Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Abstract

London School of Economics and Political Science and Universität WienWhile absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Itô process. As a consequence, we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Itô process in a one-sided manner without reflection. Introduction.Most of the literature in mathematical finance assumes that discounted prices S = (S t ) 0≤t≤T of risky assets are modelled by semimartingales. In frictionless financial markets, where arbitrary amounts of stock can be bought and sold at the same price S t , the semimartingale assumption is necessary. Otherwise, there would exist "arbitrage opportunities" (see [26], Theorem 7.2 for a precise statement) and optimal strategies for utility maximisation problems would fail to exist or yield infinite expected utility (see [2,40,42]).For non-semimartingale models based on fractional Brownian motion (B H t ) t≥0 such as the fractional Black-Scholes model S t = exp(μt + σ B H t ), where μ ∈ R, σ > 0 and Hurst parameter H ∈ (0, 1) \ {

Continuous-time duality for superreplication with transient price impact

“…, such that almost surelyX n,↑ andX n,↓ converge weakly as Borel-measures on [0, T ] to, respectively, A and B, two adapted, right-continuous, and increasing processes with A 0− = B 0− = 0. By lower-semicontinuity and convexity of X → Λ X T , see (19) in Lemma 4.1, it follows that for…”

“…We thus postpone it to the end of Section 4.2. We adopted the notion of shadow prices from the theory of optimal investment with proportional transaction costs (see, e.g., [17,35,19]) where the martingales M with the stated flat-off conditions are constructed explicitly or emerge from duality of utility maximization. In our setting, the construction of shadow prices is more challenging as the spread λ is not given exogenously.…”

“…The duality theory for these models is now developed in great detail (see [32,17,33,34,14,28]). This has been used to study utility maximization via its relation to shadow prices ( [35,24,18,19,9]) and has also been instrumental in the development of asymptotic approaches for small transaction costs ( [42] and the references therein).…”

Robust utility maximisation in markets with transaction costs

“…The condition U (0) = 0 is used only to simplify calculations. Condition (11) is mild and so is (12): as shown in Corollary 4.2(i) of [32], for every utility function U with reasonable asymptotic elasticity, its conjugate V satisfies (12). The studies [11], [23] assumed a smooth U which is strictly concave on its entire domain, we do not need either smoothness or strict concavity of U .…”

Fractional Brownian motion satisfies two-way crossing

“…The notion of optimality considered here is to maximise the expectation value of some (reasonable) utility function. Using the stickiness prop-erty, [Czichowsky and Schachermayer, 2015] proved first that geometric fBm does admit a shadow price indeed in the case the utility function is bounded from above and defined on the whole R (e.g. U (x) = 1 − e −x ).…”