Short- and long-term relative arbitrage in stochastic portfolio - PowerPoint PPT Presentation

Short- and long-term relative arbitrage in stochastic portfolio theory Martin Larsson Department of Mathematics, ETH Zurich (with J. Ruf)

Stochastic Portfolio Theory was first introduced by Robert Fernholz. One considers the market weights S i µ i t t = S 1 t + · · · + S d t where S 1 t , . . . , S d t are the market capitalizations of d stocks. 2/25

Stochastic Portfolio Theory was first introduced by Robert Fernholz. One considers the market weights S i µ i t t = S 1 t + · · · + S d t where S 1 t , . . . , S d t are the market capitalizations of d stocks. t , . . . , θ d A basic goal is to find self-financing trading strategies θ t = ( θ 1 t ) that perform well relative to the market. The relative wealth is � t V θ t = θ ⊤ t µ t = V θ θ ⊤ 0 + s dµ s . 0 There is no bank account, but holding (a constant fraction of) the market portfolio is “relatively risk-free”. 2/25

Stochastic Portfolio Theory was first introduced by Robert Fernholz. One considers the market weights S i µ i t t = S 1 t + · · · + S d t where S 1 t , . . . , S d t are the market capitalizations of d stocks. t , . . . , θ d A basic goal is to find self-financing trading strategies θ t = ( θ 1 t ) that perform well relative to the market. The relative wealth is � t V θ t = θ ⊤ t µ t = V θ θ ⊤ 0 + s dµ s . 0 There is no bank account, but holding (a constant fraction of) the market portfolio is “relatively risk-free”. The market weights are Itˆ o semimartingales dµ t = b t dt + σ t dW t valued in ∆ d = { x ∈ R d + : x 1 + · · · + x d = 1 } . 2/25

Definition. Given T ≥ 0 , a self-financing trading strategy θ is a relative arbitrage over [0 , T ] if 0 = 1 , V θ ≥ 0 , V θ V θ T ≥ 1 , P ( V θ T > 1) > 0 . 3/25

Definition. Given T ≥ 0 , a self-financing trading strategy θ is a relative arbitrage over [0 , T ] if 0 = 1 , V θ ≥ 0 , V θ V θ T ≥ 1 , P ( V θ T > 1) > 0 . Questions: ◮ When does relative arbitrage over [0 , T ] exist for some T ≥ 0 ? ◮ How small/large can/must T be? ◮ What does θ look like? How (if at all) does it depend on the probabilistic properties of S (or µ )? 3/25

Conditions for relative arbitrage over [0 , T ] Fernholz ’02: Large enough T , provided for some δ > 0 , ε > 0 , � d � 1 ≤ i ≤ d µ i max t ≤ 1 − δ, λ min dt � log S � t ≥ ε ( ∗ ) 4/25

Conditions for relative arbitrage over [0 , T ] Fernholz ’02: Large enough T , provided for some δ > 0 , ε > 0 , � d � 1 ≤ i ≤ d µ i max t ≤ 1 − δ, λ min dt � log S � t ≥ ε ( ∗ ) Fernholz, Karatzas, Kardaras ’05: Any T > 0 , still assuming ( ∗ ) . 4/25

Conditions for relative arbitrage over [0 , T ] Fernholz ’02: Large enough T , provided for some δ > 0 , ε > 0 , � d � 1 ≤ i ≤ d µ i max t ≤ 1 − δ, λ min dt � log S � t ≥ ε ( ∗ ) Fernholz, Karatzas, Kardaras ’05: Any T > 0 , still assuming ( ∗ ) . Fernholz, Karatzas ’05: Large enough T , provided for some η > 0 , d d � µ i dt � log µ i � t ≥ η ( ∗∗ ) t i =1 4/25

Conditions for relative arbitrage over [0 , T ] Fernholz ’02: Large enough T , provided for some δ > 0 , ε > 0 , � d � 1 ≤ i ≤ d µ i max t ≤ 1 − δ, λ min dt � log S � t ≥ ε ( ∗ ) Fernholz, Karatzas, Kardaras ’05: Any T > 0 , still assuming ( ∗ ) . Fernholz, Karatzas ’05: Large enough T , provided for some η > 0 , d d � µ i dt � log µ i � t ≥ η ( ∗∗ ) t i =1 Banner, D. Fernholz ’08 and Pal ’16 : Short-term relative arbitrage. 4/25

Conditions for relative arbitrage over [0 , T ] Fernholz ’02: Large enough T , provided for some δ > 0 , ε > 0 , � d � 1 ≤ i ≤ d µ i max t ≤ 1 − δ, λ min dt � log S � t ≥ ε ( ∗ ) Fernholz, Karatzas, Kardaras ’05: Any T > 0 , still assuming ( ∗ ) . Fernholz, Karatzas ’05: Large enough T , provided for some η > 0 , d d � µ i dt � log µ i � t ≥ η ( ∗∗ ) t i =1 Banner, D. Fernholz ’08 and Pal ’16 : Short-term relative arbitrage. One might suspect that ( ∗ ) is an unrealistic condition, while ( ∗∗ ) is much better. Is it sufficient for short-term relative arbitrage? Until recently the answer to this question was unknown. 4/25

Theorem ( Fernholz, Karatzas, Ruf (FKR) ’18). The condition ( ∗∗ ) is not enough to guarantee relative arbitrage over [0 , T ] for any T > 0 . 5/25

We’ll use a condition that is similar to, but not exactly the same as, the condition ( ∗∗ ) : The market weight process dµ t = b t dt + σ t dW t with values in ∆ d is admissible if tr( σ t σ ⊤ t ) ≥ 1 . 6/25

We’ll use a condition that is similar to, but not exactly the same as, the condition ( ∗∗ ) : The market weight process dµ t = b t dt + σ t dW t with values in ∆ d is admissible if tr( σ t σ ⊤ t ) ≥ 1 . We’d like to compute the smallest time horizon beyond which relative arbitrage is always possible: � � every admissible market weight process T ∗ = inf T ≥ 0: admits relative arbitrage over [0 , T ] 6/25

We’ll use a condition that is similar to, but not exactly the same as, the condition ( ∗∗ ) : The market weight process dµ t = b t dt + σ t dW t with values in ∆ d is admissible if tr( σ t σ ⊤ t ) ≥ 1 . We’d like to compute the smallest time horizon beyond which relative arbitrage is always possible: � � every admissible market weight process T ∗ = inf T ≥ 0: admits relative arbitrage over [0 , T ] For d ≥ 3 , FKR show that d ( d − 1) ≤ T ∗ ≤ 1 − 1 1 d 6/25

Trading in the market weights µ 1 t , . . . , µ d t is equivalent to trading in 1 “relatively risk-free” asset (the benchmark) and d − 1 “relatively risky” assets. We make this explicit by a change of coordinates: Q ∆ d D = Q (∆ d ) µ t = ( µ 1 t , . . . , µ d X t = ( X 1 t , . . . , X d − 1 t ) ) = Qµ t t 7/25

Correspondence between: dµ t = b t dt + σ t dW t dX t = β t dt + ν t dW t self-financing trading in µ self-financing trading in (1 , X ) � t � t V ϕ V θ t = θ ⊤ θ ⊤ ϕ ⊤ t µ t = v 0 + s dµ s t = v 0 + s dX s 0 0 µ is admissible, tr( σ t σ ⊤ X satisfies tr( ν t ν ⊤ t ) ≥ 1 t ) ≥ 1 No relative arbitrage X satisfies (NA) on [0 , T ] exists over [0 , T ] 8/25

Upper bounds on T ∗ can be derived using functionally generated portfolios . 9/25

Upper bounds on T ∗ can be derived using functionally generated portfolios . For u ∈ C 2 ( R d − 1 ) , Itˆ o’s formula states that � t � t ∇ u ( X s ) ⊤ dX s = u ( X t ) − 1 tr( ∇ 2 u ( X s ) ν s ν ⊤ u ( X 0 ) + s ) ds. 2 0 0 This is the wealth V ϕ of the self-financing trading strategy ϕ t = ∇ u ( X t ) t with initial wealth u ( X 0 ) . 9/25

Upper bounds on T ∗ can be derived using functionally generated portfolios . For u ∈ C 2 ( R d − 1 ) , Itˆ o’s formula states that � t � t ∇ u ( X s ) ⊤ dX s = u ( X t ) − 1 tr( ∇ 2 u ( X s ) ν s ν ⊤ u ( X 0 ) + s ) ds. 2 0 0 This is the wealth V ϕ of the self-financing trading strategy ϕ t = ∇ u ( X t ) t with initial wealth u ( X 0 ) . d − | x | 2 ≥ 0 on D . In an admissible model, Example: Take u ( x ) = 1 − 1 � T V ϕ T − V ϕ tr( ν s ν ⊤ s ) ds ≥ T − (1 − 1 0 = − u ( X 0 ) + u ( X T ) + d ) . 0 Hence T ∗ ≤ 1 − 1 d . This is the upper bound of FKR. 9/25

Upper bounds on T ∗ can be derived using functionally generated portfolios . For u ∈ C 2 ( R d − 1 ) , Itˆ o’s formula states that � t � t ∇ u ( X s ) ⊤ dX s = u ( X t ) − 1 tr( ∇ 2 u ( X s ) ν s ν ⊤ u ( X 0 ) + s ) ds. 2 0 0 This is the wealth V ϕ of the self-financing trading strategy ϕ t = ∇ u ( X t ) t with initial wealth u ( X 0 ) . d − | x | 2 ≥ 0 on D . In an admissible model, Example: Take u ( x ) = 1 − 1 � T V ϕ T − V ϕ tr( ν s ν ⊤ s ) ds ≥ T − (1 − 1 0 = − u ( X 0 ) + u ( X T ) + d ) . 0 Hence T ∗ ≤ 1 − 1 d . This is the upper bound of FKR. What about lower bounds on T ∗ ? 9/25

Idea: Let T > 0 and suppose X is a martingale on [0 , T ] . This model does not admit relative arbitrage on [0 , T ] , so T ≤ T ∗ . 10/25

Idea: Let T > 0 and suppose X is a martingale on [0 , T ] . This model does not admit relative arbitrage on [0 , T ] , so T ≤ T ∗ . Theorem. With the notation ζ ( X ) = inf { t ≥ 0: X t / ∈ D } , one has the representation � o martingale in R d − 1 � X is an Itˆ T ∗ = sup ess inf ζ ( X ): d with dt tr � X � t = 1 10/25

Idea: Let T > 0 and suppose X is a martingale on [0 , T ] . This model does not admit relative arbitrage on [0 , T ] , so T ≤ T ∗ . Theorem. With the notation ζ ( X ) = inf { t ≥ 0: X t / ∈ D } , one has the representation � o martingale in R d − 1 � X is an Itˆ T ∗ = sup ess inf ζ ( X ): d with dt tr � X � t = 1 But how do we find martingales that don’t slow down, yet remain in D for a deterministic amount of time? 10/25

Here is a 2-dimensional martingale that doesn’t slow down, yet stays bounded for deterministic amounts of time: � X t � � � 1 Y t d = dW t = σ t dW t Y t − X t � X 2 t + Y 2 t It satisfies d ( X 2 t + Y 2 t ) = tr( σ t σ ⊤ t ) = | σ t | 2 dt = dt and looks like this: 11/25