ON LONG-TERM ARBITRAGE OPPORTUNITIES IN MARKOVIAN MODELS OF - PowerPoint PPT Presentation

ON LONG-TERM ARBITRAGE OPPORTUNITIES IN MARKOVIAN MODELS OF FINANCIAL MARKETS asonyi, University of Edinburgh ∗ Mikl´ os R´ Based on joint work with Martin L. D. Mbele Bidima. Vienna, 15th July 2010 ∗ On leave from MTA SZTAKI, Budapest. 1

Motivation I H. F¨ ollmer and W. Schachermayer. Asymptotic arbitrage and large devia- tions. Math. Financ. Econ. 1 , 213–249, 2007. Our aims: – Understanding better which features of a financial market ensure an expo- nential growth of an investor’s wealth. – Controlling the probability of failing to achieve this. – Settling issues raised in the above paper. – Qualitative results that can be made quantitative in concrete models. 2

Motivation II For simplicity, we consider only one stock throughout this talk. dS t = Σ( S t )( dW t + φ ( S t ) dt ) , t ≥ 0 . W t : Brownian motion. Usual class of admissible strategies. Value process of strategy π t denoted by V π t . φ : market price of risk. Σ : volatility. 3

Motivation III S has a nontrivial market price of risk if � T � � 1 0 | φ ( S t ) | 2 dt < c lim = 0 . T →∞ P T for some c > 0 . Technical assumption: minimal martingale measure exists for all finite hori- zons T > 0 . 4

Motivation IV Theorem. (F& S) If S has a nontrivial market price of risk then there exists γ > 0 and for each ε > 0 there exists T ε such that for all T > T ε T ≥ e γT ) ≥ 1 − ε P ( V π T ≥ − e − γT , for some admissible trading strategy π = π ( ε, T ) , start- and V π ing from 0 initial capital. 5

Questions – Is it possible to have explicit strategies ? – Strategies in the above Theorem depend on T . Can we use the same π for all T ? – Can π be chosen Markovian ? – How large is T ε ? 6

Control of failure probabilities I The market price of risk satisfies a large deviation estimate if there are c 1 , c 2 > 0 such that � T � � 1 1 0 | φ ( S t ) | 2 dt ≤ c 1 lim sup T ln P ≤ − c 2 . T T →∞ Conjecture. This condition should imply the existence of strategies as in the above Theorem but with T ≥ e γT ) ≥ 1 − e − βT , P ( V π for some β > 0 . 7

Control of failure probabilities II In F&S paper the relationship of the above statement to large deviation the- ory is explained. They analyse the case of geometric Ornstein-Uhlenbeck process, give explicit γ, β as well as explicit formulas for long-term maximal expected utility for various utility functions. We wished to prove the conjecture and to give conditions that ensure that the market price of risk satisfies a large deviation estimate, i.e. to extend the result of F&S beyond the Ornstein-Uhlenbeck case. 8

Model description I Discrete time modelling. Positive price process S t = exp[ X t ] where X t is a Markov chain with dynamics X t +1 − X t = µ ( X t ) + σ ( X t ) ε t +1 , where ε t are i.i.d., µ, σ measurable and X 0 constant. Define F t := σ ( X 1 , . . . , X t ) . Trading strategies will be F t -predictable pro- cesses, i.e. π t is F t − 1 -measurable. 9

Model description II No short-selling. No borrowing of money. π t takes values in [0 , 1] and represents the proportion of wealth allocated to the stock. Dynamics of the wealth process: � � S t +1 V π t +1 = V π 1 − π t +1 + π t +1 t S t and V π 0 = V 0 > 0 constant. 10

Arbitrage concepts I We say that there is asymptotic exponential arbitrage (AEA) if there exist γ > 0 and a trading strategy π t , t ≥ 0 such that, for all ε > 0 there is T ε ∈ N satisfying T ≥ e γT ) ≥ 1 − ε, P ( V π for all T ≥ T ε . 11

Arbitrage concepts II Proposition. If there is AEA then for each ε > 0 there exists T ε and trading strategies π t ( ε, T ) , t ≥ 1 , such that for T ≥ T ε we have V π ( ε,T ) ≥ V 0 − e − γT/ 2 T and P ( V π ( ε,T ) ≥ e γT/ 2 ) ≥ 1 − ε. T 12

Arbitrage concepts III We say that there is asymptotic exponential arbitrage with geometrically decaying failure probabilities if there exist C, β, γ > 0 and a trading strategy π t , t ≥ 0 such that P ( V π t ≤ e γt ) ≤ Ce − βt . 13

A naive approach I r ( x ) := E exp[ µ ( x ) + σ ( x ) ε 1 ] − 1 This is the expected future excess return of the stock conditional to the present log-price x . It will play the role of φ , the market price of risk. We did not assume ε 1 Gaussian but will investigate that case later. 14

A naive approach II One may try the (Markovian) strategy ˜ π t := 1 { r ( X t − 1 ) > 0 } or rather ˜ π t := ηr ( X t − 1 )1 { r ( X t − 1 ) > 0 } with some 0 < η < 1 . 15

Main result I Theorem. Assume that µ, σ are bounded, Ee a | ε 1 | < ∞ for all a > 0 and for some c > 0 ,   T  1 r 2 ( X i − 1 )1 { r ( X i − 1 ) > 0 } < c  = 0 . � T →∞ P lim (1) T i =1 Then ˜ π with a suitable η realizes AEA. Formula (1) is the present version of the nontrivial market price of risk con- dition. The indicator appears because of prohibiting short sales. 16

Gaussian case Theorem. Assume further that ε 1 is standard Gaussian, σ ( x ) > 0 for all x and for some c > 0 ,   � 2 T � 1 σ ( X i − 1 ) + σ ( X i − 1 ) µ ( X i − 1 ) � < c �   lim 1 �  = 0 . T →∞ P   µ ( Xi − 1) σ ( Xi − 1) + σ ( Xi − 1) T 2  > 0 i =1 2 (2) Then there is AEA. Condition (2) is completely analogous to the condition of F& S paper (except the indicator). 17

Main result II Theorem. Under the conditions of the previous Theorem,   T 1  1 r 2 ( X i − 1 )1 { r ( X i − 1 ) > 0 } ≤ c 1  ≤ − c 2 � lim sup T ln P (3) T T →∞ i =1 for some c 1 , c 2 > 0 implies AEA with geometrically decaying failure proba- bilities. In the Gaussian case we may again replace r by µ σ + σ 2 . 18

Tools Law of large numbers for martingales, large deviation principle for martin- gales, respectively. Liu, Q. and Watbled, F. Exponential inequalities for martingales and asymp- totic properties of the free energy of directed polymers in a random environ- ment. Stochastic Process. Appl. 119 , 3101–3132, 2009. The bounded case was previously treated in Blackwell, D. Large deviations for martingales. Festschrift for Lucien Le Cam, 89–91, Springer, New York, 1997. 19

Sufficient conditions Theorem. Assume furthermore that ε 1 has a density (w.r.t. Lebesgue mea- sure) locally bounded away from 0 and σ is locally bounded away from 0 . Assume that for some Lyapunov-function V : R → [1 , ∞ ) the drift condition E [ V ( X 1 ) | X 0 = x ] ≤ (1 − δ ) V ( x )1 { x/ ∈ Γ } + C 1 { x ∈ Γ } holds for constants C, δ > 0 and compact set Γ . If Leb ( { x : r ( x ) > 0 } ) > 0 then there is AEA with geometrically decaying failure probabilities. 20

Geometric ergodicity When X t is irreducible and aperiodic (in a suitable sense) then E [ V ( X 1 ) | X 0 = x ] ≤ (1 − δ ) V ( x )1 { x/ ∈ Γ } + C 1 { x ∈ Γ } implies that the law of X t tends to its invariant distribution at a geometric rate as t → ∞ . Usually this drift criterion can be used to prove geometric ergodicity, it is “close” to being a necessary condition, too. 21

Drift criterion in the present setting Proposition. Assume further that ε 1 has a density (w.r.t. Lebesgue mea- sure) locally bounded away from 0 and σ is locally bounded away from 0 . If Leb ( { x : r ( x ) > 0 } ) > 0 then there is AEA with geometrically decaying failure probabilities provided that there are N + , N − > 0 such that µ ( x ) ≤ − χ for x ≥ N + and µ ( x ) ≥ χ for x ≤ − N − . for some χ > 0 large enough. 22

Tools Ergodic theory of Markov chains on general state spaces, as described in the books of Nummelin or Meyn & Tweedie. Kontoyiannis, I. and Meyn, S. P. Spectral theory and limit theorems for ge- ometrically ergodic Markov processes. Ann. Appl. Probab. , 13 , 304–362, 2003. 23

What about O-U process ? Theorem. Assume that ε 1 has a density (w.r.t. Lebesgue measure) locally bounded and locally bounded away from 0 and σ is bounded and locally bounded away from 0 . Assume further that Ee κε 2 1 < ∞ for some κ > 0 . Then the mean-reverting condition | x + µ ( x ) | lim sup < 1 | x | | x |→∞ together with Leb ( { x : r ( x ) > 0 } ) > 0 imply AEA with geometrically decaying probability of failure. Kontoyiannis, I. and Meyn, S. P. Large deviations asymptotics and the spec- tral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 , 61–123, 2005. 24

Arbitrage in the expected utility sense I Proposition. Let U ( x ) := x α with some 0 < α < 1 . If a trading strategy π realizes AEA then there is a constant b > 0 such that for t large enough, EU ( V π t ) ≥ e bt . When α < 0 and U ( x ) := − x α then, in general, AEA (even with geomet- rically decaying failure probabilities) is insufficient to get exponentially fast convergence of EU ( V π t ) to 0 = U ( ∞ ) . 25