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3.3.3 Evaluate an expectancy or eliminate the uncertainty
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The Itô lemma shows how it is possible to superpose infinitesimal
increments
to mimic the evolution of the value of a financial
derivative
, which is a known function of the stochastic
spot price
.
Starting from an initial (alt. terminal) value that is known at a
time
, a finite number of incremental changes
can in be
accumulated to approximate a single possible outcome at a later
(alt. earlier) time: the implementation of the socalled
MonteCarlo method
will discussed later with a practical example (sect.4.5).
At the end, the fair price for the derivative is calculated as the
expectancy from a large number of possible outcomes, i.e. by
performing a statistical average where each payoff is properly
weighted with the number of times this value has been reached.
The main drawback of a statistical method is the slow convergence
(
) with the number of samples. The problem can
be traced back to the difficulty of integrating the stochastic
term in the Itô differential (3.3.2#eq.2).
By combining anticorrelated assets, it is however possible to reduce
the amount of fluctuations in a portfolio. Sometimes, it is even possible
to completely eliminate the uncertainty through deltahedging,
in effect transforming the stochastic differential equation (SDE) into
a partial differential equation (PDE) that is much simpler to solve.
For that
 Create a portfolio, combining one derivative (e.g. an option) of
value
with a yet unspecified, but constant number
of the underlying asset. The initial value of this portfolio and its
incremental change per timestep are

(3.3.3#eq.1) 
where the Itô differential (3.3.2#eq.2) can be used to substitute
and the stochastic differential (3.3.1#eq.1) for
.
 Choose the right amount
of the underlying so as to exactly
cancel the random component, which is proportional to
in the
Itô differential

(3.3.3#eq.2) 
With this choice, the total value of the portfolio becomes deterministic,
i.e. the remaining equation has no term left in
.
 Noarbitrage arguments show that without taking any risk, the
portfolio has to earn the same as the riskfree interest rate

(3.3.3#eq.3) 
Indeed, if this was not the case and the earnings were larger (alt. smaller),
arbitrageurs would immediately borrow money from (alt. lend money to) the
market until the derivative expires and make a risk less profit from the
difference in the returns.
 Reassemble the small deterministic incremental values into a partial
differential equation, which can be solved more efficiently to obtain the
present value of the derivative
.
Of course, the amount
will change after a short time and the
portfolio has to be continuously rehedged to obtain a meaningful
value for the derivativewhich is not quite possible in the real world.
Two examples illustrate the procedure in the coming sections, using
deltahedging to calculate the price of derivatives in the stock and
the bond markets.
SYLLABUS Previous: 3.3.2 Itô lemma
Up: 3.3 Improved model using
Next: 3.4 Hedging an option