ARBITRAGE-FREE MODELS IN MARKETS WITH TRANSACTION COSTS

In the paper [ 7 ] , Guasoni studies ﬁnancial markets which are subject to proportional transaction costs. The standard martingale framework of stochastic ﬁnance is not applicable in these markets, since the transaction costs force trading strategies to have bounded variation, while continuous-time martingale strategies have inﬁnite transaction cost. The main question that arises out of [ 7 ] is whether it is possible to give a convenient condition to guarantee that a trading strategy has no arbitrage. Such a condition was proposed and studied in [ 6 ] and [ 1 ] , the so-called stickiness property, whereby an asset’s price is never certain to exit a ball within a predetermined ﬁnite time. In this paper, we deﬁne the multidimensional extension of the stickiness property, to handle arbitrage-free conditions for markets with multiple assets and proportional transaction costs. We show that this condition is sufﬁcient for a multi-asset model to be free of arbitrage. We also show that d -dimensional fractional Brownian models are jointly sticky, and we establish a time-change result for joint stickiness.


Introduction
In [7], a market with multiple risky assets and proportional transaction costs were studied. In the setting of [7], the market contains one risk free asset, used as a numeraire and hence assumed identically equal to 1, and d risky assets, given by an R d −valued process Y t = (Y 1 t , Y 2 t , · · · , Y d t ) 1 SUPPORTED BY NSF GRANT DMS 0907321.
614 DOI: 10.1214/ECP.v16-1671 that is càdlàg (right-continuous with left-limits), adapted, and quasi-left continuous (i.e., Y i τ = Y i τ− , 1 ≤ i ≤ d for all predictable stopping times τ). Transaction costs are proportional and each unit of numeraire traded in the risky assets generates a transaction cost of k units that are charged to the riskless asset account.
Trading strategies are given by adapted, left-continuous, R d −valued processes θ = (θ 1 t , θ 2 t , · · · , θ d t ) that are of finite variation and satisfy the following admissibility condition: for some determistic M > 0 and all t ≥ 0. Here Dθ i is the derivative of θ i t in the sense of distribution, and |Dθ i | t is the total variation measure associated to Dθ i in [0, t]. In (1) [7] and the quasi-left continuity assumption on the price processes, left-continuity of the trading strategies θ can be relaxed to predictablity.

Remark 1. Due to Proposition 2.5 of
In the case when there is only one risky asset, the model (1) reduces to (2) This model was studied in the recent papers [8,1]. In [8], the notion of stickiness (see definition 2.9 of [8] and also Proposition 1 of [1]) was introduced as a sufficient for no-arbitrage in the model (2). It was also shown that a large class of Markov processes and models with full support in the Wiener space are sticky. In [1] stickiness was further studied and other classes of sticky processes were provided. In this note, we introduce a condition, which we call joint stickiness, and show that it is sufficient for no-arbitrage in the model (1), see proposition 1. Then we show joint stickiness remains unchanged under composition with continuous functions, see proposition 2. As an example, we show the joint sticky property for independent fractional Brownian motions with possibily different Hurst parameters, see Proposition 3. Lastly, we show a time change result on joint stickiness and provide non-semimartingale joint sticky processes by using time change, see Proposition 4 and corollaries thereafter.

Main Results
Let X t = (X 1 t , X 2 t , · · · , X d t ) be a càdlàg process adapted to the filtration F = ( t ) t∈[0,T ] . For any F stopping time τ ≤ T , let A τ,ε

Definition 2.
We say that X t = (X 1 t , X 2 t , · · · , X d t ) is jointly sticky with respect to F if for any F stopping time τ ≤ T , and any ε > 0.
In the following proposition, we show that joint stickiness implies no arbitrage in the model (1).
be a jointly sticky, adapted, and càdlàg process . Then, the market (1, e X 0 t , e X 1 t , · · · , e X d t ) does not admit arbitrage with propotional transaction costs k for any k > 0.
If τ = t almost surely, then the left-continuity of the paths and the definition of τ implies θ s = 0 on [0, t] for almost all ω, thus V t (θ ) = 0 almost surely and this contradicts with the assumption P(V t (θ ) > 0) > 0. Therefore we assume that the event A = {τ < t} has positive probability. Let We can write (1) as following on A for any s ∈ [τ, t]. Thus (4) becomes and for all s ∈ [τ, t]. From (5) (6), and (7) we conclude that on A ε Note that almost surely on A (this follows from the definitions of A and τ ). Therefore from (7) it follows that V t (θ ) < 0 on A ε 1 ⊂ A whenever ε < ln(1 + k). This contradicts with the assumption P(V t (θ ) ≥ 0) = 1, since P(A ε 1 ) > 0 for all ε > 0. This shows that θ can not be an arbitrage strategy. This completes the proof.
is jointly sticky with respect to F. To see this, let τ be any The first equality above follows from the independence of L i τ+t − L i τ with τ for each 1 ≤ i ≤ d, the second equality follows from the independence assumption on L i t , i = 1, 2, · · · , d, and the last inequality follows from stickiness of lévy processes (the stickiness of Lévy processes was shown in [8]).
be a jointly sticky process with respect to the filtration F.
is also jointly sticky with respect to F.

Proof. Fix any ε > 0. For any stopping time
This completes the proof.

Example 2. Let B = (B 1
In the following Proposition shows that any finite sequence of independent fractional Brownian motions with possibily different Hurst parameters is jointly sticky.
Then B are independent fractional Brownian motions in the filtered probability t ≥ 0, therefore F B stopping times are also F Ω d stopping times. Now, let τ be any stopping time of F B and let for each 1 ≤ i ≤ d and for any ε > 0. To show the jointly stickiness of B, we need to show We divide the proof of (11) into two steps.