Fractional Stochastic Volatility Models

Transcription

1 Fractional Stochastic Volatility Models Option Pricing & Statistical Inference Alexandra Chronopoulou Industrial & Enterprise Systems Engineering University of Illinois, Urbana-Champaign May 21, 2017 Conference for the 10 th Anniversary of the Center for Financial Mathematics & Actuarial Research

2 Outline Introduction to fractional Stochastic Volatility models Pricing with fractional Stochastic Volatility models Inference for fractional Stochastic Volatility models Summary & Conclusion 1/61

3 A well-known Stochastic Volatility Model Log-Returns: {Y t, t [0, T ]} ( ) dy t = r σt 2 2 dt + σ t dw t. σ = σ(x t) is a deterministic function and X t is modeled by a diffusion driven by noise other than W t, such as a Mean Reverting OU: with dw t, dz t = ρ dt. dx t = α (m X t) dt + β dz t 2/61

4 Volatility is LRD or Rough? Long-Range Dependent/ Persistent Volatility It is shown in Willinger and Taqqu, Bayraktar, Poor and Sircar, Hall, Härdle, Kleinow and Schmidt that S& P 500 exhibits a slowly decaying autocorrelation function, with decay of order (dt) 2H with H > 1/2. In C. and Viens, it is shown that when calibrating a fractional stochastic volatility model, the value of the Hurst parameter that makes the calculated option price better match the realized prices to be close to /61

5 Volatility is LRD or Rough? ACF of S&P 500 Squared Log-Returns 4/61

6 Volatility is LRD or Rough? Rough/ Anti-Persistent Volatility It is shown in Gatheral et al. that log-volatility time series behave in fact like a fractional Brownian motion, with Hurst parameter of order 0.1. From Alos, Fukasawa and Bayer et al., we know that such model also enables us to reproduce very well the behavior of the implied volatility surface, in particular the at-the-money skew (without jumps). 5/61

7 Volatility is LRD or Rough? Rough Volatility: SPX (top) - Simulated Rough Model (bottom) Reference: Gatheral et al. (2014) 6/61

8 Fractional Stochastic Volatility Models Log-Returns: {Y t, t [0, T ]} ( ) dy t = r σt 2 2 dt + σ t dw t. where σ t = σ(x t) and X t is described by: dx t = α (m X t) dt + β db H t B H t is a fractional Brownian motion with Hurst parameter H (0, 1). X t is a fractional Ornstein-Uhlenbeck process (fou). 7/61

9 Fractional Brownian Motion Definition A centered Gaussian process B H = {Bt H, t 0} is called fractional Brownian motion (fbm) with selfsimilarity parameter H (0, 1), if it has the following covariance function E ( ) } Bt H Bs H 1 = {t 2H + s 2H t s 2H. 2 and a.s. continuous paths. When H = 1 2, BH is a standard Brownian motion. 8/61

10 Properties of fbm The increments of fbm, {B n B n 1} n N, are stationary, i.e. E[(B n B n 1) (B n+h B n+h 1 )] = γ(h). H selfsimilar, i.e. c H (B n B n 1) D (B c n B c(n 1) ) dependent When H > 1, the increments exhibit long-memory, i.e. 2 E [B 1 (B n B n 1 )] = +. When H < 1, the increments exhibit antipersistence, i.e. 2 E [B 1 (B n B n 1 )] < +. 9/61

11 Properties of fbm FBM... has a.s. Hölder-continuous sample paths of any order γ < H. its 1 -variation on [0, t] is finite. In particular, fbm has an infinite H quadratic variation for H < 1/2. When H 1 2, BH t is not a semimartingale. 10/61

12 Representation of fbm on an Interval where B H (t) = t 0 K H (t, s)d W s, { W t; t 0} is a standard Brownian motion K H (t, s) is a non-random kernel ( with c H = t K H (t, s) = c H s 1 2 H (u s) H 3 2 u H 1 2 du H(2H 1) 1/2. β(2 2H;H 1/2)) s 11/61

13 Fractional Stochastic Volatility Model { ( ) dy t = r σ2 (X t ) 2 dt + σ(x t) dw t, dx t = α (m X t) dt + β db H t Some Properties Hölder continuity: Hölder continuous of order γ, for all γ < H. Self-similarity: Self-similar in the sense that {B H ct; t R} D {c H B H t ; t R}, c R This property is approximately inherited by the fou, for scales smaller than 1/α. 12/61

14 Literature : Long Memory Stochastic Volatility Comte and Renault (1996) Comte, Coutin and Renault (2010) C. and Viens (2010, 2012) Gulisashvili, Viens and Zhang (2015) Guennoun, Jacquier, and Roome (2015) Garnier & Solna (2015, 2016) Bezborodov, Di Persio & Mishura (2016) Fouque & Hu (2017) 13/61

15 Literature: Rough Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) Bayer, Friz, and Gatheral (2015) Forde, Zhang (2015) El Euch, Rosenbaum (2016, 2017) 14/61

16 Outline Introduction to fractional Stochastic Volatility models Pricing with fractional Stochastic Volatility models Inference for fractional Stochastic Volatility models Summary & Conclusion 15/61

17 Pricing in classical SV models Let x t = ln ( S t/ke r(t t)). Conditionally on F t (i.e. the filtration generated by W t and Z t up to t) and conditionally on the volatility path {σ t; 0 t T }, we can compute [ ( C t = S t {E Q xt t Φ + U )] [ ( t,t e xt E Q xt t Φ U )]} t,t U t,t 2 U t,t 2 T where U t,t = σudu. 2 t 16/61

18 Natural extension in a fractional SV model For x t = ln ( S t/ke r(t t)) defined as before. Conditionally on F t (i.e. the filtration generated by W t and Bt H up to t) and conditionally on the volatility path {σ t; 0 t T }, we can compute C t = S t {E Q t [ Φ ( xt U t,t T where U t,t = σudu. 2 (Comte & Renault (1998)) t + U )] t,t e xt E Q t 2 [ Φ ( xt U t,t U )]} t,t 2 17/61

19 An integral transformation of FBM Observed process: X t = B H t, t 0. Integral transformation: Z t := t 0 k H (t, s) dx s, t 0, where k H (t, s) := c 1 H s 1 2 H (t s) 1 2 H and c H := 2H Γ ( 3 2 H) Γ ( H + 1 2). Theorem (Norros, Valkeila, Virtamo) The integral transformation is 1 1, thus the filtration generated by X t and the filtration generated by Z t coincide. In fact, Z t is a martingale. 18/61

20 An integral transformation of FBM with drift Observed process: X t = B t + t θs ds, t 0. τ Decomposition of Z t: with t t Z t = k H (t, s) db s + Q s dm s 0 0 Q t := d t k H (t, s) θ s ds, t 0. dm t 0 Theorem (Kleptsyna & Le Breton) The integral transformation is 1 1, thus the filtration generated by X t and the filtration generated by Z t coincide and Z t is a semimartingale. 19/61

21 Discrete Time Analog Observations: X 1, X 2,... Discrete transformation: Z n := n k(n, m)x m, t 0, m=1 where k(n, m) is a deterministic kernel. k(n, m) is invertible with inverse K(n, m). Both matrices are obtained from the Choleski decomposition of the covariance and inverse of covariance matrices of the process, i.e. Γ n = K T n DnKn, Γ 1 = kn T D 1 n kn Proposition (Brouste & Kleptsyna) The discrete transformation is 1 1, thus the filtration generated by the two processes coincide. 20/61

22 Option Pricing Estimate the empirical stochastic volatility distribution. Construct a multinomial recombining tree to compute the option prices or use a Monte-Carlo type approach. 21/61

23 Simulated Example Option Price Strike Price 22/61

24 Outline Introduction to fractional Stochastic Volatility models Pricing with fractional Stochastic Volatility models Inference for fractional Stochastic Volatility models Summary & Conclusion 23/61

25 Which parameters to estimate? { ( ) dy t = µ σ2 (X t ) 2 dt + σ(x t) dw t, dx t = α (m X t) dt + β db H t We can also assume that Corr(B H t, W t) = ρ (leverage effects). Parameters to estimate: θ = (α, m, β, µ, ρ) and H. 24/61

26 Estimation of parameters given H { ( ) dy t = µ σ2 (X t ) 2 dt + σ(x t) dw t, dx t = α (m X t) dt + β db H t Remark The estimation of H is decoupled from the estimation of the drift components, but not from the estimation of the diffusion terms. Methods Extension of classical methods Sequential Monte Carlo method Discrete time Kalman filtering 25/61

27 Non-Markovian dynamic models Denote θ the vector of all parameters, except for H. Observation equation State equation Initial distribution f (Y t X t; θ) f (X t X t 1,..., X 1; θ) f (X 0 θ) 26/61

28 Filtering for states & parameters Non-Markovian Case f (X 1:t ; θ Y 1:t ) f (X 1:t, Y 1:t, θ) t f (X 1 ) f (X 2 X 1 ; θ)... f (X n X n 1,..., X 1 ; θ) f (Y i X i ; θ) f (θ Y t), where f (θ Y t) is a prior density for the parameter vector θ. i=1 If the parameter is known, then the density is degenerate. Otherwise, we need to compute or approximate the theoretical density function f (θ Y t). 27/61

29 Learning θ offline Two step strategy: Approximate f (θ Y ) by f N (θ Y ) = f N (Y θ)f (θ) p(y) f N (Y θ)f (θ) where f N (Y θ) is a SMC approximation to f (Y θ). Sample θ via an MCMC scheme or an SIR scheme. Disadvantages SMC looses its appealing sequential nature. Overall sampling scheme is sensitive to f N (Y θ) 28/61

30 Learning θ sequentially Filtering for states and parameter(s): Learning X t and θ sequentially. Posterior at t : f (X t θ, Y t) f (θ Y t) Prior at t + 1 : f (X t+1 θ, Y t) f (θ Y t) Posterior at t + 1 : f (X t+1 θ, Y t+1) f (θ Y t+1) Advantages Sequential updates of f (θ Y t), f (X t Y t) and f (θ, X t Y t). Sequential h-step ahead forecasts f (Y t+h Y t) Sequential approximations for f (Y t Y t 1). 29/61

31 Artificial Evolution of θ Gordon et al (1993) θ t+1 = θ t + ζ t+1 ζ t+1 N (0, W t+1) West et al. (1993, 1998) t update f (θ Y t) Compute a Monte Carlo approximation of f (θ Y t), by using samples θ (j) t and weights w (j) t. Smooth kernel density approximation f (θ Y t) N j=1 ω (j) t N (θ m (j) t, h 2 V t) 30/61

32 Parameter Learning Algorithm At time t = 1 Sample X (i) 1,1 q 1( ) and θ (i) (1) f 1( ). ( Compute the weights w 1 X (i) 1,1 ; θ(i) (1) normalize W (i) 1 = w (i) N 1 ) f (θ (i) (1) ) µ (X (i). i=1 w (i) 1 Resample to obtain X (i) 1,1 N j=1 W (j) 1 δ X (j) (dx 1 ). (1,1) 1,1 θ(i) (1) ( (i) q 1 X 1,1 ) ) f (y 1 X (i) 1,1 ) and 31/61

33 Parameter Learning Algorithm At time t, t 2 (step t 1 t) Set and sample X (i) t,1:t 1 = X (i) t 1,1:t 1, θ (i) (t) N (m(i) t 1 h2 V t 1 ), m (i) t 1 = αθ(i) (t 1) + (1 α) θ (t 1) ( ) X (i) t,t qt (i) X 1:t 1,t 1 ; θ(i) (t) ) ( Compute the weights w (i) t = w t X (i) (i) 1:t 1,t 1, X t,t ; θ(i) (t) Resample X 1:t,t πt N(dx 1:t), where πt N(dx N 1:t) = j=1 W (j) t and set θ N (t) = i=1 W (i) t θ (i) (t). and normalize. δ (j) X 1:t 1,t 1, X (j) t,t (dx 1:t ) 32/61

34 Parameter Learning Algorithm Output The filtering distribution f (dx 1:t y 1:t) is approximated by or π N (dx 1:t) = and the estimator for θ is N j=1 π N (dx 1:t) = 1 N θ (t) = W (j) t δ (j) X N i=1 N j=1 1:t 1,t 1, X (j) (dx 1:t) t,t δ (j) X (dx 1:t) 1:t,t W (i) t θ (i). (t) 33/61

35 Filter convergence Let φ : X R be an appropriate test function and assume that we want estimate φ t = φ t(x 1:t)p(x 1:t, θ (t) Y 1:t)dx 1:tdθ (t). The SISR algorithm provides us with the estimator ˆφ N t = φ t(x 1:t)π N (dx 1:t) = CLT for the filter (C. and Spiliopoulos) N ( ˆφN t φ t ) N ( 0, σ 2 (φ t) ) N i=1 ( ) Wt i φ t X (i) 1:t 1,t 1, X (i) t,t as N. 34/61

36 Convergence of the Parameter θ N (t) = N i=1 with m N t 1 = αθ(n,i) (t 1) + (1 α) θ N (t 1), V N t 1 = 1 N 1 CLT for the parameter Assuming that E π θ t W t 2+δ < : W (i) t θ (N,i) (t), where θ (N,i) (t) N (m (N,i) t 1 h2 Vt 1) N N i=1 (W (i) t 1 θ(n,i) (t 1) θ N (t 1)) 2. N ( θ(n) (t) θ (t) ) N ( 0, σ 2 (θ (t) ) ), as N Moreover, if the model P θ is identifiable, then the posterior mean θ (t) consistently estimates the true parameter value θ, as t. 35/61

37 Challenges Computational complexity increases significantly as N increases. Take advantage of the exact solution of the fractional OU and sample directly from its distribution. Sample impoverishment occcurs as in the usual case. Choice of appropriate importance density is crucial. 36/61

38 Real Data Example S& P 500 Data: 252 observations, starting in January 2010 until December 2010 Fractional Ornstein-Uhlenbeck Model { ( ) dy t = µ σ2 (X t ) dt + σ(x 2 t) dw t, dx t = α (m X t) dt + β db H t The long-memory parameter H for the particular data set is estimated to be 0.55 using the GPH (Geweke and Porter-Hudak) method. We apply the SMC algorithm to estimate: the unobserved volatility distribution, and the remaining unknown parameters of the model, µ, α, m, β. 37/61

39 Output 1: Volatility Particle Filter Density /61

40 Output 2: Parameter Estimators alpha mu (a) Estimator of µ (b) Estimator of α sigma m (c) Estimator of m (d) Estimator of β 39/61

41 Model Validation: 1-Step Ahead Prediction 40/61

42 Model Validation: Residuals (a) Residuals (b) ACF of Residuals 41/61

43 Leverage Effects Leverage Effect: The phenomenon of (negative) correlation between the stock returns and the volatility. Mathematical Definition The stochastic parameter of the contemporaneous leverage effect is defined as the quadratic co-variation between Y t and F (σ t): Y, F (σ 2 ) where x F (x) is a twice differentiable and monotone on (0, ). A typical choice for F is F (x) = log x. 42/61

44 Quadratic Covariation Definition Mathematically, we define the quadratic covariation to be Y, F (σ 2 ) = 2 ρ β T 0 2F (σ 2 t )σ 2 t (dt) H+ 1 2 It is easy to see this result is true using Itô s formula for fbm: dσ 2 t = 2σt dσt + β2 t 2H 1 dt df (σ 2 t ) = F (σ 2 t ) dσ 2 t β2 F (σ 2 t ) t 2H 1 dt d Y, F (σ 2 t ) = 2 β σ2 t F (σ 2 t ) dwtdbh t 43/61

45 Discrete Quadratic Covariation Assume we obtain equidistant observations of Y t at times 0 = t n,0 < t n,1 <... < t n,n = T. Define the discrete quadratic covariation as follows: n 1 ( ) ( ( ) ( )) Y, F (σ 2 ) n = 2 Ytn,i+1 Y tn,i F σ 2 tn,i+1 F σ 2 tn,i i=0 n 1 := i=0 v i 44/61

46 Consistency Theorem For processes Y and σ defined as before: 1 lim n n(h+ 2 ) Y, F (σ2 ) n = lim n n(h+ 1 2 ) = Y, F (σ 2 ) n i=1 v i almost surely. 45/61

47 Sketch of Proof 1. Discretize the processes Y and F (σ 2 ) to compute the increment. Using an Euler discretization of the fractional SDEs, we have that For Y : Y ti+1 Y ti = (µ 12 ) σ2ti t i + σ ti W ti For F (σ 2 ): ( ) ( ) F σ 2 (t i+1 ) F σ 2 (t i ) [ ( 2ασ ) =2 β σ ti F (σt 2 i ) B H 2 (t i ) + t i + β 2 t 2H 1 i F (σt 2 i ) + 1 ] 2 β2 t 2H 1 i F (σt 2 i ) t i, where t i = t i+1 t i, W (t i) = W ti+1 W ti and B H t i = B H t i+1 B H t i. 46/61

48 Sketch of Proof 2. Approximate the sample quadratic co-variation. Substitution leads to v i = { } ( µ 1 ) 2 σ2 t i t i + σ ti W ti { [( 2ασ ) 2 t i + β 2 t 2H 1 i F (σt 2 i ) + 1 ] 2 β2 t 2H 1 i F (σt 2 i ) t i + 2 β σ ti F (σ 2 t i ) B H t i } 47/61

49 Discrete Quadratic Covariation Y, F (ˆσ 2 ) n = 2 := ˆσ 2 τ n,i = K n 2 i=0 K n 2 v i i=0 1 M n t ( Yτn,i+1 Y τn,i τ n,j (τ n,i,τ n,i+1 ] ) ( F (ˆσ 2 τn,i+1 ) F (ˆσ 2 τn,i )) ( Yτn,j+1 2 Y τn,j + Y τn,j 1) 2 48/61

50 Variations-based Estimator The estimator of ρ is defined as ˆρ n = Y, F (ˆσ2 ) n {2β T 0 2F (ˆσ 2 t )ˆσ 2 t (dt) H+ 1 2 } 1 When F (x) = log x, the estimator of ρ is simplified to ˆρ n = Y, F (ˆσ 2 ) n 2βT H /61

51 Consistency Proposition For the process ˆσ as before, we have lim n M n ˆσ 2 n = Y, Y Theorem The variations-based estimator for the correlation coefficient ρ is strongly consistent, that is lim ˆρ n = ρ a.s. n 50/61

52 Asymptotic Normality Theorem By choosing K n = [ n M n ] and Mn = [c n] as before, and letting X n = n 4 1 (H+ 1 2 ) ˆρ n ρ, Var(ˆρn) we have that X n D (0, 1), or in other words the estimator of the correlation coefficient ρ, ˆρ is asymptotically Normal, as n. This is a generalization of the result in the case of H = 1/2 (Mykland & Wang). 51/61

53 Sketch of Proof (Asymptotic Normality) We have X n = n 1 4 (H+ 1 2 ) ˆρ n ρ. Var(ˆρn) Let D denote the Malliavin derivative and L the OU semigroup operator. We, then, compute G Xn = DX n, D(L 1 X n) L 2 Finally, we show that based on which we conclude G Xn L1 (Ω) 1. X n N(0, 1), as n. 52/61

54 Sampling Frequencies We have the two following sampling frequencies: Sampling frequency: Subsampling frequency: [ ] n K n = M n M n = [c n] Rate of Convergence: ( ) ( ) K 1 2 (H+ 2 1 ) 1 n 2 (H+ 1 2 n = ) 1 n 2 (H+ 2 1 ) = M n c = c n 4 1 (H+ 1 2 ) n 53/61

55 Controlling Sampling Frequencies Remarks We can control the rate of convergence, by properly choosing K n and M n. Specifically, if we choose M n = [c n α(h) ], this implies that ( ) 2 K 1 (H+ 1 2 ) 1 n 2 (H+ 1 ( ) 2 n = ) 1 n = M n c n α(h) = c n α(h) ( H 2 +1) 2 (H+ 1 2 ) For example, by choosing α(h) = c (H ) 1 we obtain K 1 2 (H+ 1 2 ) n = c n c. 54/61

56 Simulation Study We simulated the following model: { ( ) dy t = µ σt 2 2 dt + σ t dw t dσ t = α (m σ t) dt + β dbt H with α = 0.03, m = 1, β = 0.2, µ = For the simulations, we considered H = 0.55, 0.65, 0.75, We used an equidistant discretization scheme with N = 1000 points. 55/61

57 Histogram of ˆρ n Frequency rho_hat 56/61

58 Asymptotic Variance of ˆρ n 57/61

59 Time Series of Leverage Effect 58/61

60 Outline Introduction to fractional Stochastic Volatility models Pricing with fractional Stochastic Volatility models Inference for fractional Stochastic Volatility models Summary & Conclusion 59/61

61 Summary We discussed pricing under a fractional stochastic volatility framework. We employed a Sequential Monte Carlo Algorithm for the estimation of unknown parameters in a LMSV model. We defined and estimated leverage effects in a LMSV setup. 60/61

62 Thank you! 61/61