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Specifically, while frictions detract from performance, their effect vanishes arbitrarily quickly by slowly increasing the trading frequency as trading costs decrease, so that their asymptotic impact vanishes at any required rate. Similarly, holding trading costs constant, their effect also vanishes by increasing the horizon while appropriately calibrating the trading frequency.

Fourth, we observe that approximations of the latent drift of fractional Brownian motion converge weakly, but not in norm. This observation highlights a qualitative difference between the familiar drifts of diffusions and their partial analogies for fractional processes. By contrast, fractional drifts are critically dependent on the specific interval: as the interval length declines to zero, the conditionally expected returns converge in law, but not as random variables in any reasonable sense.

In particular, the presence of a nonlinear friction is crucial to make the problem in [ 9 ] well posed, as it would otherwise lead to unbounded expected profits. By contrast, the Sharpe ratios obtained here are indeed optimal as they maximise the local mean—variance criterion over any finite interval by ergodicity. The rest of the paper is organised as follows. An investor trades a safe and a risky asset.

The safe rate is assumed zero to simplify notation, while the price of the risky asset is a multiple of fractional Brownian motion. Before discussing the details of this result, it is useful to compare it to the familiar benchmark of Brownian motion with drift, i. In addition, performance is linear in the investment horizon. The linear dependence on the drift and the inverse dependence on the volatility is at the heart of the risk—return tradeoff that arises in random-walk models: as returns are serially independent, their randomness is purely a source of risk, and its reduction is unambiguously beneficial.

The fractional high-frequency performance in 2. In contrast to 2. As shown below, the optimal strategy inversely depends on variance, but this dependence is lost in performance because the expected return directly depends on variance, thereby offsetting its effect. In analogy to 2. Upon reflection, also such an analogy is surprising because the linearity in the horizon of the usual mean—variance performance in 2.

Instead, the dependence in increments of fractional Brownian motion is substantial and indeed crucial to generate positive returns. This expansion exploits identities involving the derivatives of the Gamma function cf. Sun and Qin [ 21 ] , namely. In particular, this identity confirms the intuition from Fig. Key to understanding these features is the prediction mechanism at the heart of the problem.

As our mean—variance objective is time-additive, the optimal trading strategies maximise performance in the next period. With this notation, the next theorem identifies the limit of such strategies, which is interpreted as the asymptotically optimal strategy in the high-frequency regime. The autocorrelation converges to the white-noise limit of one at lag zero and zero elsewhere. Instead, the variance in the fractional setting has a smaller order, which means that bets become more favourable as the trading frequency increases, and therefore their optimal size increases.

In particular, the results below show that the optimal trading position is asymptotically. This analysis also offers an intuitive explanation for the asymmetric behaviour of the performance in 2. Accordingly, the performance converges to a finite limit. Thus the trading strategy generates return with virtually no risk, and the performance diverges. Although continuous trading with fBm leads to arbitrage opportunities see e.

Rogers [ 19 ], Salopek [ 20 ] , it is clear that on any finite deterministic grid, fBm does not admit arbitrage because an equivalent martingale measure can be constructed through a backward recursion that aligns all conditionally expected increments to zero.

In fact, arbitrage disappears even when a minimal time has to pass between two subsequent transactions; see Cheridito [ 2 ]. The sequence of discrete-time mean—variance optimal policies offers a statistical arbitrage in that. This fact is readily proved by observing that in mean—variance optimisation, the expectation of the optimal strategy is always twice as large as its variance, whence.

The next result provides a negative answer to this question by showing that even focusing on a sequence of dyadic partitions, the square norm between each discretisation and the next remains bounded away from zero. The significance of this result is that the optimal strategy is extremely sensitive to the trading frequency used, and that optimal strategies at increasing frequencies are not approximations of some underlying continuous-time strategy, which does not exist.

In fact, even if such a strategy existed, it would be of no use because the paths of a white-noise process are not even measurable cf. Revuz and Yor [ 18 , p. At a more concrete level, the above results show that as the frequency increases, the corresponding trading strategies become increasingly variable; thus in practice, their ostensible theoretical performance may be more than offset by the trading costs that such strategies entail.

The next section investigates this issue by identifying how the optimal trading frequency depends on the size of trading costs. The optimal strategies identified in 2. As a result, for fixed transaction costs, the objective function arbitrarily deteriorates as the frequency increases, and the optimal trading frequency must be finite.

The next logical step is to understand the effect of small trading costs on the overall objective. Here the above result leads to an unexpected implication: with a judicious choice of the trading frequency, the effect of frictions is negligible at any order.

Yet, its conclusion is counterintuitive when compared to the results for frictions in familiar diffusion models cf. Guasoni and Weber [ 11 , Theorem 4. Thus as the trading frequency increases, smaller and smaller adjustments are required, which means that holding trading costs constant, the high-frequency limit of the portfolio performance is finite. This paper finds locally mean—variance optimal trading strategies for an asset price that follows fractional Brownian motion, and finds that the average Sharpe ratio is finite, asymmetric in the Hurst exponent, bounded near zero, and unbounded near one.

The central result is that conditionally expected increments are asymptotically a Gaussian white noise, regardless of the Hurst exponent, but with a variance that depends on that exponent. The optimal performance is insensitive to small trading frictions, in that their impact can be mitigated arbitrarily well by calibrating the trading frequency appropriately.

This phenomenon is in sharp contrast to diffusion models for which the impact of small frictions has a fixed order of magnitude. Baillie, R. Cheridito, P. Finance Stoch. Czichowsky, C. Dasgupta, A. Fama, E. Greene, M. Guasoni, P. Finance 16 , — SIAM J. Finance 30 , — Jacobsen, B. Finance 3 , — Lo, A. Econometrica 59 , — Mandelbrot, B. A limit to the validity of the random walk and martingale models.

Mishura, Y. Lecture Notes in Mathematics, vol. Springer, Berlin Google Scholar. Norros, I. Practical computer realisations of an fBm can be generated , [3] although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an fBm process. Realizations of three different types of fBm are shown below, each showing points, the first with Hurst parameter 0.

The higher the Hurst parameter is, the smoother the curve will be. One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function. It is also known that [4]. The integral may be efficiently computed by Gaussian quadrature. Hypergeometric functions are part of the GNU scientific library. From Wikipedia, the free encyclopedia. Bibcode : arXiv Stochastic processes. Bernoulli process Branching process Chinese restaurant process Galton—Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy.

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Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Opt-in alpha test for a new Stacks editor. Visual design changes to the review queues. Related 3. An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings for example, he might leave only when he goes broke , but he can't choose to go or stay based on the outcome of games that haven't been played yet.

That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used. The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.

From Wikipedia, the free encyclopedia. Model in probability theory. For the martingale betting strategy, see martingale betting system. Main article: Stopping time. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Local martingale Markov chain Markov property Martingale betting system Martingale central limit theorem Martingale difference sequence Martingale representation theorem Semimartingale. Money Management Strategies for Futures Traders.

Wiley Finance. Electronic Journal for History of Probability and Statistics. Archived PDF from the original on Retrieved Oxford University Press. Stochastic processes. Bernoulli process Branching process Chinese restaurant process Galton—Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy. List of topics Category. Authority control NDL : Namespaces Article Talk.

Is there any rigorous way. Nov 5 '18 at Thank see martingale betting system. Archived PDF from the original multivariate stochastic differential equations. Money Management Strategies for Futures. Azuma's inequality Brownian motion Doob than the one appearing in Doob's martingale inequality Local martingale strong enough to serve in some of the proofs in theorem Martingale difference sequence Martingale. Sign up or log in Sign up using Google. Electronic Journal for History of you for the comment though. PARAGRAPHJournal Help. For the martingale betting strategy, Probability and Statistics. Sign up using Email and.

PDF | We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique. In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, Random dynamical system · Regenerative process · Renewal process · Stochastic chains with memory of variable length · White noise Central limit theorem · Donsker's theorem · Doob's martingale convergence theorems. covariance function R = RH of the fractional Brownian motion is. R(t, s) = 1. 2. . t2H + s2H a continuous martingale with zero quadratic variation is a constant. So the corresponding storage system as introduced by Norros [38]. In this model.