Simple Examples of Pure-Jump Strict Local Martingales

Simple Examples of Pure-Jump Strict Local Martingales

Martin Keller-Ressel

TU Dresden

Workshop on Stochastic Analysis,Controlled Dynamical Systems and Applications, Jena

March 9th, 2015

Martin Keller-Ressel (TU Dresden) Pure-Jump Strict Local Martingales March 9th, 2015 1 / 28

Strict Local Martingales (1)

Strict local martingales are local martingales which are no truemartingales

Interesting to probabilists, because. . .

they illustrate the differences between true and local martingales,

the distinction between true and strictly local martingales plays a keyrole in Girsanov’s theorem,

without them Novikov’s condition and generalizations would bepointless.



Strict local martingales are local martingales which are no truemartingales

Interesting to probabilists, because. . .

they illustrate the differences between true and local martingales,

the distinction between true and strictly local martingales plays a keyrole in Girsanov’s theorem,

without them Novikov’s condition and generalizations would bepointless.



Interesting in financial mathematics, because they are . . .

examples of arbitrage-free markets where put-call-parity fails, marketprices deviate from fundamental prices, etc.

often considered as models of asset pricing bubbles.


1 Economic Motivation

2 The Setting of Continuous Local Martingales

3 A first example of a pure-jump strict local martingale

4 Generalizations of the Example


Section 1

Economic Motivation


Fundamental Theorem of Asset Pricing

Theorem (FTAP; Delbaen & Schachermayer (1998))

Let S be a locally bounded semimartingale on a given filtered probabilityspace. The following are equivalent:

1 The Financial Market described by (S ,P) does not allow for arbitragein the sense of No Free Lunch with Vanishing Risk (NFLVR).

2 There exists Q ∼ P such that S is a local Q-martingale.

Any ‘reasonable’ model for a stock price S has the local martingaleproperty under Q.

If ‘locally bounded’ is dropped, the implication (2) ⇒ (1) remainsvalid.


Pricing Bubbles (1)

Definition (Price Bubble; Heston, Loewenstein & Willard (2007))

The Financial Market (S ,Q) with time horizon T contains a price bubble,if for some t ∈ [0,T ) the current stock price St exceeds the fundamentalprice EQ [ST | Ft ], i.e., if

St > EQ [ST | Ft ] .

Clearly, for locally bounded processes, an arbitrage-free financialmarket (S ,Q) contains a bubble iff S is a strict local Q-martingale.

If ‘locally bounded’ is dropped, the strict local martingale property isstill sufficient for the appearance of a bubble in an arbitrage freemarket model.


Pricing Bubbles (2)

In a similar way, price bubbles of Put & Call options, bond prices etc.can be studied.

In a strict local martingale model, also (risk-neutral) call pricescontain bubbles.

In a strict local martingale model put-call-parity may fail and otherpathologies appear.


Section 2

The Setting of Continuous Local Martingales


Novikov’s and Kazamaki’s condition

Typical setting: S given as stochastic exponential of a continuous localmartingale X :

St = E(X )t = exp

(Xt − X0 −

1

2〈X ,X 〉t

).

The following (well-known) conditions are sufficient for the true martingaleproperty of S on [0,T ]:

Novikov’s condition:

E[

exp

(1

2〈X ,X 〉T

)]<∞.

Kazamaki’s condition:

exp

(1

2Xt

)is submartingale on [0,T ].


Novikov’s and Kazamaki’s condition

Typical setting: S given as stochastic exponential of a continuous localmartingale X :

St = E(X )t = exp

(Xt − X0 −

1

2〈X ,X 〉t

).

The following (well-known) conditions are sufficient for the true martingaleproperty of S on [0,T ]:

Novikov’s condition:

E[

exp

(1

2〈X ,X 〉T

)]<∞.

Kazamaki’s condition:

exp

(1

2Xt

)is submartingale on [0,T ].


Blei-Engelbert condition (1)

Consider the one-dimensional diffusion setting:

Let X be unique weak solution of

Xt = x +

∫ t

0b(Xs)dBs .

where b satisfies the Engelbert-Schmidt conditions, i.e.,

b(y) 6= 0 ∀y ∈ R, b−2 is locally integrable.


Blei-Engelbert condition (2)

Theorem (Blei and Engelbert (2009))

S = E(X ) is a true martingale if and only if∫ ∞x−ε

b−2(y)dy =∞

for all ε > 0.

Corollary

S = E(X ) is a strict local martingale if and only if there is ε > 0 such that∫ ∞x−ε

b−2(y)dy <∞.


Delbaen-Schachermayer Construction

Let M be a continuous local martingale starting at M0 = 1 and τ itsfirst hitting time of 0.

Assume P(τ ≤ T ) > 0

Define a probability measure Q P by

dQdP

∣∣∣∣FT

= MT∧τ .

Theorem (Delbaen & Schachermayer (1995))

The process S = 1/M is a strict local Q-Martingale on [0,T ].

Example: If M is a P-Brownian motion, then 1/M is the inverse Besselprocess of dimension 3 under Q.


Final Remarks

My personal starting point for this topic: A passing remark by PhilippProtter that . . .

There are few good examples of strict local martingales withjumps.

Exceptions can be found in:

Kallsen and Muhle-Karbe (2010)

Mijatovic and Urusov (2011)

Protter (2014) – based on filtration shrinkage


Section 3

A first example of a pure-jump strict local martingale


A First Example (1)

Define Levy measure

µ(dξ) =1

2√πe−ξξ−3/2.

J(x , dξ, ds). . . Poisson random measure with compensator x µ(dξ)ds

J(x , dξ, ds) = J(x , dξ, ds)− x µ(dξ)ds

Theorem (K.-R. (2014))

Let X be the (weak) solution of

Xt = X0 +

∫ t

0

∫ ∞0

ξ J(Xs−, dξ, ds)− 1

2

∫ t

0Xs−ds.

Then S = eX is a strict local martingale.


A First Example (2)

Can also be defined as semimartingale X with characteristics∫ t

0Xs−ds ·

(−1

2, 0, µ(dξ)

)relative to truncation function h(x) = x .

Drift chosen such that S = eX is local martingale.

X is a pure-jump process of finite variation but infinite activity.

Jumps are self-exciting: The arrival rate of jumps is proportional tothe level of the process.


A First Example (2)


0Xs−ds ·

(−1

2, 0, µ(dξ)






A First Example (2)


0Xs−ds ·

(−1

2, 0, µ(dξ)






A First Example (2)


0Xs−ds ·

(−1

2, 0, µ(dξ)






A First Example (3)

The process does not explode (but has huge excursions).

X is an affine process in the sense of Duffie, Filipovic andSchachermayer (2003). From there we obtain existence of X andseveral other properties.

Removing the exponential factor e−ξ from µ leads to explosion of Xin finite time (see Duffie, Filipovic and Schachermayer (2003))


A First Example (3)





A First Example (3)





The Proof (1)

Associate to X the following function (for u ∈ (−∞, 1]).

R(u) =

∫ ∞0

(euξ − 1− uξ

)µ(dξ)− u

2= (1− u)−

√1− u.

and the ODEg ′(t, u) = R(g(t, u)), g(0, u) = u (1)

By Ito’s formula for jump processes:

Lemma

The process Mut = eXtg(T−t,u) is a local martingale if and only if g(t, u) is

a solution of the ODE (1).

Key observation: For u = 1 the solution of (1) is not unique! This leadsdirectly to a proof of the strict local martingale property of eX .


The Proof (1)

Associate to X the following function (for u ∈ (−∞, 1]).

R(u) =

∫ ∞0

(euξ − 1− uξ

)µ(dξ)− u

2= (1− u)−

√1− u.

and the ODEg ′(t, u) = R(g(t, u)), g(0, u) = u (1)

By Ito’s formula for jump processes:

Lemma

The process Mut = eXtg(T−t,u) is a local martingale if and only if g(t, u) is

a solution of the ODE (1).

Key observation: For u = 1 the solution of (1) is not unique! This leadsdirectly to a proof of the strict local martingale property of eX .


The Proof (2)

One possible solution at of the ODE starting at u = 1 is g+(t, u) = 1,but there is another non-constant solution g−(t, u) ≤ 1.

Since they are different there is T > 0 such that g−(T , u) < 0.

Set M+t = eX = eXtg+(T−t,u) and M−t = eXtg−(T−t,u)

M± are local martingales with the same terminal value M±T = euXT

and different initial values M+0 > M−0 .


The Proof (2)







The Proof (2)







The Proof (2)







The Proof (3)

Both M+ and M− are non-negative, hence supermartingales.

Assume M+ is a true martingale, then

M+0 = E

[M+

T

]= E

[M−T

]≤ M−0

in contradiction to M+0 > M−0 . Voila!


The Proof (3)

Both M+ and M− are non-negative, hence supermartingales.

Assume M+ is a true martingale, then

M+0 = E

[M+

T

]= E

[M−T

]≤ M−0

in contradiction to M+0 > M−0 . Voila!


Generalization (1)

Replace µ by an arbitrary Levy measure on R>0 that satisfies∫R>0

(eξ ∧ ξ2

)µ(dξ) <∞.

Define J(x , dξ, ds) and J(x , dξ, ds) as above.

Define X as the (weak) solution of

Xt = X0 +

∫ t

0

∫ ∞0

ξ J(Xs−, dξ, ds)− b

∫ t

0Xs−ds,

where b =∫∞0 (eξ − 1− ξ)µ(dξ) > 0.

Finally, define

R(u) :=

∫ ∞0

(euξ − 1− uξ

)µ(dξ)− bu.


Generalization (2)


The following are equivalent:

(a) S = eX is a strict local martingale

(b) The ordinary differential equation

g ′(t) = R(g(t)), g(0) = 1

has more than one solution.

(c) The function 1/R(u) is integrable in a left neighborhood of u = 1, i.e.there exists ε > 0 such that

−∫ 1

1−ε

dη

R(η)<∞.


Generalization (3)


Let the measure µ be given by

µ(dξ) = ce−ξξ−α`(ξ) dξ, ξ ≥ 0

where c > 0, α ∈ (1, 2), ` is slowly varying at infinity, and∫∞0 (eξ ∧ ξ2)µ(dξ) <∞.

Define the associated Poisson random measure and the process X asbefore. Then S = eX is a strict local martingale.

Only the large jumps matter!


Alternative Construction (1)

Define the Levy measure

µ(dξ) =1

2√πξ−3/2dξ

This measure is equal to µ from above without the exponential factore−ξ.

On a probability space (Ω,F ,P) we define X as the Feller processwith generator

Af (x) = −xf ′(x) + x

∫ ∞0

(f (x + ξ)− f (x)) µ(dξ)



Following Duffie, Filipovic and Schachermayer (1995) X explodes infinite time with positive probability.

The function h(x) = e−x is harmonic for X , i.e. A(e−x) = 0. Itfollows that

Mt = exp(−(Xt − X0

))is a true martingale that reaches zero in finite time with positiveprobability.

Define a measure Q, absolutely continuous with respect to P, bysetting

dQdP

∣∣∣∣Ft

= Mt .




The process Lt = 1Mt

is a strict local martingale under Q.

This construction parallels the construction of continuous strict localmartingales by Delbaen & Schachermayer.

The process L under Q is identical in law to the process S = eX

considered in the previous section, i.e. this is an alternativeconstruction of the same processes.


Thank you for your attention!

M. Keller-Ressel. Simple Examples of Pure-Jump StrictLocal Martingales (2014). arXiv:1405.2669.


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Simple Examples of Pure-Jump Strict Local Martingales