Rough volatility models: When population processes become a new tool for trading and risk management

Transcription

1 Rough volatility models: When population processes become a new tool for trading and risk management Omar El Euch and Mathieu Rosenbaum École Polytechnique 4 October 2017 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 1

2 Table of contents 1 Introduction 2 3 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 2

3 Table of contents 1 Introduction 2 3 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 3

4 One financial asset, 2 years Bund Time. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 4

5 A celebrated model The Black-Scholes model (1973) The most famous model in quantitative finance for a stock price : ds t = S t σdw t. Pros and cons Fractal property of Brownian motion, as that observed on stock prices (up to some scale). Semi-martingale process no arbitrage. Very simple, log normal model explicit formulas for many quantities, including prices of complex financial products. Risk of financial product can be perfectly hedged. But not realistic : A constant volatility coefficient σ is not consistent with neither historical data (stock prices data), nor implied data (complex products price data). Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 5

6 A well-know stochastic volatility model The Heston model (1993) A very popular stochastic volatility model for a stock price is the Heston model : ds t = S t Vt dw t dv t = λ(θ V t )dt + λν V t db t, dw t, db t = ρdt. Popularity of the Heston model Reproduces several important features of low frequency price data : leverage effect, time-varying volatility, fat tails,... Provides quite reasonable dynamics for the volatility surface. Explicit formula for the characteristic function of the asset log-price very efficient model calibration procedures. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 6

7 But...Volatility is rough! Figure : The log volatility log(σ t ) as a function of t, S&P, 3500 days. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 7

8 Rough volatility Volatility as a rough fractional process In Heston model, volatility follows a Brownian diffusion. 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. More precisely, basically all the statistical stylized facts of volatility are retrieved when modeling it by a rough fractional Brownian motion. A universal finding established for more than 5000 assets. Not aware of any reasonable asset for which volatility is not rough. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 8

9 Fractional Brownian motion (I) Definition The fractional Brownian motion with Hurst parameter H is the only process W H to satisfy : Self-similarity : (W H at ) L = a H (W H t ). Stationary increments : (W H t+h W H t ) L = (W H h ). Gaussian process with E[W H 1 ] = 0 and E[(W H 1 )2 ] = 1. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 9

10 Fractional Brownian motion (II) Proposition For all ε > 0, W H is (H ε)-hölder a.s. Proposition The absolute moments satisfy E[ W H t+h W H t q ] = K q h Hq. Mandelbrot-van Ness representation t Wt H = 0 dw 0 ( s (t s) H 1 (t s) H ( s) 1 2 H ) dw s. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 10

11 Evidence of rough volatility Volatility of the S&P Everyday, we estimate the volatility of the S&P at 11am (say), over 3500 days. We study the quantity m(, q) = E[ log(σ t+ ) log(σ t ) q ], for various q and, the smallest being one day. In the case where the log-volatility is a fractional Brownian motion : m(, q) c qh. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 11

12 Example : Scaling of the moments Figure : log(m(q, )) = ζ q log( ) + C q. The scaling is not only valid as tends to zero, but holds on a wide range of time scales. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 12

13 Example : Monofractality of the log-volatility Figure : Empirical ζ q and q Hq with H = 0.14 (similar to a fbm with Hurst parameter H). Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 13

14 Modifying Heston model Rough Heston model It is natural to modify Heston model and consider its rough version : ds t = S t Vt dw t V t =V 0 + t (t s) H 1/2 λ(θ V s )ds + λν 0 with dw t, db t = ρdt. t (t s) H 1/2 V s db s, 0 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 14

15 Pricing in Heston models Classical Heston model From simple arguments based on the Markovian structure of the model and Ito s formula, we get that in the classical Heston model, the characteristic function of the log-price X t = log(s t /S 0 ) satisfies E[e iaxt ] = exp ( g(a, t) + V 0 h(a, t) ), where h is solution of the following Riccati equation : t h = 1 2 ( a2 ia)+λ(iaρν 1)h(a, s)+ (λν)2 h 2 (a, s), h(a, 0) = 0, 2 and g(a, t) = θλ t 0 h(a, s)ds. Rough Heston models Pricing in rough Heston models is much more intricate : no efficient Monte-Carlo, no Ito calculus, no Markov property... Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 15

16 In this presentation Our goal Understanding why volatility is rough. Deriving a Heston like risk management formula in the rough case. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 16

17 Table of contents 1 Introduction 2 3 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 17

18 Microstructural foundations High frequency finance We wish to model typical behaviors of market participants at the high frequency scale (time horizon of an hour or a day). We want to investigate the consequences of these behaviors on the long run, in particular on the volatility. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 18

19 One financial asset, 1 hour value time. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 19

20 Building the model Necessary conditions for a good microscopic price model We want : A tick-by-tick model. A model reproducing the stylized facts of modern electronic markets in the context of high frequency trading. A model helping us to understand the rough dynamics of the volatility from the high frequency behavior of market participants. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 20

21 Building the model Stylized facts 1-2 Markets are highly endogenous, meaning that most of the orders have no real economic motivations but are rather sent by algorithms in reaction to other orders, see Bouchaud et al., Filimonov and Sornette. Mechanisms preventing statistical arbitrages take place on high frequency markets, meaning that at the high frequency scale, building strategies that are on average profitable is hardly possible. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 21

22 Building the model Stylized facts 3-4 There is some asymmetry in the liquidity on the bid and ask sides of the order book. In particular, a market maker is likely to raise the price by less following a buy order than to lower the price following the same size sell order, see Brennan et al., Brunnermeier and Pedersen, Hendershott and Seasholes. A large proportion of transactions is due to large orders, called metaorders, which are not executed at once but split in time. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 22

23 Building the model Hawkes processes Our tick-by-tick price model is based on Hawkes processes in dimension two, very much inspired by the approaches in Bacry et al. and Jaisson and R. A two-dimensional Hawkes process is a bivariate point process (N + t, N t ) t 0 taking values in (R + ) 2 and with intensity (λ + t, λ t ) of the form : ( ) ( λ + t µ + = λ t µ ) t + 0 ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns ). Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 23

24 Building the model The microscopic price model Our model is simply given by P t = N + t N t. N t + corresponds to the number of upward jumps of the asset in the time interval [0, t] and Nt to the number of downward jumps. Hence, the instantaneous probability to get an upward (downward) jump depends on the location in time of the past upward and downward jumps. By construction, the price process lives on a discrete grid. Statistical properties of this model have been studied in details. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 24

25 Encoding the stylized facts The right parametrization of the model Recall that ( ) ( λ + t µ + = λ t µ ) t + 0 ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns High degree of endogeneity of the market L 1 norm of the largest eigenvalue of the kernel matrix close to one. No arbitrage ϕ 1 + ϕ 3 = ϕ 2 + ϕ 4. Liquidity asymmetry ϕ 3 = βϕ 2, with β > 1. Metaorders splitting ϕ 1 (x), ϕ 2 (x) ). x K/x 1+α, α 0.6. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 25

26 About the degree of endogeneity of the market L 1 norm close to unity For simplicity, let us consider the case of a Hawkes process in dimension 1 with Poisson rate µ and kernel φ : λ t = µ + φ(t s)dn s. (0,t) N t then represents the number of transactions between time 0 and time t. L 1 norm of the largest eigenvalue close to unity L 1 norm of φ close to unity. This is systematically observed in practice, see Hardiman, Bercot and Bouchaud ; Filimonov and Sornette. The parameter φ 1 corresponds to the so-called degree of endogeneity of the market. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 26

27 About the degree of endogeneity of the market Population interpretation of Hawkes processes Under the assumption φ 1 < 1, Hawkes processes can be represented as a population process where migrants arrive according to a Poisson process with parameter µ. Then each migrant gives birth to children according to a non homogeneous Poisson process with intensity function φ, these children also giving birth to children according to the same non homogeneous Poisson process, see Hawkes (74). Now consider for example the classical case of buy (or sell) market orders. Then migrants can be seen as exogenous orders whereas children are viewed as orders triggered by other orders. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 27

28 About the degree of endogeneity of the market Degree of endogeneity of the market The parameter φ 1 corresponds to the average number of children of an individual, φ 2 1 to the average number of grandchildren of an individual,... Therefore, if we call cluster the descendants of a migrant, then the average size of a cluster is given by k 1 φ k 1 = φ 1/(1 φ 1 ). Thus, the average proportion of endogenously triggered events is φ 1 /(1 φ 1 ) divided by 1 + φ 1 /(1 φ 1 ), which is equal to φ 1. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 28

29 The scaling limit of the price model Limit theorem After suitable scaling in time and space, the long term limit of our price model satisfies the following rough Heston dynamics : P t = t 0 Vs dw s 1 t V s ds, 2 t V t = V 0 + c H (t s) H 1 2 λ(θ Vs )ds + c H λν with 0 d W, B t = 0 t (t s) H 1 2 Vs db s, 1 β dt and H = α 1/2. 2(1 + β 2 ) 0 Hence stylized facts of modern market microstructure naturally give rise to rough volatility! Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 29

30 Table of contents 1 Introduction 2 3 Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 30

31 Deriving the characteristic function Strategy We derive characteristic functions for our microscopic Hawkes-based price model model. We then pass to the limit. Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 31

32 We write : I 1 α f (x) = 1 x Γ(1 α) 0 f (t) (x t) α dt, Dα f (x) = d dx I 1 α f (x). Theorem The characteristic function at time t for the rough Heston model is given by ( t exp g(a, s)ds + V ) 0 0 θλ I 1/2 H g(a, t), with g(a, ) the unique solution of the fractional Riccati equation : D H+1/2 g(a, s) = λθ 2 ( a2 ia) + λ(iaρν 1)g(a, s) + λν2 2θ g 2 (a, s). Omar El Euch and Mathieu Rosenbaum Population processes and rough volatility 32