Y t = y, Z t = z ( 1 ε L + 1 ε L 1 + L 2 + δm 1 + δm 2 + P ε,δ (T, x, y, z) = h(x) ) δ ε M 3 P ε,δ = L = α y β2 2 y 2 L 1 = β (ρ 1 fx 2 x y Λ ) y L 2 = t + 1 ( 2 f2 x 2 2 x 2 + r x ) x M 1 = g M 2 = c z + g2 2 (ρ 2 fx 2 x z Γ ) z M 3 = ρ 12 βg 2 y z 2 z 2 4