arXiv:1005.0194v1 [q-fin.PR] 3 May 2010 Delta Hedging in Financial Engineering: Towards a Model-Free Approach Michel FLIESS, C´ edric J OIN Abstract— Delta hedging, which plays a crucial rˆ ole in modern financial engineering, is a tracking control design for a “risk-free” management. We utilize the existence of trends in financial time series (Fliess M., Join C.: A mathematical proof of the existence of trends in financial time series, Proc. Int. Conf. Systems Theory: Modelling, Analysis and Control, Fes, 2009. Online: http://hal.inria.fr/inria-00352834/en/ ) in order to propose a model-free setting for delta hedging. It avoids most of the shortcomings encountered with the now classic Black-Scholes-Merton framework. Several convincing computer simulations are presented. Some of them are dealing with abrupt changes, i.e., jumps. Keywords—Financial engineering, delta hedging, dynamic hedging, trends, quick fluctuations, abrupt changes, jumps, tracking control, model-free control. I. I NTRODUCTION Delta hedging, which plays an important rˆ ole in financial engineering (see, e.g., [36] and the references therein), is a tracking control design for a “risk-free” management. It is the key ingredient of the famous Black-Scholes-Merton (BSM) partial differential equation ([3], [33]), which yields option pricing formulas. Although the BSM equation is nowadays utilized and taught all over the world (see, e.g., [24], [39]), the severe assumptions, which are at its bottom, brought about a number of devastating criticisms (see, e.g., [7], [21], [22], [23], [31], [37], [38] and the references therein), which attack the very basis of modern financial mathematics, and therefore of delta hedging. We introduce here a new dynamic hedging, which is influ- enced by recent advances in model-free control ([10], [12]), 1 and bypass the shortcomings due to the BSM viewpoint: • In order to avoid the study of the precise probabilistic nature of the fluctuations (see the comments in [11], [13], and in [20]), we replace the various time series of prices by their trends [11], like we already did for redefining the classic beta coefficient [14]. • The control variable satisfies an elementary algebraic equation of degree 1, which results at once from the dynamic replication and which, contrarily to the BSM equation, does not need cumbersome final conditions. • No complex calibrations of various coefficients are required. Michel FLIESS is with INRIA-ALIEN & LIX (CNRS, UMR 7161), ´ Ecole polytechnique, 91128 Palaiseau, France. Michel.Fliess@polytechnique.edu C´ edric J OIN is with INRIA-ALIEN & CRAN (CNRS, UMR 7039), Nancy-Universit´ e, BP 239, 54506 Vandoeuvre-l` es-Nancy, France. cedric.join@cran.uhp-nancy.fr 1 See, e.g., [26] for a most convincing application. Remark 1.1: Connections between mathematical finance and various aspects of control theory has already been exploited by several authors (see, e.g., [2], [34], [35] and the references therein). Those approaches are however quite far from what we are doing. Our paper 2 is organized as follows. The theoretical back- ground is explained in Section II. Section III displays several convincing numerical simulations which • describe the behavior of Δ in “normal” situations, • suggest new control strategies when abrupt changes, i.e., jumps, occur, and are forecasted via techniques from [16] and [13], [14]. Some future developments are listed in Section IV. II. THE FUNDAMENTAL EQUATIONS A. Trends and quick fluctuations in financial time series See [11], and [13], [14], for the definition and the existence of trends and quick fluctuations, which follow from the Cartier-Perrin theorem [4]. 3 Calculations of the trends and of its derivatives are deduced from the denoising 4 results in [17], [32] (see also [18]), which extend the familiar moving average techniques in technical analysis (see, e.g., [1], [28], [29]). B. Dynamic hedging 1) The first equation: Let Π be the value of an elementary portfolio of one long option position V and one short position in quantity Δ of some underlying S: Π= V - ΔS (1) Note that Δ is the control variable: the underlying asset is sold or bought. The portfolio is riskless if its value obeys the equation dΠ= r(t)Πdt where r(t) is the risk-free rate interest of the equivalent amount of cash. It yields Π(t) = Π(0) exp  t 0 r(τ )dτ (2) Replace Equation (1) by Π trend = V trend - ΔS trend (3) 2 See [15] for a first draft. 3 The connections between the Cartier-Perrin-theorem (see [30] for an introductory explanation) and technical analysis (see, e.g., [1], [28], [29]) are obvious (see [11] for details). 4 The Cartier-Perrin theorem permits to give a new definition of noises in engineering [9].