On the Martingale Property of Exponential Local Martingales c 1 - - PowerPoint PPT Presentation

On the Martingale Property

• f Exponential Local Martingales

Aleksandar Mijatovi´ c1 Mikhail Urusov2

1University of Warwick 2Ulm University

Analysis, Stochastics, and Applications. A Conference in Honour of Walter Schachermayer — Vienna University, July 12–16, 2010

Outline

Introduction Formulation of the main result Some examples Ways of proving the theorem and their restrictions

Outline

Introduction Formulation of the main result Some examples Ways of proving the theorem and their restrictions

Description of the main result

Diffusion Y: dYt = µ(Yt) dt + σ(Yt) dWt Zt = E . b(Ys) dWs

• t

= exp t b(Ys) dWs − 1 2 t b2(Ys) ds

• Z nonnegative local martingale =

⇒ supermartingale Z martingale ⇐ ⇒ EZt = 1, t ∈ [0, ∞) Input: functions µ, σ, and b Output: deterministic necessary and sufficient conditions for Z to be a true martingale in terms of µ, σ, and b

Description of the main result

Diffusion Y: dYt = µ(Yt) dt + σ(Yt) dWt Zt = E . b(Ys) dWs

• t

= exp t b(Ys) dWs − 1 2 t b2(Ys) ds

• Z nonnegative local martingale =

⇒ supermartingale Z martingale ⇐ ⇒ EZt = 1, t ∈ [0, ∞) Input: functions µ, σ, and b Output: deterministic necessary and sufficient conditions for Z to be a true martingale in terms of µ, σ, and b Questions Literature Where applies?

Literature

Sufficient conditions Novikov (1972), Kazamaki (1977) Many participants of AnStAp10 Cheridito, Filipovi´ c, and Yor (2005) Necessary and sufficient conditions Blei and Engelbert (2009) Mayerhofer, Muhle-Karbe, and Smirnov (2009)

Where applies?

SA: examples and counterexamples MF: characterizations of NFLVR, NGA, and NRA in

• ne-dimensional diffusion setting

MF: other problems also reduce to this setting with appropriately chosen µ, σ, b

Precise formulation of the problem

J = (l, r), −∞ ≤ l < r ≤ ∞ dYt = µ(Yt) dt + σ(Yt) dWt, Y0 = x0 ∈ J ζ the explosion time of Y

◮ σ(x) = 0 ∀x ∈ J ◮ 1/σ2, µ/σ2 ∈ L1 loc(J)

Zt = exp{ t∧ζ b(Ys) dWs − (1/2) t∧ζ b2(Ys) ds}

◮ b2/σ2 ∈ L1 loc(J) ◮ Zt := 0 for t ≥ ζ on {

ζ

0 b2(Ys) ds = ∞}

Precise formulation of the problem

J = (l, r), −∞ ≤ l < r ≤ ∞ dYt = µ(Yt) dt + σ(Yt) dWt, Y0 = x0 ∈ J ζ the explosion time of Y

◮ σ(x) = 0 ∀x ∈ J ◮ 1/σ2, µ/σ2 ∈ L1 loc(J)

Zt = exp{ t∧ζ b(Ys) dWs − (1/2) t∧ζ b2(Ys) ds}

◮ b2/σ2 ∈ L1 loc(J) ◮ Zt := 0 for t ≥ ζ on {

ζ

0 b2(Ys) ds = ∞}

Question Why considering possibility of explosion?

Why explosions?

◮ Examples and counterexamples ◮ In MF there are models, where explosion happens.

E.g. CEV: for α ∈ R, dYt = cY α

t dWt,

Y0 = x0 ∈ J := (0, ∞). Y explodes at 0 ⇐ ⇒ α < 1

Outline

Introduction Formulation of the main result Some examples Ways of proving the theorem and their restrictions

Terminology

s: (l, r) → R scale function of diffusion Y, ρ := s′ r is good if s(r) < ∞ and (s(r) − s)b2 ρσ2 ∈ L1

loc(r−)

l is good if . . . Auxiliary diffusion (with the same state space J = (l, r)): d Yt = (µ + bσ)( Yt) dt + σ( Yt) d Wt,

• Y0 = x0
• s: J → R scale function of diffusion

Y, ρ := s′

Useful facts

• 1. “r is good” means

s(r) < ∞ and (s(r) − s)b2 ρσ2 ∈ L1

loc(r−)

• r, equivalently,
• s(r) < ∞ and (

s(r) − s)b2

• ρσ2

∈ L1

loc(r−)

• 2. If one of the diffusions Y and

Y explodes at r and the other does not, then r is bad These facts are often helpful in the application of the theorem below to specific situations

Main result

Theorem Z martingale ⇐ ⇒ ((a) or (b)) and ((c) or (d)) (a) Y does not explode at r (b) r is good (c) Y does not explode at l (d) l is good Theorem above together with Fact 2 on the previous slide imply Corollary Suppose Y is non-explosive. Then Z is a martingale ⇐ ⇒ Y is non-explosive

Outline

Introduction Formulation of the main result Some examples Ways of proving the theorem and their restrictions

Example: funny

Fix α > −1 and define diffusion Y by dYt = |Yt|α dt + dWt, Y0 = x0 ∈ J := R. Let Z be the local martingale given by Zt = exp t∧ζ Ys dWs − 1 2 t∧ζ Y 2

s ds

• .

Our results imply the following classification: α ∈ (−1, 1]: Z martingale, not u.i. α ∈ (1, 3]: Z strict local martingale α > 3: Z u.i. martingale

Example: bubbles and not only

dYt = σ(Yt) dWt, Y0 = x0 ∈ J := (0, ∞). We stop Y after it reaches 0 Corollary Y is a martingale ⇐ ⇒ x/σ2(x) / ∈ L1

loc(∞−)

Delbaen and Shirakawa (2002) Carr, Cherny, and Urusov (2007) Reduction to our setting Yt = x0E . σ(Ys) Ys dWs

• t∧ζ

Example: bubbles and not only

dYt = σ(Yt) dWt, Y0 = x0 ∈ J := (0, ∞). We stop Y after it reaches 0 Corollary Y is a martingale ⇐ ⇒ x/σ2(x) / ∈ L1

loc(∞−)

Delbaen and Shirakawa (2002) Carr, Cherny, and Urusov (2007) Reduction to our setting Yt = x0E . σ(Ys) Ys dWs

• t∧ζ

Why interesting? SA: nice problem, simple explicit answer MF: characterization of existence/absence of bubbles in

• ne-dimensional diffusion models

Why interesting — another MF argument

Stock price Y (complete market), interest rate := 0 (YT − K)+ − (K − YT)+ = YT − K Ct − Pt = E(YT|Ft) − K (1) Ct − Pt = Yt − K (2) (1) holds always, (2) holds iff Y is a martingale Arbitrageurs = ⇒ in practice (2) tends to hold, not (1) = ⇒ practitioners would not work with a model, where Y is not a martingale

Further applications in MF

Characterizations of NFLVR, NGA, and NRA in

• ne-dimensional diffusion setting, i.e.

dYt = µ(Yt) dt + σ(Yt) dWt, Y0 = x0 ∈ J := (0, ∞)

◮ For NFLVR, b(x) := −µ(x)/σ(x) does only a part of the job ◮ Recall Delbaen and Schachermayer (1998)

A simple counterexample to several problems in the theory

• f asset pricing

◮ For NRA, b(x) := σ(x)/x − µ(x)/σ(x)

Outline

Introduction Formulation of the main result Some examples Ways of proving the theorem and their restrictions

A possible way

Reduce the problem to the canonical setting 1 ? = EPZt = limn EP(ZtI(τn > t)) = . . . = P(ζ > t). Done! Recall the talk by Damir Filipovi´ c Sin (1998), Carr, Cherny, and Urusov (2007) This method works only if the coordinate process is nonexplosive under P. Otherwise lim

n EP(ZtI(τn > t)) = EP(ZtI(ζ > t)), which may be < EPZt

Our approach

= ⇒ we needed to refuse this argument and elaborate a different one in order to consider the possibility of explosion under P This is needed in MF, as they sometimes consider models with explosions (e.g. CEV) From the viewpoint of SA, the reward that we get is a possibility to construct pathological (counter)examples

On my website

• A. Mijatovi´

c and M. Urusov (2010). On the martingale property

• f certain local martingales. To appear in Probability Theory

and Related Fields.

• A. Mijatovi´

c and M. Urusov (2010). Deterministic criteria for the absence of arbitrage in diffusion models. To appear in Finance and Stochastics. The talk covered a part of the first paper.

Dear Walter, happy birthday and many happy returns!

Blei, S. and H.-J. Engelbert (2009). On exponential local martingales associated with strong Markov continuous local martingales. Stochastic Process. Appl. 119(9), 2859–2880. Carr, P ., A. Cherny, and M. Urusov (2007). On the martingale property of time-homogeneous diffusions. Preprint, available at: http://www.uni-ulm.de/mawi/finmath/people/urusov.html. Cheridito, P ., D. Filipovi´ c, and M. Yor (2005). Equivalent and absolutely continuous measure changes for jump-diffusion processes.

• Ann. Appl. Probab. 15(3), 1713–1732.

Delbaen, F . and W. Schachermayer (1998). A simple counter-example to several problems in the theory

• f asset pricing.

Mathematical Finance 8(2), 1–11. Delbaen, F . and H. Shirakawa (2002).

No arbitrage condition for positive diffusion price processes. Asia-Pacific Financial Markets 9, 159–168. Kazamaki, N. (1977). On a problem of Girsanov. Tˆ

• hoku Math. J. 29(4), 597–600.

Mayerhofer, E., J. Muhle-Karbe, and A. Smirnov (2009). A characterization of the martingale property of exponentially affine processes. Preprint, available at: http://www.mat.univie.ac.at/˜muhlekarbe/. Novikov, A. A. (1972). A certain identity for stochastic integrals. Theory Probab. Appl. 17, 761–765. Sin, C. A. (1998). Complications with stochastic volatility models.

• Adv. in Appl. Probab. 30(1), 256–268.