On the Martingale Property of Exponential Local Martingales c 1 Mikhail Urusov 2 Aleksandar Mijatovi´ 1 University of Warwick 2 Ulm 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 : dY t = µ ( Y t ) dt + σ ( Y t ) dW t �� . � Z t = E b ( Y s ) dW s 0 t �� t � � t b ( Y s ) dW s − 1 b 2 ( Y s ) ds = exp 2 0 0 Z nonnegative local martingale = ⇒ supermartingale Z martingale ⇐ ⇒ E Z t = 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 : dY t = µ ( Y t ) dt + σ ( Y t ) dW t �� . � Z t = E b ( Y s ) dW s 0 t �� t � � t b ( Y s ) dW s − 1 b 2 ( Y s ) ds = exp 2 0 0 Z nonnegative local martingale = ⇒ supermartingale Z martingale ⇐ ⇒ E Z t = 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 one-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 ≤ ∞ dY t = µ ( Y t ) dt + σ ( Y t ) dW t , Y 0 = x 0 ∈ J ζ the explosion time of Y ◮ σ ( x ) � = 0 ∀ x ∈ J ◮ 1 /σ 2 , µ/σ 2 ∈ L 1 loc ( J ) � t ∧ ζ � t ∧ ζ b 2 ( Y s ) ds } Z t = exp { b ( Y s ) dW s − ( 1 / 2 ) 0 0 ◮ b 2 /σ 2 ∈ L 1 loc ( J ) � ζ 0 b 2 ( Y s ) ds = ∞} ◮ Z t := 0 for t ≥ ζ on {
Precise formulation of the problem J = ( l , r ) , −∞ ≤ l < r ≤ ∞ dY t = µ ( Y t ) dt + σ ( Y t ) dW t , Y 0 = x 0 ∈ J ζ the explosion time of Y ◮ σ ( x ) � = 0 ∀ x ∈ J ◮ 1 /σ 2 , µ/σ 2 ∈ L 1 loc ( J ) � t ∧ ζ � t ∧ ζ b 2 ( Y s ) ds } Z t = exp { b ( Y s ) dW s − ( 1 / 2 ) 0 0 ◮ b 2 /σ 2 ∈ L 1 loc ( J ) � ζ 0 b 2 ( Y s ) ds = ∞} ◮ Z t := 0 for t ≥ ζ on { 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 , dY t = cY α t dW t , Y 0 = x 0 ∈ 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 ) b 2 ∈ L 1 loc ( r − ) ρσ 2 l is good if . . . Auxiliary diffusion (with the same state space J = ( l , r ) ): d � Y t = ( µ + b σ )( � Y t ) dt + σ ( � Y t ) d � � W t , Y 0 = x 0 s : J → R scale function of diffusion � � ρ := � s ′ Y , �
Useful facts 1. “ r is good” means s ( r ) < ∞ and ( s ( r ) − s ) b 2 ∈ L 1 loc ( r − ) ρσ 2 or, equivalently, s ) b 2 s ( r ) < ∞ and ( � s ( r ) − � ∈ L 1 � loc ( r − ) ρσ 2 � 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 dY t = | Y t | α dt + dW t , Y 0 = x 0 ∈ J := R . Let Z be the local martingale given by �� t ∧ ζ � t ∧ ζ � Y s dW s − 1 Y 2 Z t = exp s ds . 2 0 0 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 dY t = σ ( Y t ) dW t , Y 0 = x 0 ∈ J := ( 0 , ∞ ) . We stop Y after it reaches 0 ⇒ x /σ 2 ( x ) / ∈ L 1 Corollary Y is a martingale ⇐ loc ( ∞− ) Delbaen and Shirakawa (2002) Carr, Cherny, and Urusov (2007) Reduction to our setting �� . � σ ( Y s ) Y t = x 0 E dW s Y s 0 t ∧ ζ
Example: bubbles and not only dY t = σ ( Y t ) dW t , Y 0 = x 0 ∈ J := ( 0 , ∞ ) . We stop Y after it reaches 0 ⇒ x /σ 2 ( x ) / ∈ L 1 Corollary Y is a martingale ⇐ loc ( ∞− ) Delbaen and Shirakawa (2002) Carr, Cherny, and Urusov (2007) Reduction to our setting �� . � σ ( Y s ) Y t = x 0 E dW s Y s 0 t ∧ ζ Why interesting? SA: nice problem, simple explicit answer MF: characterization of existence/absence of bubbles in one-dimensional diffusion models
Why interesting — another MF argument Stock price Y (complete market), interest rate := 0 ( Y T − K ) + − ( K − Y T ) + = Y T − K C t − P t = E ( Y T |F t ) − K (1) C t − P t = Y t − 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 one-dimensional diffusion setting, i.e. dY t = µ ( Y t ) dt + σ ( Y t ) dW t , Y 0 = x 0 ∈ 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 of 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 ? = E P Z t = lim n E P ( Z t I ( τ n > 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 E P ( Z t I ( τ n > t )) = E P ( Z t I ( ζ > t )) , which may be < E P Z t
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 of 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. ., D. Filipovi´ Cheridito, P 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 of 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 ohoku 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.
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