Analyse du Risque Financier et modélisation de la ... · Propose a long-run risk co-movement...

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Motivations ARFIMA-J ARFIMA-J : Step 1 ARFIMA-J : Step 2 ARFIMA-J : Step 3

Analyse du Risque Financier et modélisation de lapersistance en présence de jumps

Banulescu, D., Batakis, A., de Truchis, G., Desgraupes, B., Dumitrescu, E.

Colloque de restitution du défi INFINITI

November 3, 2017


Analyse du Risque Financier et modélisation de la persistance en présence de jumps

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Financial market risk

risk lies at the heart of financial analysis ; it is fondamental for investmentdecision and financial regulation

risk is unobserved and stochastic

a simple and highly used measure of risk is volatility



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Volatility dynamics

it is complex, unobserved and requires specific methods to model and forecast

it is mainly driven by two stylized facts :

presence of jumps in asset prices→ estimation

persistence→ forecasting

07 08 09 10 11 12 13 14 15 16 176.5

7

7.5

8

07 08 09 10 11 12 13 14 15 16 17-0.1

-0.05

0

0.05

0.1

0.15

Standard & Poor's 500 prices

Standard & Poor's 500 returns



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Volatility dynamics

A strand of literature focuses on volatility modeling in presence of jumps

e.g. Barndorf-Nielsen et al., (2006). Limit theorems for multipower variation inthe presence of jumps. Stochastic Processes and their Aplications.

Another strand of literature gauges the long-range dependence of volatility

e.g. Anderson et al., (2001). The distribution of realized exchange rate volatility.Journal of the American Statistical Association.

Few empirical papers merge these two stylized facts

e.g. Boudt et al., (2011). Robust estimation of intraweek periodicity in volatilityand jump detection. Journal of Empirical Finance.

But no theoretical paper tackles these issues at the same time



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ARFIMA-J Project outline

Sequential approach in three steps :

Step 1 : Study volatility dynamics in presence of jumps and persistence in atheoretical framework

Step 2 : Develop an estimator of volatility persistence robust to the presence ofjumps

Step 3 : Extend this analysis to a multi-market framework and propose an estimatorof risk co-movement (co-persistence)



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Project outline

Sequential approach in three steps :

Step 1 : Study volatility dynamics in presence of jumps and persistence in atheoretical framework

Step 2 : Develop an estimator of volatility persistence robust to the presence ofjumps

Step 3 : Extend this analysis to a multi-market framework and propose an estimatorof risk co-movement (co-persistence)



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Theoretical framework

The assumed data generating process :

dS(t) = µ(µS, λ?(t), ζS)S(t)dt + σ(t)S(t)dWS(t) + (ζS − 1)S(t)dN(t)

σ2(t) = η +∫ t

−a

(t− s)δ

Γ(1 + δ)dV(s)

dV(t) = κ(ϑ−V(t))dt + ς√

V(t)dWσ(t)

λ?(t) = λ̄ +∫ t

−∞ν(t− u)dN(u)

with

S(t) the true price process of a given asset at a continuous time t

µ the compensated drift term

σ2(t) fractionally integrated variance process

WS(t) and Wσ(t) two independent standard Brownian motions

(ζS − 1)dN(t) a finite activity jump process

N(t) a point process with possibly time varying intensity parameter λ∗(t)



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Theoretical framework

Under the continuous time process introduced, the true ex-post measure ofreturns variability over a day is the quadratic variation

QVt+1 =

t+1∫t

σ2τ dτ

︸︷︷︸(a)

+ ∑t<τ<t+1

Jump2τ︸︷︷︸

(b)

(a) represents the integrated variance, IVt which measures the daily variancearising from the continuous part of the log-price process

(b) represents the contribution of the cumulated squared jumps to the quadraticvariation

The persistence in QV does not reflect the persistence in IV



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Objective

Provide theoretical evidence on the diminishing QV persistence (in presence ofjumps)

Standard hypothesis : second-order stationarity of the volatility and jumpprocesses

⇒ Analyse the asymptotic behaviour of the QV autocovariance function



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Results

ACF(h) =CovQVc (h) + CovQVd (h)

VQVc + VQVd

=CovQVc (h)

VQVc + VQVd

+CovQVd (h)

VQVc + VQVd

0 50 100 150 200

0.2

0.4

0.6

0.8

1.0

h

Aut

ocor

rela

tion

func

tion

0 50 100 150 200

0.4

0.5

0.6

0.7

0.8

0.9

1.0

h

Aut

ocor

rela

tion

func

tion

Cov (h) /

ACF(h)

QV

Hawkes Jumps Poisson Jumps

cVQVc

Cov (h) /QVc( VQVc

+ V )QVd

Cov (h) /

ACF(h)

QVcVQVc

Cov (h) /QVc( VQVc

+ V )QVd



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Results

Persistence parameter estimates (DGP : δ=0.25)

parametric semi-parametric

IV Q̂V IV Q̂VContinuous Process 0.257 0.220 0.242 0.232CP + Poisson Jumps 0.257 0.183 0.242 0.161CP + Hawkes Jumps 0.257 0.050 0.242 0.078

⇒ underestimation of risk persistence in presence of jumps



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Persistence estimator

Step 2 of the project aims at developing an unbiased persistence estimator inpresence of jumps

Our estimator operates in the frequency domain

Frequency domain analysis allows one to easily deal with very large datasets

It takes into account the data-agreggation process

With this DGP, we already developed a Monte-Carlo simulation framework

Adapted to each of the two types of jump processes

Handling the discretization of fractional integrals

Able to generate price dynamics over very long samples (1000 days of 3901-minute data × 1000 simulations)



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Preliminary results

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2IV and QV without jump

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700 800 900 10000

5

10

0

1

2IV and QV in presence of jumps (Hawkes)ˆ ˆ

QV

IV

ˆ QV

IV

ˆ



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To be continued

Step 3 will extend Step 2 to a multi-market framework

Develop a multivariate version of our fractional Heston model

Propose a long-run risk co-movement estimator

Account not only for the presence of jumps but also for that of co-jumps

Implication in terms of hedging and portfolio management



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Thank you for your attention !



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