We consider an economy containing a single asset that is traded on two different mar- kets b and s. The trading activity on these markets is exogenously given and we as- sume that agents can continuously monitor the quotes of the asset across all markets. We assume that marketi∈ {b, s}continuously provides marginal buy quotes (asks)Ai t
and sell quotes (bids)Bi
t(withBti ≤Ait) for one marginal unit of the asset at timet. We
address the possibility to trade more than one marginal unit of the asset and the con- sideration of transaction costs in the next section and show that these generalizations do not affect our main insights.
Our sole agent is an arbitrageur who aims to exploit observed price differences across markets. The arbitrageur continuously monitors the quotes on marketsbands
and considers the following strategy: if buying on one market and selling on the other market implies a profit, she intends to buy a marginal unit of the asset on the market with the lower buy quote, transfer the asset to the market with a higher sell quote and sell it as soon as the transfer is settled.
We assume that (margin-based) short-selling is too costly to render a short-based strategy profitable or, alternatively, that margin constraints, which would allow for short sales, bind. Similarly, we assume that inventory holdings on any of the markets are too risky or are exhausted. In that sense, we consider a scenario where, upon observing the quotes, any cross-market price differences have already been absorbed up to the point where the arbitrageur is forced to physically transfer the asset between markets.
market band sells on market s. The converse case of selling on marketb and buying on marketscan be handled analogously. Hence, in case of frictionless trading and no latency in settlement, the arbitrageur exploits observed price differences if
Bts > Abt, (2.1)
as she can buy the asset on marketbatAbt, instantaneously transfer the asset to markets
and sell it again at priceBs t.
An instantaneous transfer is not possible, however, whenever the settlement of the transaction is time-consuming. Such a (possibly random) latency constitutes a fun- damental element of distributed ledger systems that do not rely on central clearing entities. It should not be confused, however, with latency inorder executionas heavily discussed in the context of high-frequency trading (e.g., Hasbrouck and Saar, 2013; Foucault et al., 2017). Such latencies are in the order of milliseconds and thus of sev- eral magnitudes smaller than settlement latencies. Therefore, without loss of general- ity, we refrain from latency in order execution and assume that markets process orders instantaneously.
Let latencyτ denote the random waiting time until a transfer of the asset between markets is settled. If the buy transaction on market b takes place at time t and the transfer of the asset to market sis settled att+τ, the arbitrageur faces the sell quote
Bs
t+τ. The profit of the arbitrageur’s trading decision is thus at risk if the probability of
losing money is non-zero, that is, if
P Bts+τ < A
b t
>0. (2.2) In this case, a risk-averse arbitrageur faces limits to (statistical) arbitrage whenever the associated risk exceeds the expected return (see, e.g., Bondarenko, 2003). To formalize the trading decision of the arbitrageur, denote the log quotes by abt := log Abt and
bst := log (Bts), respectively, to cast the payoff in log returns. The log return resulting
from buying on marketbat time t and selling on markets at timet+τ is then given by r(b,st:t+τ):=bst+τ −abt = δb,st |{z} instantaneous return +bst+τ −bst | {z } exposure to price risk , (2.3) whereδtb,s:=bs
2.2 Settlement Latency and Limits to Arbitrage
settlement, that is, in the absence of any latency. The second part of the decomposi- tion captures the risk of adverse price movements on the sell-side market. As the instantaneous returnδb,st is observable and thus known int, the arbitrageur only faces uncertainty about the evolution of prices on the sell-side market. The price process on the sell-side market is given as follows.
Assumption 1. For a given latencyτ, we model the log price change on the sell-sidebs t+τ−bst as a Brownian motion with driftµst such that
rb,s(t:t+τ) =δtb,s+τ µst + t+τ Z t σtsdWks, (2.4) where σs
t denotes the spot volatility of the bid quote process on market s, andWks denotes a Wiener process. We assume that σs
t is constant over the interval [t, t+τ] and rule out any jumps.2
The dynamics of the sell price thus expose the arbitrageur to uncertainty about her profits. The uncertainty is triggered by the spot volatility σts and the latencyτ. We require only weak assumptions regarding the stochastic nature of the latency.
Assumption 2. The stochastic latency τ ∈ R+ is a random variable equipped with a con-
ditional probability distribution πt(τ) := π(τ|It), where It denotes the set of available in- formation at time t. We assume that the moment-generating function of πt(τ), defined as mτ(u) :=Et(euτ)foru∈R, is finite on an interval around zero.
Assumptions 1 and 2 allow us to fully characterize the return distributionπt
rb,s(t:t+τ)
through the interval of random length from t to t + τ for a wide range of latency distributions.
Lemma 4. Under Assumptions 1 and 2, the returns follow a normal variance-mean mixture with probability distribution
πt r(b,st:t+τ) = Z R+ πt rb,s(t:t+τ)τ πt(τ)dτ, (2.5)
2Time-varying and stochastic volatility can be incorporated by means of a change of the time-scale
of the underlying Brownian motion. We provide the corresponding derivations in Appendix B.2.
However, both the time-variability ofσstand the presence of jumps would further increase the price
and corresponding characteristic function3 ϕrb,s (t:t+τ) (u) = eiuδb,st m τ iuµst − 1 2u 2(σs t) 2 . (2.6)
Proof. See Appendix B.1.
For any valid distributionπt(τ), Lemma 4 characterizes the impact of stochastic latency
on the return distribution. In Appendix B.3, we illustrate the special case whereπt(τ)
follows an exponential distribution and show that the resulting return distribution follows an asymmetric Laplace distribution.