continuous expansion of a filtration with a stochastic ...collaborators : philip protter, dan wang...

Continuous expansion of a filtration with a stochastic

process: the information drift

Leo Neufcourt

Michigan State University

[email protected]

February 26, 2020

Seminar of Financial and Actuarial Mathematics

University of Michigan

Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 1 / 41

Introduction

Collaborators : Philip Protter, Dan Wang (Columbia U), Ruihua Ruan, Luyi

Shen (Ecole Polytechnique)

L. N., PhD Thesis, Columbia University (2017)

L. N. & P. Protter, Expansion of a filtration with a stochastic process: the

information drift, Submitted (2019)

L. N., R. Ruan & L. Shen, A strict local martingale study of the bitcoin

bubble, Preprint (2019)


Roadmap

1 A suggestive example from Ito

2 Semimartingales, filtrations and arbitrage

3 Expansion of a filtration with a stochastic process

4 Strict local martingales


Market model

We consider

a financial market described by a complete probability space (Ω,A,P)

(i.e. N P ∈ A)

a price process W given by a P-(standard) Brownian motion

F its natural (completed) filtration (which is here right continuous):

Ft := σ(Wu, u ≤ t)

G the filtration expanded with W1, i.e.

Gt := Ft ∨ σ(W1)

(which is here complete and right-continuous)


Suggestive example: information drift

Proposition

Let αt := W1−Wt

1−t . Then the process

Wt := Wt −∫ t

0

αudu

defines a G-Brownian motion.

1

E∫ 1

0

|W1 −Wu|1− u

du =

√2

π

∫ 1

0

du√1− u

<∞

2

E[Wt −Ws |Gs ] = E[Wt −Ws |Fs ,W1] = E[Wt −Ws |Fs ,W1 −Ws ]

= E[Wt −Ws |W1 −Ws ] =t − s

1− s(W1 −Ws)


Suggestive example: information drift II

3

E[

∫ t

s

W1 −Wu

1− udu|Gs ] =

∫ t

s

E[W1 −Wu|Gs ]

1− udu

=

∫ t

s

E[W1 −Ws |Gs ]− E[Wu −Ws |Gs ]

1− udu

=

∫ t

s

(W1 −Ws) + u−s1−s (W1 −Ws)

1− udu

= (W1 −Ws)

∫ t

s

du

1− u

=t − s

1− s(W1 −Ws)

3 And we conclude from the Levy characterization.


Suggestive example: risk neutral probability

From the perspective of the filtration G, the price process W is the Ito process

dWt := dWt + αtdt

Can this drift be canceled by an equivalent change of probability? Recall that

there is no arbitrage opportunities (NFLVR) if and only if there exist a risk neutral

probability.

[Girsanov] The change of probability candidate is the stochastic exponential

dQdP

= Zt := E(

∫ t

0

αudWu) := exp(

∫ t

0

αudWu −1

2

∫ t

0

α2udu)


Suggestive example: risk neutral probability II

In general Zt := E(∫ t

0αtdWt) is a local martingale (whence supermartingale). It

is a true martingale on [0, t] if and only if E[Zt ] = 1, e.g. under Novikov’s

criterion Ee∫ t

0α2

udu <∞. Here,∫ t

0

α2udu =

∫ 1

1−t

W ′2uu2

du

where W ′1−t = W1 −Wt is a new BM.

When t < 1,∫ t

0α2udu ≤ 1

1−t∫ 1

0W 2

u du which has a finite Laplace transform

(e.g. Erdos and Kac). Thus Z is a true martingale on [0, t].

When t = 1, Ee∫ 1

0α2

udu ≤ eE∫ 1

0α2

udu = e∫ 1

0duu =∞ so Novikov criterion cannot

hold. [LIL] When t → 0, supu>tW 2

t

t2 ∼ 2t log log 1

t which is not integrable at 0.

Is it surprising ? There is an obvious G-arbitrage on [0, 1]...


Roadmap






Market model

We consider

a financial market described by a complete probability space (Ω,A,P)

a public filtration (Ft)t∈[0,T ] satisfying the usual hypotheses

a price process (St)t∈[0,T ] given by a F-semimartingale

an expansion expansion Gt := Ft ∨Ht of F

F-semimartingale : St = S0 + Mt + At , with

M : F-local martingale, i.e. E[Mt |Fs ] = Ms , up to an increasing sequence of

stopping time (“volatility”)

A : process of finite variation, i.e.∫ T

0|dAt | <∞ (“drift”)

A portfolio is characterized by an admissible 1 strategy H and has at any time a

value Vt := V0 +∫ t

0HsdSs .


Absence of arbitrage

No Arbitrage (NA) means that there is no admissible strategy with terminal

value V such that V ≥ 0 and P(V > 0) > 0.

No Free Lunch with Vanishing Risk (NFLVR) means that no sequence of

portfolios (Hn)n with value (V n)n dominated at all times by a sequence of

admissible portfolio can converge in probability to an “arbitrage” value V

such that V ≥ 0 and P(V > 0) > 0.

Theorem (Delbaen & Schachermayer, 1994)

NFLVR holds if and only if there exist an Equivalent Local Martingale Measure

(ELMM), or risk neutral probability, i.e. a probability Q ∼ P under which S is a

local martingale, i.e. of a positive and uniformly integrable martingale Z such that

ZS is a P-local martingale


No arbitrage of the first kind

NA1 or NUPBR is a weaker form of no arbitrage.

Theorem (Kardaras, Fontana)

NA1 is equivalent to the existence of a local martingale deflator, i.e. a positive

local martingale Z such that ZS is a local martingale. Additionally, NFLVR ⇐⇒(NA and NA1).

NA1 also means that any non-trivial contingent claim ξ ≥ 0 has a strictly positive

superhedging value (minimal initial wealth for which there exist a superhedging

strategy)


Characterizing the existence of the information drift

Theorem

NA1 holds for G if and only if there exists a process α ∈ S(G) such that M is a

G-semimartingale with decomposition

M =: M +

∫ .

0

αsd [M,M]s .

α is called the information drift of M (between the filtrations F and G).

A fortiori, NFLVR for ”the insider” requires existence of the information drift; it

can be seen as a general non-degeneracy condition.


Information drift of Brownian motion

Theorem

Let W be an F-Brownian motion. If there exists a process α ∈H 1(G) such that

W −∫ .

0αsds is a G-Brownian motion, then we have:

(i) αs = limt→st>s

E[Wt−Ws

t−s |Gs ]

(ii) αs = ∂∂tE[Wt |Gs ]

∣∣∣t=s

.

Conversely if there exists a process α ∈H 1(G) satisfying (i) or (ii), then and α is

the information drift of W , i.e. W −∫ .

0αsds is a G-Brownian motion.


Proof.

Since W is G-adapted it is clear that (i) and (ii) are equivalent. Suppose that

M −∫ .

0αudu is a G-martingale. It follows that, for every s ≤ t,

E[Wt −Ws −

∫ t

sαudu|Gs

]= 0, hence

E[

∫ t

s

αudu|Gs]

=

∫ t

s

E[αu|Gs ]du.

By differentiating with respect to t we obtain E[αt |Gs ] = ∂∂tE[Wt |Gs ] which

establishes (ii). Conversely if (ii) holds it is also clear that

E[Wt −Ws −∫ t

s

αudu|Gs ] = E[Wt −Ws − (E[Wt |Gt ]− E[Ws |Gs ])|Gs ] = 0.


Statistical arbitrage: a hard cash interpretation

We consider a market agent with initial wealth x and admissible strategies

H ∈ S : x + H ·M ≥ 0.

Example 1 : maximization of logarithmic utility

Let u(x ,Y) := supE log(x + H ·M)T s.t. H ∈ S(Y) and x + H ·M ≥ 0.

Proposition (Karatzas & Pikovsky, 1993 / Ankirchner & al, 2004)

u(x ,G)− u(x ,F) = E∫ T

0

α2sd [M,M]s

Example 2 : constrained maximization of returns

vλ(x ,Y) := sup x +E(H ·M)T −λVar(H ·M)T s.t. H ∈ S(Y) and x +H ·M ≥ 0.

Proposition (2017)

vλ(x ,G)− vλ(x ,F) = E∫ T

0

α2s

4λd [M,M]s


Roadmap






Jacod’s condition

Definition (Jacod’s condition)

A random variable L satisfies Jacod’s condition if for a.e. t ∈ I there exists a

(non-random) σ-finite measure ηt (which can be taken invariant with time or P0)

that dominates its conditional distribution Pt , i.e.

Pt(ω, L ∈ .) ηt(.) a.s.

Proposition (Jacod)

Under Jacod’s condition we can define qLt (ω, x) := dPt(ω,.)dη(.)

∣∣∣∣σ(L)

(L)−1(x) which

satisfies qLt (., L(.)) > 0, dP× dPt-a.s.

In other words, Jacod’s condition means that the knowledge of Ft does not

impact on whether events involving L are possible or not.


Initial expansions

Gt :=⋂

u>t

(Fu ∨ σ(L)

)Theorem (Jacod)

Suppose that L satisfies Jacod’s condition. Then:

(i) every F-semimartingale is a G-semimartingale, and

(ii) a continuous F-local martingale M has a decomposition

M = M +

∫ .

0

γs(ω, L(ω))d [M,M]s

where M is a G-local martingale and γ is a G-predictable process with

marginals

γt(., x) =1

qt(., x)

d [q(., x),M]td [M,M]t

.

Examples: L = W1, L = supt≤T Wt [Aksamit & Jeanblanc]


Successive initial expansions

Theorem (Kchia, Larsson & Protter)

Let M be a continuous F-local martingale and suppose that L := (L1, ..., Ln)

satisfies Jacod’s condition. Let 0 =: τ0 < τ1 < ... < τn < τn+1 := T be an

increasing sequence of (fixed) times and

Gnt :=⋂u>t

(Fu ∨ σ(Lk1t≥τk , k = 1...n)).

(i) M is a Gn-semimartingale, and

(ii) M has decomposition M =: M +∫ .

0αns d [M,M]s , where

αns (ω) :=

n∑k=1

1τk≤s≤τk+1

1

qk,ns− (ω, .)

(d [qk,n(ω, .),M]s

d [M,M]s

)(Lk(ω)).

The conditional densities qk,n(ω) are given by qk,ns := Ps (ω,(L1,...,Lk )∈dx)ηn(dx×Ωn−k )

.Leo Neufcourt (MSU) Expansions of filtrations February 26, 2020 20 / 41

Expansion with a stochastic process: how far can we go?

Gt := Ft ∨ σ(Xu, u ≤ t)

1 Static anticipative signal Xt := W1: initial expansion

2 Static anticipative signal + dynamic noise Xt := W1 + εt [Corcuera, ...]

3 Dynamic anticipative signal Xt = Wt+δ: W is not a semimartingale

(Wt+δ is ...); besides, lim sup/inf E(Wt−Ws |Gs )t−s = ±∞

4 Dynamic anticipative signal + noise Xt = Wt+δ + εt : if εt are i.i.d. centered

random variables (white noise), then for (hn)n≥1 non-decreasing with

hn →∞ and hnn → 0 we have [SLLN]

1n

∑ni=1(W

t− hin +δ

+ εt− hi

n

)a.s.−−−→

n→∞Wt+δ.

Since all terms of the sequence are Gt-measurable, so is Wt+δ which thus

cannot be a G-semimartingale


Expansion with a stochastic process : setup

X : cadlag process

Gt := Ft ∨ σ(Xu, u ≤ t) and Gt :=⋂

u>t Gu

(πn)n≥1 : refining sequence of subdivisions of [0,T ] with |πn| →n→∞ 0

πn =: (tni )`(n)i=0 , with 0 = tn0 < ... < tn`(n) < tn`(n)+1 = T

(X n)n≥1 : sequence of cadlag (pure) jump process given by

X nt :=

∑`(n)i=0 Xttn

i1tni ≤t<tni+1

(Hn)n≥1 : non-decreasing sequence of filtration generated by X n, namely

Hnt := σ(X n

s , s ≤ t) = σ(Xtn0,Xtn1

− Xtn0, ...,Xtn

`(n)+1− Xtn

`(n))

Gn and Gn : expansions of F given by

Gnt := Ft ∨Hnt and Gnt =

⋂u>t Gnu


Convergence of filtrations in Lp

DefinitionLet p > 0.

1. A sequence of σ-algebras (Yn)n≥1 converges in Lp to a σ-algebra Y if

∀Y ∈ Lp(Y,P),E[Y |Y]Lp

−−−→n→∞

Y

2. We say that a sequence of filtrations (Yn)n≥1 converges weakly in Lp to a

filtration Y if Ynt

Lp

−−−→n→∞

Yt for every t ∈ I .


Convergence of filtrations in Lp II

Let (Gn)n≥1 be a non-decreasing sequence of filtrations and M a continuous

Gn-semimartingale with decomposition M =: Mn +∫ .

0αns d [M,M]s for every n ≥ 1

for some αn ∈ H1(Gn).

Theorem (Stability of the semimartingale property)

If Gn L2

−−−→n→∞

G and supn E∫ t

0|αn

u|d [M,M]u <∞, then M is a G-semimartingale on

[0, t].

Theorem (Convergence of information drifts)

If Gn L2

−−−→n→∞

G and supn≥1 E∫ t

0(αn

u)2d [M,M]u <∞ then M is a G-semimartingale

on [0, t] with decomposition M =: M +∫ .

0αsd [M,M]s , where α ∈ S2(G,M) and

αn L2

−−−→n→∞

α.


Definition (Class Xπ)

We say that the cadlag process X is of Class Xπ, or of Class X if there is no

ambiguity, if

∀t ∈ I ,Hnt−

L2

−−−→n→1

Ht− .

Proposition

The cadlag process X is of Class X if one of the following holds:

(i) P(∆Xt 6= 0) = 0 for any fixed time t > 0.

(i’) X is continuous

(ii) H is (quasi-) left continuous.

(ii’) X is a Hunt process (e.g a Levy process)

(iii) X jumps only at totally inaccessible stopping times

(iv) π contains all the fixed times of discontinuity of X after a given rank


Expansion with a process: convergence of info. drifts

TheoremLet X be a stochastic process of Class Xπ for some sequence of subdivisions

πn := (tni )`(n)i=1 of [0,T ] and M a continuous F-local martingale. Suppose that for

every n ≥ 1 the random variable(X0,Xtn1

− Xtn0, ...,XT − Xtn

`(n)

)satisfies Jacod’s

condition and let αn be a Gn-predictable version of the process

αnt (ω) :=

∑`(n)k=0 1tnk≤t<tnk+1

1

qk,ns− (.,xk )

d [qk,n(.,xk ),M]sd [M,M]s

∣∣∣∣xk=(Xt0

,Xt1−Xt0

,··· ,Xtk−Xtk−1

)

.

1. If supn≥1E∫ T

0|αn

t |d [M,M]t <∞, M is a continuous G-semimartingale.

2. If supn E∫ T

0(αn

s )2d [M,M]s <∞, M is a continuous G-semimartingale with

decomposition M =: M +∫ .

0αsd [M,M]s , where α ∈ S2(G,M) and

E∫ T

0(αn

t − αt)2d [M,M]t −−−→

n→∞0.


Existence of an approximating sequence of expansions

TheoremLet M be a continuous square integrable F-martingale, H another filtration and

Gt :=⋂

u>t(Fu ∨Hu). Suppose that Hn is a refining sequence of filtrations such

that Ht =∨

n≥1Hnt , and let Gnt :=

⋂u>t(Fu ∨Hn

u). Then, the following

statements are equivalent :

(i) There exists a predictable process α such that Mt := M −∫ t

0αsds defines a

continuous G-local martingale and E∫ T

0α2sds <∞.

(ii) For every n ≥ 1 there exists a predictable process αn such that

Mnt := M −

∫ t

0αnuds is a continuous Gn-local martingale and

supn≥1 E∫ T

0(αn

u)2d [M,M]u <∞.

In that case, M (resp. Mn) is a G- (resp. Gn-) square integrable martingale and

we have∫ t

0(αn

u − αu)2d [M,M]u −−−→n→∞

0.


Example: A model for HF trading

W : F-Brownian motion

δ : non-negative stochastic process independent from W

ε : Markov process with stationary increments with density κt−s with respect

to Lebesgue measure

Xt := Wt+δ + εt

Gt := Ft ∨ σ(Xu, u ≤ t)

Then W has an information drift α given by

αt = lims→t

∫Xt + u −Ws

t − sκt−s(u − εs)du.

This highlights that the decay speed of the noise plays a key role in whether the

information drift exists.


Example: the Bessel-3 process

Let us now consider the counterexample of the Bessel-3 process: let Z be a

Bessel-3 process and F its natural filtration; it is classical that

Wt := Zt −∫ t

0dsZs

is a F- Brownian motion.

Let G the expansion of F with Xt := inft>s Zs . The formula of Pitman shows

that W is a G-semimartingale with decomposition Wt − (2Xt −∫ t

0dsZs

) defines

a G-Brownian motion.

This implies that W is a G-semimartingale but cannot admit an information

drift since the finite variation component is singular with respect to Lebesgue

measure. We can recover this conclusion from our theorem.


Example: the Bessel-3 process

Using the discretization induced by the refining family of random times

τnp := supt : Zt = pεn we have

αns =

1

Zs−∞∑p=0

1τnp<s1Zs≤(p+1)εn

1

Zs − pεn

∫ t

0

|αnt |dt = 2E

∫ t

0

ds

Zs<∞

∫ t

0

E[(αns )2]ds

∫ t

0

E[∞∑p=0

1pεn≤Zs≤(p+1)εn

pεnZ 2s (Zs − pεn)

]ds ≥∫ t

0

E[1εn≤Zs

1

Z 2s

]ds →∞

Thus supn≥1 E∫∞

0(αn

s )2ds =∞, αn cannot converge in H 2 and there cannot

exist an information drift α ∈H 2.


Roadmap






Strict local martingales

A strict local martingale is a local martingale which is not a martingale (e.g.

Bessel-3 process Bt := 1

|W (3d)t |2

).

[Dandapani] There exists (initial) expansions G of F for which a risk neutral

probability exists but turns (some) true martingales into strict local

martingales ! This is shown for Markov stochastic volatility models under a

coupling condition on the drift and volatility of the volatility (+ abstract

conditions on the information drift)

How to detect strict local martingales?


The bitcoin bubble

2013 2014 2015 2016 2017 2018 2019

0

2500

5000

7500

10000

12500

15000

17500

20000

Total capitalization > $100bn

adaaeaionantb10b20b40batbchbnbbsvbtcbtcpbtgbtknbtmcennzctxccvcdaidash

dcrdgbdogedrgndxyelfengeosetcethethosfungasgnogntgoldgusdicnicxkcsknc

liborusdloomlrclskltcmaidmanamtlnasneoomgpaxpaypivxpolypowrpptqashreprhocsalt

sntsp500srntrxtusdusdcusdtvenverivtcwaveswtcxemxlmxmrxrpxvgzeczilzrx

Price history of the main cryptocurrencies (www.coinmetrics.io)


Characterization of strict local martingales

Suppose that under a risk-neutral probability the local martingale S follows a

diffusion

dSt = σ(St)StdWt ,

where σ > 0 on (0,∞).

Theorem (Delbaen & Shirakawa, 1997)

S is a strict local martingale if and only if∫ T

01

xσ(x)2 <∞.

A natural candidate to estimate x 7→ σ(x) : the Florens-Zmirou estimator, where

Xt := log(St).

SFZn (x) :=

∑ni=1 1|Sti

−x|<hnn(Xti+1 − Xti )2∑n

i=1 1|Sti−x|<hn

.


Bitcoin and microstructure noise

Bitcoin price paths exhibit typical patterns of microstructure noise

0 200000 400000 600000 800000 1000000number of points

800000

1000000

1200000

1400000

1600000

1800000

QV (y

r2 )

Quadratic variation of log-prices between 01/2014 and 12/2016 according to discretization size

QVy = 0.584 + 1.21e6QV / 2ny = 0.898

0

10

20

30

40

50

[ 2]

Model : Yti := Xti + εti [Zhang, Mykland & Aıt-Sahalia, 2004]

E[ε2] = limn→∞QVn(Y )

2n a.s.


Subsampling against microstructure noise

When microstructure noise is present its variance dominates the FZ

estimator: limn→∞SFZn (x)2n = E[ε2]

This can be corrected using a subsampled FZ estimator :

SFZ ,subn (x) :=

∑[n/λ]i=1 1|St[λi ]−x|<hn n[(Xt[λ(i+1)]

−Xt[λi ])2− 1

λ

∑[λ(i+1)]−1

j=[λi ](Xtj+1

−Xtj)2]∑n

i=1 1|Sti−x|<hn.

SFZ ,subn (x) converges at the same rate as the FZ estimator in absence of

noise, i.e. if nh2n →∞ then SFZ ,sub

n (x)P−−−→

n→∞σ2(x) + o(E[ε2]) [L.N., Ruan &

Shen].


A bubble detection procedure

Following [Jarrow, Kchia & Protter]

At every time t:

Step 1 Compute SFZ ,sub to estimate x 7→ σ2(x).

Step 2 Estimate the right-tail asymptotics x 7→ σ2(x) via a regression

(parametric or RKHS) and decide whether Bt is 0 or 1

Step 3 Smoothen the bubble estimator t 7→ Bt with a Hidden Markov

Chain model


Results: volatility estimate

Birth:6 8 10 12

price0.0

0.2

0.4

0.6

0.8

1.0

1.2

vola

tility

1e9

6 8 10 12 6 8 10 12 6 8 10 12 6 8 10 120

50

100

150

200

Death:

price0

1

2

3

4

5

6

vola

tility

1e10

100 2000

1

2

3

4

5

6

1e10

100 200 100 200 100 200 100 2000

25

50

75

100

125

150


Results: bubbles

2012 2013 2014 2015 2016 2017 20180

2500

5000

7500

10000

12500

15000

17500

20000

price

(USD

)

Bubbles in bitcoin prices

price1bubble

log-price

2

4

6

8

10

log-

price

(USD

)


Conclusion

There are various situations in which asset prices can differ significantly from

martingales

This has major implications in term of arbitrage opportunities

Expansions of filtrations can change deeply the properties of semimartingales

These new expansion models can be applied to find new statistics on

stochastic processes which can help identify the information structure of the

market.


Thank you!


continuous expansion of a filtration with a stochastic ...collaborators : philip protter, dan wang...

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