SYLLABUS Previous: 3.3.2 Itô lemma
Up: 3.3 Improved model using
Next: 3.4 Hedging an option
3.3.3 Evaluate an expectancy or eliminate the uncertainty
[ SLIDE
portfolio -
no random -
no arbitrage -
Black-Scholes -
properties ||
VIDEO
modem -
LAN -
DSL]
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 so-called
Monte-Carlo 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 anti-correlated 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 delta-hedging,
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 time-step 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
.
- No-arbitrage arguments show that without taking any risk, the
portfolio has to earn the same as the risk-free 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 re-hedged to obtain a meaningful
value for the derivative-which is not quite possible in the real world.
Two examples illustrate the procedure in the coming sections, using
delta-hedging 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