A model-free approach to delta hedging

2.2 Dynamic hedging

2.2.1 The first equation

Let Π be the value of an elementary portfolio of one long option position V and
one short position in quantity ∆ of some underlying S:

Π=V -∆S                      (1)

Note that ∆ 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.
yields

Π(t) = Π(0) exp [ r(τ)dτ

0

∆(o) = Vtrend(⁰) ^{- r} (⁰)Vtrend ⁽⁰⁾

Strend(⁰) ^{- r}(⁰)Strend ⁽⁰⁾

and

(7)

Remark 2.1 Let us emphasize once more that the derivation of Equations (5),
(6) and (7) does not necessitate any precise mathematical description of the sto-
chastic process S and of the volatility. The numerical analysis of those equations
is moreover straightforward.

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