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
It
(2)
Π(t) = Π(0) exp [ r(τ)dτ
0
inria-00457222, version 1 - 16 Feb 2010
Replace Equation (1) by
and Equation (2) by
Πtrend = Vtrend
- ∆Strend
Πtrend = Πtrend(0) exp r(τ)dτ
0
Combining Equations (3) and (4) leads to the tracking control strategy
Vtrend — ∏trend(0)eʃθ r(T)dT
Strend
(3)
(4)
(5)
We might again call delta hedging this strategy, although it is of course an
approximate dynamic hedging via the utilization of trends.
2.2.2 Initialization
In order to implement correctly Equation (5), the initial values ∆(0) and Πtrend(0)
of ∆ and Πtrend have to be known. This is achieved by equating the logarithmic
derivatives at t = 0 of the right handsides of Equations (3) and (4). It yields
∆(o) = Vtrend(0) - r (0)Vtrend (0)
(6)
Strend(0) - r(0)Strend (0)
and
Πtrend(0) = Vtrend(0) - ∆(0)Strend(0)
(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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