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Nearly unstable linear processes
In Chapters II, III and IV, we study the long time behavior of linear processes (Hawkes and autoregressive processes) when their stability condition is almost saturated, that is when the L1 norm of their kernel is close to one.
Hawkes processes
Hawkes processes were introduced in 1971 in [Haw71a] and [Haw71b] to reproduce the clustering of earthquakes. A Hawkes process N is defined as a point process whose intensity ‚ writes as the sum of an exogenous term „ and a linear regression on the past of the process t ‚t ˘ „ ¯ `(t ¡ s)d Ns . ¡1. Over the last decade, these processes have been applied to several fields where it is necessary to have models which reproduce the clustering of some phenomena and thus where Poisson processes are inadapted. For example, Hawkes processes have been successfully used to re-produce the correlation of earthquakes, see [Ada76], genomic sequences, see [RBS¯10], neural spikes, see [RBRGTM14], terrorist attacks, see [LMBB12], crime, see [MSB¯11], price moves, see [BDHM13] or financial defaults, see [ASCDL10]. In Part II, we motivate and illustrate our results by using Hawkes processes to model the arrival of market orders.
There are many reasons for this recent popularity of Hawkes processes. First of all, Hawkes processes are tractable and very easy to simulate by thinning, see [Oga81]. Moreover, we will see that Hawkes processes can be efficiently fitted to reproduce the cross-correlations between events, see [BM14b]. Finally, let us mention the nice population dynamics interpretation of Hawkes processes that we will explicit later. This representation enables us to measure the endogeneity and causality between events in the Hawkes model, see [HO74]. In particular, we will see that the L1 norm of the kernel corresponds to the proportion of endogenous points of the process.
Light tailed nearly unstable Hawkes processes
An important property of Hawkes processes is that under the following stability condition ¯1 k`k1 ˘ `(s)d s ˙ 1 0. the process admits a version with a stationary intensity. Moreover, it is shown in [BDHM12] that under this condition, Hawkes processes present weak dependence in the sense that they asymptotically behave as their expectation at long time scales: NT t „ ¡! t. T T ` !¯1 1 ¡ j j.
However, in practice, for financial order flows that we want to model as Hawkes processes, the asymptotic behavior above is not satisfied. Indeed, the cumulated order flow does not behave as a linear function of time. In fact, it presents a multiscaling behavior in the sense that whatever the time scale at which one is looking at the order flow, one observes periods of high market activity (with a lot of trades) and periods of low market activity (with few trades). Moreover, statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly, see [FS12] and [HBB13]. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. In Chapter II we formally study such processes for which the stability condition is almost violated and the kernel has a light tail.
Asymptotic framework
In order to have a mathematical framework in which we can study the long time behavior of Hawkes processes whose kernel’s norm is close to one, we consider a sequence of Hawkes processes (NtT )t‚0 indexed by T which corresponds to the observation scale. For a given T , (NtT ) satisfies N0T ˘ 0 and its intensity process (‚Tt ) is defined for t ‚ 0 by t ‚Tt ˘ „ ¯ `T (t ¡ s)d NsT , 0. where „ is a positive real number and `T a non negative measurable function on R¯. Let us now give more specific assumptions on the asymptotic behavior of the function `T . For 2 R¯, `T (t) ˘ aT `(t). where (aT )T ‚0 is a sequence of positive numbers smaller than one and converging to one and is a non negative measurable function which we call the shape of the kernel and does not depend on T such that¯1`(s)d s ˘ 1.0. Thus, the form of the function `T depends on T varies with T . For a given T , this L1 norm is equal that the stability condition is in force. Since aT unstable Hawkes processes. so that its shape is fixed but its L1 norm to aT and so is smaller than one, implying tends to one, we call our sequence nearly.
Table of contents :
Introduction
Motivations
Outline
1 Part I: How regular and persistent is really volatility?
1.1 Regularity of the volatility
1.2 Volatility properties
1.3 Option pricing
2 Part II: Nearly unstable linear processes
2.1 Hawkes processes
2.2 Light tailed nearly unstable Hawkes processes
2.3 Heavy tailed Hawkes processes
2.4 Nearly unstable autoregressive processes
2.5 Beyond long memory
3 Part III: Multiscale Hawkes estimation
3.1 Hawkes processes and causality
3.2 The Wiener-Hopf equation
3.3 Application to slowly decreasing kernels
3.4 The Hawkes order book model
3.5 The likelihood of Hawkes processes
4 Part IV: Market impact throughout time scales
4.1 The impact of trades and metaorders
4.2 Fair markets and the fair price
4.3 A model independent impact equation
Part I Rough volatility modeling
I Volatility is rough
1 Introduction
1.1 Volatility modeling
1.2 Fractional volatility
1.3 The shape of the implied volatility surface
1.4 Main results and organization of the paper
2 Smoothness of the volatility: empirical results
2.1 Estimating the smoothness of the volatility process
2.2 DAX and Bund futures contracts
2.3 S&P and NASDAQ indices
2.4 Other indices
2.5 Distribution of the increments of the log-volatility
2.6 Does H vary over time?
3 A simple model compatible with the empirical smoothness of the volatility
3.1 Specication of the RFSV model
3.2 RFSV model autocovariance functions
3.3 RFSV versus FSV again
3.4 Simulation-based analysis of the RFSV model
4 Spurious long memory of volatility?
5 Forecasting using the RFSV model
5.1 Forecasting log-volatility
5.2 Variance prediction
6 The microstructural foundations of the irregularity of the volatility
7 RFSV and pricing
8 Conclusion
I.A Technical results
I.A.1 Proof of Proposition 1
I.A.2 Proof of Corollary 1
I.B Estimations of H
I.B.1 On dierent indices
I.B.2 On dierent time intervals
I.C The eect of smoothing
I.D Further results
I.D.1 RFSV model at intra day time scales
I.D.2 From metaorders distribution to persistence in the order ow
I.D.3 RFSV and the distribution of returns
I.D.4 An empirical time irreversibility
I.D.5 Test of multifractality
I.D.6 Historical skew measurement
I.D.7 Rough market activity
Part II Nearly unstable linear processes 6
II Limit theorems for nearly unstable Hawkes processes
1 Introduction
2 Scaling limits of nearly unstable Hawkes processes
2.1 Assumptions and asymptotic framework
2.2 Observation scales
2.3 Non degenerate scaling limit for nearly unstable Hawkes processes
2.4 Discussion
3 Extension of Theorem 2 to a price model
3.1 A Hawkes based price model
3.2 Scaling limit
4 Proofs
4.1 Proof of Theorem 1
4.2 Proof of Proposition 1
4.3 Proof of Theorem 2
4.4 Proof of Theorem 3
III Rough fractional diusions as scaling limits of heavy-tailed Hawkes processes
1 Introduction
2 Assumptions and intuitions for the results
3 Main results
3.1 The function f ®,¸
3.2 The limiting behavior of nearly unstable heavy tailed Hawkes processes
3.3 The limiting volatility process
3.4 Discussion
4 Proofs
4.1 Proof of Proposition 2
4.2 Proof of Theorem 1
4.3 Regularity of the solutions of Equation (8)
4.4 Proof of Theorem 2
III.A Technical results
IV The dierent asymptotic regimes of nearly unstable autoregressive processes
1 Introduction
2 Geometric sums
2.1 Geometric sums and convolution
2.2 Light tail case
2.3 Heavy tail case
3 The asymptotic setting
4 Main results
4.1 The light tail case
4.2 The heavy tail case
5 Application to volatility modeling
5.1 The case ® È 1/2
5.2 The case ® Ç 1/2
5.3 The case ® Æ 1/2
IV.A Technical results
IV.A.1 Proof of Proposition 3
IV.A.2 Proof of Proposition 4
Part III Multiscale Hawkes estimation and application to nance
V Estimation of slowly decreasing Hawkes kernels
1 Introduction
2 Multidimensional Hawkes processes and estimation principles
2.1 Basic denitions
2.2 Second-order properties and Wiener-Hopf system
2.3 Existence and uniqueness of solutions of the Wiener-Hopf system
2.4 Model independent origin of the Wiener-Hopf equation
2.5 Estimation principles
3 Non parametric estimation of slowly decaying kernels
3.1 Using an adapted sample-grid for g
3.2 Towards an adapted quadrature scheme
3.3 The adapted quadrature scheme for slow-decaying kernels
3.4 The adapted estimation procedure
4 Hawkes model for Level I Order book events
4.1 Denition of the model
4.2 Description of the database
4.3 Conditional law estimations
4.4 Exogenous intensities
4.5 Matrix of kernel norms jjÁjj1
4.6 Matrix of norms jjÃjj1
4.7 Shape of the kernels
4.8 The impact of the order ows on the price
4.9 The impact of price jumps on the order ows
4.10 The impact of market orders on liquidity and vice-versa
4.11 The impact of the limit order ow on the cancel order ow and vice-versa
5 Discussion and concluding remarks
6 An alternative approach: Dirac Hawkes processes
6.1 Dealing with Kernels involving Diracs
6.2 A Hawkes impact model
6.3 Estimation results
V.A Multiscale estimation of the conditional laws
V.A.1 Estimation procedure of g
V.A.2 Choice of the grid
V.B Estimation results for the full order book model: the cumulated kernel matrix
V.B.1 Inuence of price moves on price moves
V.B.2 Inuence of trades on trades
V.B.3 Inuence of limit orders on limit orders
V.B.4 Inuence of cancel orders on cancel orders
V.B.5 Inuence of trades on price changes
V.B.6 Inuence of limit orders on price changes
V.B.7 Inuence of cancel orders on price changes
V.B.8 Inuence of price changes on trades
V.B.9 Inuence of price changes on limit orders
V.B.10 Inuence of price changes on cancel orders
V.B.11 Inuence of trades on limit orders
V.B.12 Inuence of trades on cancel orders
V.B.13 Inuence of limit orders on trades
V.B.14 Inuence of cancel orders on trades
V.B.15 Inuence of limit orders on cancel orders
V.B.16 Inuence of cancel orders on limit orders
V.C Innitesimal covariance for Dirac Hawkes processes
V.C.1 A multidimensional branching process
V.C.2 Back to Hawkes estimation
V.D Error analysis
V.D.1 Bootstrap error analysis
V.D.2 Other simulations
V.D.3 Dependence with respects to the parameters
VI Estimation of Hawkes processes with a stochastic gradient descent
1 Introduction
2 The stochastic gradient descent
2.1 Rewriting the Hawkes likelihood
2.2 The algorithm
2.3 SVRG modication
3 Performance on simulated data
3.1 The Majoration-Minimization estimation method
3.2 Performance comparison
4 Multiscale Hawkes processes
4.1 The algorithm
4.2 Numerical test
5 Conclusion and extensions
Part IV Market impact throughout time scales
VII Liquidity and impact in fair markets
1 Introduction
2 Fair market and fair price
2.1 General framework
2.2 The example of the MRR model
2.3 General denition of the fair price
2.4 Verication of Proposition 2 on data
2.5 The fair price as a reference price
3 Bid-ask spread, fair price and impact
3.1 Necessary and sucient conditions on the bid and ask prices
3.2 Verication of Proposition 5 on data
3.3 Response functions, time horizon and market making
3.4 On tick value, market making and liquidity
4 Conclusion
VIII Market impact as anticipation of the order ow imbalance
1 Introduction
2 Computing price moves from the order ow
2.1 Linear permanent impact under the mechanical impact assumption
2.2 A toy investor model
2.3 Impact dynamics
3 Application to the Hawkes order ow example
3.1 Impact of individual market orders
3.2 Impact of metaorders
4 Conclusion
VIII.AProofs
VIII.A.1 Proof of Proposition 1
VIII.A.2Proof of Theorem 3
VIII.A.3Proof of Proposition 3
VIII.A.4 Proof of Theorem 4
VIII.BNumerical example for the toy model of Section 2.2
Bibliography
