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

ON LONG-TERM ARBITRAGE OPPORTUNITIES IN MARKOVIAN MODELS OF FINANCIAL MARKETS Mikl´

• s R´

asonyi, University of Edinburgh∗ Based on joint work with Martin L. D. Mbele Bidima. Vienna, 15th July 2010

∗On leave from MTA SZTAKI, Budapest.

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Motivation I

• H. F¨
• llmer 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.

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Motivation II For simplicity, we consider only one stock throughout this talk. dSt = Σ(St)(dWt + φ(St)dt), t ≥ 0. Wt: Brownian motion. Usual class of admissible strategies. Value process of strategy πt denoted by V π

t .

φ: market price of risk. Σ: volatility.

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Motivation III S has a nontrivial market price of risk if lim

T→∞ P

• 1

T

T

0 |φ(St)|2dt < c

• = 0.

for some c > 0. Technical assumption: minimal martingale measure exists for all finite hori- zons T > 0.

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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 Tε such that for all T > Tε P(V π

T ≥ eγT ) ≥ 1 − ε

and V π

T ≥ −e−γT , for some admissible trading strategy π = π(ε, T), start-

ing from 0 initial capital.

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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 Tε ?

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Control of failure probabilities I The market price of risk satisfies a large deviation estimate if there are c1, c2 > 0 such that lim sup

T→∞

1 T ln P

• 1

T

T

0 |φ(St)|2dt ≤ c1

• ≤ −c2.
• Conjecture. This condition should imply the existence of strategies as in the

above Theorem but with P(V π

T ≥ eγT ) ≥ 1 − e−βT ,

for some β > 0.

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Control of failure probabilities II In F&S paper the relationship of the above statement to large deviation the-

• ry 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.

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Model description I Discrete time modelling. Positive price process St = exp[Xt] where Xt is a Markov chain with dynamics Xt+1 − Xt = µ(Xt) + σ(Xt)εt+1, where εt are i.i.d., µ, σ measurable and X0 constant. Define Ft := σ(X1, . . . , Xt). Trading strategies will be Ft-predictable pro- cesses, i.e. πt is Ft−1-measurable.

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Model description II No short-selling. No borrowing of money. πt takes values in [0, 1] and represents the proportion of wealth allocated to the stock. Dynamics of the wealth process: V π

t+1 = V π t

• 1 − πt+1 + πt+1

St+1 St

• and V π

0 = V0 > 0 constant.

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

T ≥ eγT ) ≥ 1 − ε,

for all T ≥ Tε.

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Arbitrage concepts II

• Proposition. If there is AEA then for each ε > 0 there exists Tε and trading

strategies πt(ε, T), t ≥ 1, such that for T ≥ Tε we have V π(ε,T)

T

≥ V0 − e−γT/2 and P(V π(ε,T)

T

≥ eγT/2) ≥ 1 − ε.

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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 πt, t ≥ 0 such that P(V π

t ≤ eγt) ≤ Ce−βt.

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A naive approach I r(x) := E exp[µ(x) + σ(x)ε1] − 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 ε1 Gaussian but will investigate that case later.

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A naive approach II One may try the (Markovian) strategy ˜ πt := 1{r(Xt−1)>0}

• r rather

˜ πt := ηr(Xt−1)1{r(Xt−1)>0} with some 0 < η < 1.

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Main result I

• Theorem. Assume that µ, σ are bounded,

Eea|ε1| < ∞ for all a > 0 and for some c > 0, lim

T→∞ P

 1

T

T

• i=1

r2(Xi−1)1{r(Xi−1)>0} < c

  = 0.

(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.

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Gaussian case

• Theorem. Assume further that ε1 is standard Gaussian, σ(x) > 0 for all x

and for some c > 0, lim

T→∞ P

   

1 T

T

• i=1
• µ(Xi−1)

σ(Xi−1) + σ(Xi−1) 2

2

1

µ(Xi−1) σ(Xi−1)+σ(Xi−1) 2

>0

< c     = 0.

(2) Then there is AEA. Condition (2) is completely analogous to the condition of F& S paper (except the indicator).

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Main result II

• Theorem. Under the conditions of the previous Theorem,

lim sup

T→∞

1 T ln P

 1

T

T

• i=1

r2(Xi−1)1{r(Xi−1)>0} ≤ c1

  ≤ −c2

(3) for some c1, c2 > 0 implies AEA with geometrically decaying failure proba- bilities. In the Gaussian case we may again replace r by µ σ + σ 2.

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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.

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Sufficient conditions

• Theorem. Assume furthermore that ε1 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 (X1)|X0 = x] ≤ (1 − δ)V (x)1{x/

∈Γ} + C1{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.

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Geometric ergodicity When Xt is irreducible and aperiodic (in a suitable sense) then E[V (X1)|X0 = x] ≤ (1 − δ)V (x)1{x/

∈Γ} + C1{x∈Γ}

implies that the law of Xt 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.

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Drift criterion in the present setting

• Proposition. Assume further that ε1 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.

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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-

• metrically ergodic Markov processes. Ann. Appl. Probab., 13, 304–362,

2003.

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What about O-U process ?

• Theorem. Assume that ε1 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 Eeκε2

1 < ∞ for some κ > 0.

Then the mean-reverting condition lim sup

|x|→∞

|x + µ(x)| |x| < 1 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.

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Arbitrage in the expected utility sense I

• Proposition. Let U(x) := xα 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 ) ≥ ebt.

When α < 0 and U(x) := −xα 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(∞).

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Arbitrage in the expected utility sense II

• Theorem. Assume that ε1 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 Eeκε2

1 < ∞ for some κ > 0.

Then the mean-reverting condition lim sup

|x|→∞

|x + µ(x)| |x| < 1 together with Leb({x : r(x) > 0}) > 0 imply the existence of α0 < 0 such that for all α0 < α < 0, |E(V π

t )α| ≤ Ke−βt,

(4) for some constants K = K(α), β = β(α) > 0.

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Arbitrage in the expected utility sense III The converse is always true: (4) for some α implies that the strategy π pro- vides AEA with geometrically decreasing probability of failure. Next step: non-Markovian settings should also be studied, in a quantitative way. THANK YOU.

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