Arbitrage Bounds for Risk-Averse Arbitrageurs

To quantify the arbitrageur’s assessment of risk, we have to equip her with a corre- sponding utility function.

Assumption 3. The arbitrageur has a utility functionUγ(r)with risk aversion parameterγ, whererare the log returns implied by her trading decision. Furthermore, we assumeU_γ0(r)>0

andU_γ00(r)<0.

The arbitrageur maximizes the expected utilityEt(Uγ(r)), which we express in terms

of the certainty equivalent (CE). We derive the CE of exploiting concurrent cross- market price differences in the following theorem.

Theorem 1. Under Assumptions 1–3, the certainty equivalent (CE) resulting from the arbi- trage trade is given by

CE =δb,s_t +_Et(τ)µst + ∞ X k=2 Uγ(k) δ_tb,s+_Et(τ)µst k!U0 γ δ_tb,s+_Et(τ)µst Et r₍b,s_t:t+τ)−δ_tb,s−_Et(τ)µst k , (2.7) whereUγ(k)(r) := ∂ k ∂rkUγ(r). Proof. See Appendix B.1.

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2.2 Settlement Latency and Limits to Arbitrage

Theorem 1 allows us to compare the expected utility of making the arbitrage trade versus staying idle (which yields a riskless return of zero). The arbitrageur is will- ing to exploit cross-market price differences if and only if the CE of trading given by Equation (2.7) is positive. A positive CE corresponds to a statistical arbitrage oppor- tunity in the sense of positive expectedrisk-adjusted profits. Whenever the observed price differences δ_tb,s are positive butCE is negative, the arbitrageur does not trade. In this case, although the trade would be profitable under the possibility of instanta- neous settlement, limits to (statistical) arbitrage arise due to stochastic latency. Hence, the arbitrageur is indifferent between trading and staying idle if the observed price differencesδb,s_t implyCE = 0.

Definition 1. We define the arbitrage bound ds_t as the minimum price difference necessary such that the arbitrageur prefers to trade. Formally,ds_tis the maximum of zero and the (unique) root4_of F(d) =d+_Et(τ)µst + ∞ X k=2 Uγ(k)(d+Et(τ)µst) k!U0 γ(d+Et(τ)µst) Et rb,s_(t:t+τ)−d−_Et(τ)µst k . (2.8) Price differences below the arbitrage boundds_t might persist as the arbitrageur prefers not to trade in such a scenario.

More intuitive representations of the arbitrage bound can be derived by assuming that the arbitrageur is equipped with absolute or relative risk aversion. In particular, we follow Schneider (2015) in ignoring the impact of higher order moments above the fourth degree of the Taylor representation in Equation (2.8) and assume that the price process has a drift of µs

t of zero. These two additional assumptions yield an

analytically tractable formulation of the arbitrage bound. The following lemma gives the analytical closed-form expression for ds

t under the assumption of a power utility

function.

Lemma 5. If, in addition to Assumptions 1 and 2, the arbitrageur has an isoelastic utility functionUγ(r) := (1+r)

1−γ

1−γ with risk aversion parameterγ >1, the arbitrage bound forµ s t = 0

4_{By definition of the CE, we haveF(d) = U}−1

γ Et Uγ d+µs tτ+ Rt+τ t σ s tWks . SinceU_γ0(r) > 0,

the expectation is increasing ind. Moreover, sinceUγ00(r)<0, the inverseUγ−1(r)>0is also strictly

is given by ds_t = 1 2σ s t r γ_Et(τ) + q γ2 Et(τ)2+ 2γ(γ+ 1)(γ+ 2) (Vt(τ) +Et(τ)2). (2.9)

Proof. See Appendix B.1. Hence, ds

t positively depends on (i) the arbitrageur’s risk aversion γ, (ii) the local

volatility on the sell-side market, σs

t, (iii) the (conditionally) expected waiting time

until settlement, Et(τ), and (iv) the conditional variance of the waiting time, Vt(τ).

We show in Appendix B.4 that the special case of an exponential utility function with constantabsoluterisk aversionγyields a similarly tractable expression.

The arbitrage bound obviously depends on the arbitrageur’s risk aversion γ. Ac- cordingly, for a risk-neutral arbitrageur, we have ds

t = 0 and she would exploit any

positive price differenceδ_tb,s > 0. In this case, any price differences between the two markets should be absorbed immediately. Hence, in the absence of any other frictions, the existence of persistent price differences between two markets (which are not traded away) indicates that the markets are populated by risk-averse arbitrageurs who do not exploit price differences below the thresholdds_t. We thus denote the interval[0, ds_t]as a

no-trade regionin which price differences between marketsbandsare not exploited.5

The lower bound ds

t is a fundamental pillar of markets with settlement latency, as

the implied costs of settlement latency affect the entire action and contracting space of market participants. The only possibility to circumvent this latency would be the ability to sell instantaneouslyat the more expensive market to lock in the price differ- ence. This is only possible, however, if the arbitrageur already has an inventory of the asset on the expensive (sell) market or if she can borrow the asset on that market. The first alternative bears considerable additional risks. To be able to exploit instantaneous price differences whenever they arise, arbitrageurs have to keep inventory on the sell- side market over longer periods. Only investors who do not require a premium for this inventory risk (e.g., due to hedging needs or diversification benefits) could act as risk- neutral arbitrageurs.6 _{The second strategy requires that short-selling is offered by an}

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retically, repeatedly exploiting price differences may lead to a vanishing variance of the arbitrageurs’ aggregate returns which is equivalent to a contraction of the relevant bounds. From an empirical perspective, however, high autocorrelation in the resulting individual returns due to the latency questions the feasibility of such a law of large numbers.

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