A model-free approach to delta hedging

1 Introduction

Delta hedging, which plays an important role in financial engineering (see, e.g.,
[24] and the references therein), is a tracking control design for a “risk-free” man-
agement. It is the key ingredient of the famous Black-Scholes-Merton (BSM)
partial differential equation ([3, 22]), which yields option pricing formulas. Al-
though the BSM equation is nowadays utilized and taught all over the world
(see, e.g., [18]), the severe assumptions, which are at its bottom, brought about
a number of devastating criticisms (see, e.g., [6, 16, 17, 20, 25, 26] and the ref-
erences therein), which attack the very basis of modern financial mathematics
and therefore of delta hedging.

We introduce here a new dynamic hedging, which is influenced by recent
works on model-free control ([8, 10]), and bypass the shortcomings due to the
BSM viewpoint:

• In order to avoid the study of the precise probabilistic nature of the fluc-
tuations (see the comments in [9, 11]), we replace the various time series
of prices by their trends [9], like we already did for redefining the classic
beta coefficient [12].

• 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.

Remark 1.1 Connections between mathematical finance and various aspects of
control theory has already been exploited by several authors (see, e.g., [2, 23] and
the references therein). Those approaches are however quite far from what we
are doing.

Our paper is organized as follows. The theoretical background is explained in
Section 2. Section 3 displays several convincing numerical simulations which

• describe the behavior of ∆ in “normal” situations,

• suggest new control strategies when abrupt changes, i.e., jumps, occur,
and are forecasted via techniques from [13] and [11, 12].

Some future developments are listed in Section 4.

2 The fundamental equations

2.1 Trends and quick fluctuations in financial time series

See [9], and [11, 12], for the definition and the existence of trends and quick
fluctuations, which follow from the Cartier-Perrin theorem [4].¹ Calculations
of the trends and of its derivatives are deduced from the denoising results in
[14, 21] (see also [15]), which generalize the familiar moving average techniques
in technical analysis (see, e.g., [1, 19]).

¹The connections with technical analysis (see, e.g., [1, 19]) are obvious (see [9] for details).

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