How persistent and regular is really volatility? The Rough FSV model. Jim Gatheral, Thibault Jaisson and Mathieu Rosenbaum. Monday 17 th November 2014

Transcription

1 How persistent and regular is really volatility?. Jim Gatheral, and Mathieu Rosenbaum Groupe de travail Modèles Stochastiques en Finance du CMAP Monday 17 th November 2014

2 Table of contents 1 Elements on volatility modeling 2 The scaling of log-volatility increments 3 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data 4 Log-volatility prediction Variance prediction 5

3 Table of contents 1 Elements on volatility modeling

4 Main classes of volatility models Prices are often modeled as: dp t = P t (µ t dt + σ s dw s ) where the volatility process σ s is the most important ingredient of the model. Practitioners essentially use three classes of volatility models: Deterministic volatility (Black and Scholes 1973). Local volatility (Dupire 1994). Stochastic volatility (Heston 1993, Hull and White 1987 or Hagan et al. 2002). In term of regularity, in these models, the volatility is either very smooth or with a smoothness similar to that of a Brownian motion.

5 Long memory in volatility Definition A stationary process is said to be long memory if the sum of its autocovariance function is infinite: + t=1 A power law long memory Cov[σ t+x, σ x ] = +. Cov[σ t+x, σ x ] C t γ, with γ < 1, has been observed in volatility by Ding and Granger 1993 (using extra day data). Andersen, Bollerslev, Diebold and Labys 2001 and 2003 (using intra day data).

6 Fractional Brownian motion (I) To take into account the long memory property and to allow for a wider range of smoothness, some authors have introduced the fractional Brownian motion in volatility modeling. Definition The fractional Brownian motion (fbm) with Hurst index H is the only process W H to satisfy: Self-similarity: (W H at ) L = a H (W H t ). Stationarity: (W H t+h W H t ) L = (W H h ). Gaussianity with E[W H 1 ] = 0 and E[(W H 1 )2 ] = 1.

7 Fractional Brownian motion (II) Proposition For all ε > 0, W H is (H ε)-hölder. Proposition The moments of the increments of the fbm check Proposition E[ W H t+h W H t q ] = K q h Hq. If H > 1/2, the fbm presents long memory in the sense: Cov[W H t+1 W H t, W H 1 ] C t 2 2H.

8 Long memory volatility models Some models have been built using a fractional Brownian motion, with Hurst index H > 1/2 to reproduce the empirical long memory of the volatility: Comte and Renault 1998 (FSV model): d log(σ t ) = νdw H t + α(m log(σ t ))dt. Comte, Coutin and Renault 2012 where they define some sort of fractional CIR process.

9 Table of contents The scaling of log-volatility increments 1 Elements on volatility modeling 2 The scaling of log-volatility increments 3 4 5

10 Intraday volatility estimation The scaling of log-volatility increments We are interested in the dynamics of the (log)-volatility process. We will use two proxies of the spot volatility of a day. A 5 minutes realized variance estimation taken over the full trading day (8 hour). A one hour uncertainty zone variance estimator, see Robert and Rosenbaum When taking this kind of proxies one is not looking at a spot volatility but an integrated volatility. These issues are discussed in our paper. From now on, I will consider the realized variance estimation on the S&P on 3500 days.

11 The log-volatility The scaling of log-volatility increments Figure : log(σ t ) as a function of t, S&P. It is visually much rougher than a classical Brownian motion.

12 The scaling of log-volatility increments Measure of the regularity of the log-volatility The starting point of this work is to look at the scaling of the moments of the increments of the log-volatility: or its empirical value. m(, q) = E[ log(σ t+ ) log(σ t ) q ] Close to = 0, this should give us a indication of the regularity of the noise driving the volatility.

13 Scaling of the moments The scaling of log-volatility increments Figure : log(m(q, )) = ζ q log( ) + C q. This scaling is not only asymptotic as tends to zero but holds on a wide range of time scales.

14 Monofractality of the log-volatility The scaling of log-volatility increments Figure : ζ q = Hq with H = 0.14 (as a fbm of Hurst index H).

15 The scaling of log-volatility increments Distribution of the log-volatility increments Figure : The distribution of the log-volatility increments is close to Gaussian.

16 A geometric fbm model? The scaling of log-volatility increments These empirical findings suggest we model the log-volatility as a fractional Brownian motion: σ t = σe νw H t. However, this model is not stationary. In particular, the empirical autocovariance function of the (log-)volatility (which will be of interest) does not make much sense.

17 A geometric fou model The scaling of log-volatility increments We make it formally stationary by considering a fractional Ornstein-Uhlenbeck model for the log-volatility denoted by X t This process satisfies dx t = νdw H t X t = ν t + α(m X t )dt. e α(t s) dw H t + m where T r = 1/α is the reversion time scale that we take very long compared to the observation time scale. This model is a particular case of the FSV model. However, in strong contrast to FSV, we take H small and 1/α large. Thus we call our model Rough FSV (RFSV).

18 m(2, ) and the parameters of the FSV The scaling of log-volatility increments Figure : log(m(2, )) as a function of log( ) in our data (black) and in the FSV model (there is a closed formula) (blue). On real data, the scaling is not only valid as tends to zero but holds on a wide range of. In the FSV, the slope at the beginning of the graph is governed by the parameter H and then stationarity kicks in.

19 The scaling of log-volatility increments Behavior of our model at reasonable time scales When the reversion time scale becomes large (α 0), the mean reverting term is negligible and our model looks like a fbm. Proposition As α tends to zero, [ E sup t [0,T ] ] Xt α X0 α νw H 0. t

20 Table of contents Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data 1 Elements on volatility modeling 2 3 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data 4 5

21 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Autocorrelogram of the (log-)volatility in our model Proposition Let q > 0, t > 0, > 0. As α tends to zero, Cov[X α t, X α t+ ] = Var[X α t ] 1 2 ν2 2H + o(1). Proposition As α tends to zero, E[σ t+ σ t ] = e 2E[X t α]+2var[x t α] 2H ν2 e 2 + o(1). We will now check these relations on the data.

22 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Empirical autocorrelogram of the log-volatility Figure : Cov[log(σ t ), log(σ t+ )] as a function of 2H. This fits the predictions of our model.

23 Empirical autocorrelogram of the volatility Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Figure : log(e[σ t σ t+ ]) as a function of 2H. This fits the predictions of our model.

24 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data General remarks on taking empirical averages and ACF In any model, when replacing a theoretical average by an empirical average, one always makes the underlying assumption that an ergodicity time scale is much smaller that the observation time scale. In the RFSV model, the ergodicity time scale is of the order 1/α therefore it is not the case. For example, one cannot estimate m or α (empirically: 1/ˆα T ). However, on should be able estimate local properties of the model such as ν and H. On simulations, we can also empirically retrieve the stylized facts presented above about the various autocovariance functions.

25 Long memory in volatility Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data It is widely believed that the (log-)(squared-)volatility presents a power law long memory with γ < 1. Cov[σ x+t, σ x ] t + We will review two tests of this long memory property. We will show that they wrongly deduce a power law long memory on data generated from our model and are thus too fragile. k t γ

26 Log-log autocovariance of the volatility. Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Figure : log(cov[σ x+, σ x ]) as a function of log( ). The autocorrelation function does not behaves as a power law function as it is often stated.

27 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Scaling of the variance of the cumulated volatility Figure : V ( ) = Var( t=1 σ t) as a function of log( ) on empirical (above) and simulated (below) data. Power law long memory implies that it should behave as 2 γ as we observe on the data and the model.

28 Fractional differentiation of the log-volatility Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Figure : ACF of the log-volatility (blue) and of the fractionally integrated log-volatility ε = (1 L) d log(σ), with d = 0.4 (green) on empirical (above) and simulated (below) data.

29 Multiscaling in finance Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data An important property of financial time series is their multiscaling behavior, see Mantegna and Stanley 2000 and Bouchaud and Potters That is, one observes essentially the same law whatever the time scale. In particular, there are periods of high and low market activity at different time scales. Very few models mathematically reproduce this property, see Bacry, Delour and Muzy 2003.

30 Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Figure : Empirical volatility over 10, 3 and 1 years.

31 Our model on different time intervals Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Figure : Simulated volatility over 10, 3 and 1 years. We observe the same alternations of periods of high market activity with periods of low market activity.

32 Apparent multiscaling in our model Autocorrelograms in the RFSV model and on the data Long memory in volatility? Multiscaling in our model and on the data Let denote L H,ν the law on [0, 1] of the process e νw H t. Then the law of the volatility process on [0, T ] renormalized on [0, 1] : σ tt /σ 0 is L H,νT H. If one observes the volatility on T = 10 years (2500 days) instead of T = 1 day, the volatility parameter of the law of the volatility is only multiplied by 2500 H 3. Therefore, one observes loosely the same properties on a very wide range of time scales. The roughness of the volatility process (H = 0.14) implies a multiscaling behavior of the volatility.

33 Table of contents Log-volatility prediction Variance prediction 1 Elements on volatility modeling Log-volatility prediction Variance prediction 5

34 Log-volatility prediction Variance prediction Prediction of a fractional Brownian motion There is a nice prediction formula for the fractional Brownian motion. Proposition (Nuzman and Poor 2000) For H < 1/2 E[Wt+ H F t] = cos(hπ) t H+1/2 π Ws H ds. (t s + )(t s) H+1/2

35 Our prediction formula Log-volatility prediction Variance prediction We apply the previous formula to the prediction of the log-volatility: E [ log σ 2 t+ F t ] cos(hπ) t = H+1/2 log σs 2 ds π (t s + )(t s) H+1/2 or more precisely its discrete version E [ log σ 2 t+ F t ] = cos(hπ) π H+1/2 N k=0 We compare it to usual predictors using the criterion log σ 2 t k (k + + 1/2)(k + 1/2) H+1/2. P = N k=1 ( log(σ 2 k+ ) log(σk+ 2 ))2 N k=1 (log(σ2 k+ ). E[log(σ2 t+ )])2

36 Log-volatility prediction Variance prediction AR(5) AR(10) HAR(3) RFSV SPX2.rv = SPX2.rv = SPX2.rv = FTSE2.rv = FTSE2.rv = FTSE2.rv = N2252.rv = N2252.rv = N2252.rv = GDAXI2.rv = GDAXI2.rv = GDAXI2.rv = FCHI2.rv = FCHI2.rv = FCHI2.rv =

37 Regression window and horizon Log-volatility prediction Variance prediction After a simple change of variable, the prediction of the log-volatility can be written: E[log(σ 2 t+ ) F t] = cos(hπ) π + 0 log(σ 2 t u ) (u + 1) u H+1/2 du. The only time scale that appears in the above regression is the horizon. As it is known by practitioners: If trying to predict volatility one week ahead, one should essentially look at the volatility over the last week. If trying to predict the volatility one month ahead, one should essentially look at the volatility over the last month.

38 Log-volatility prediction Variance prediction Conditional distribution of the fractional Brownian motion and prediction of the variance Proposition (Nuzman and Poor 2000) In law, W t+ F t = N (E[W t+ F t ], c 2H ) with c = sin(π(1/2 H))Γ(3/2 H)2. π(1/2 H)Γ(2 2H) Therefore, our predictor of the variance writes: E[σ 2 t+ F t] = e E[log(σ2 t+ ) Ft]+2ν2 c 2H.

39 Log-volatility prediction Variance prediction AR(5) AR(10) HAR(3) RFSV SPX2.rv = SPX2.rv = SPX2.rv = FTSE2.rv = FTSE2.rv = FTSE2.rv = N2252.rv = N2252.rv = N2252.rv = GDAXI2.rv = GDAXI2.rv = GDAXI2.rv = FCHI2.rv = FCHI2.rv = FCHI2.rv =

40 Table of contents 1 Elements on volatility modeling

41 Introducing smile in our model An fbm can be written as the stationary fractional integration of a Brownian motion (Mandelbrot and Van Ness 1968) t Wt H = 0 dw 0 [ s (t s) γ + 1 (t s) γ 1 ( s) γ ] dw s. We can introduce smile by (anti-)correlating the BM driving the price with the BM driving the fbm of the volatility. Fukasawa (2010) showed that when doing so we have a smile which decreases as 1/T H 1/2 for small maturities as observed on the data. There is no need to introduce jumps to reproduce this stylized fact as it is often stated.

42 Hawkes processes as models for the order flow The starting point of our microstructural analysis is the modeling of the order flow though Hawkes processes. A Hawkes process (N t ) t 0 is a self exciting point process, whose intensity at time t, denoted by λ t, is of the form λ t = µ + φ(t J i ) = µ + φ(t s)dn s, 0<J i <t (0,t) where µ is a positive real number, φ a regression kernel and the J i are the points of the process before time t. These processes have been introduced in 1971 by Hawkes in the purpose of modeling earthquakes and their aftershocks and are nowadays very popular in finance.

43 Hawkes processes in practice When trying to calibrate such models on high frequency data, two main phenomena almost systematically occur: The L 1 norm of φ close to one high degree of endogeneity of the market due to high frequency trading, see Bouchaud et al. 2013, Filimonov and Sornette The function φ has a power law tail metaorders splitting.

44 Limit theorems for nearly unstable Hawkes processes We show that the macroscopic scaling limit of Hawkes processes with power law tail and kernel with L 1 norm close to one can be seen as an integrated fractional process, with Hurst parameter H smaller than 1/2. This means that at large sampling scales, the dynamics of the cumulated order flow is well approximated by an integrated fractional process with H < 1/2.

45 Agent based explanation for the behavior of the volatility There is a linear relationship between cumulated order flow and integrated variance. Thus endogeneity of the market together with order splitting lead to a superposition effect which explains (at least partly) the rough nature of the observed volatility.

46 Conclusion The log-volatility is empirically monofractal. We built a model where it is close to a fractional Brownian motion. This model reproduces the empirical structure of the autocorrelation of the volatility. This model sheds new light on the supposed long memory in the volatility. It also reproduces the apparent multiscaling of the volatility. The simple predictors of the (log-)variance issued from this model outperforms classical predictors using only one parameter H!

47 Thank you for your attention!