The portfolio of an investor at time , i.e., the market value
of the single equities (contracts) in his portfolio at time
,
is dependent on the development of the price
$S$
,
$S$
up to
time
, that is, on the information that is available at that
particular time point. Given this it is obvious that his strategy,
i.e., the development of his portfolio's value over time, should
also be modelled as a
adapted
-dimensional
stochastic process
. In doing so
represents how much is the security
in his portfolio at time
in state
, where negative values indicate a short sell
of the corresponding contract.
The corresponding market model is thus
$S$
where
$S$
The two-dimensional stochastic process
now describes a portfolio strategy in which
gives
the number of stocks and
gives the number of bonds
in the portfolio at time
in state
. The value of the
portfolio at time
is then a random variable
A particularly important portfolio strategy is that once it is implemented it does not result in any cash flows over time, i.e., when the portfolio is re-balanced no payments are necessary. This means that eventual income (through selling securities, receiving dividends, etc.) is exactly offset by required payments (through buying additional securities, transaction costs, etc.) This is referred to as a self-financing strategy. One gets the impression that the change in value of the portfolio only occurs as the price of the participating securities changes.
In the following the Black-Scholes model will be considered. The
subsequent specification shows that arbitrage is not possible in
such a market: There is no admissible self-financing strategy with
a starting value of
$&phis#phi;$
, whose end value
$&phis#phi;$
is positive with a positive probability.
The explicit specification of the corresponding strategy can be
left out when it is clear from the context and can be
written as
. With the help of the Girsanov theorem a
equivalent measure
can be constructed, under which the
process of the discounted stock prices is a martingale. Using
(A.2) one obtains
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represents with respect to
a
equivalent martingale measure. It can be shown that given this
form, it can be uniquely determined.
From the Definition of and with the help of
(A.2) one obtains
As a result of the -martingale properties of
, due to Lemma A.1, the discounted value of a
self-financing strategy
is itself a local
-martingale. Thus the value of every admissible
self-financing strategy that is a non-negative local
-martingale is a
-super martingale. Consequently it
holds that: If the starting value of an admissible self-financing
strategy is equal to zero, then its value at all later time points
must also be equal to zero. Thus in using an admissible
self-financing strategy, there is no riskless profit to be made:
The Black-Scholes market is free of arbitrage.
The following theorem represents the most important tool used to value European options with the help of the Black-Scholes model. It secures the existence of an admissible self-financing strategy that duplicates the option, thus the value of which can be calculated using martingale theory.
represents at the same time the natural filtration
for the process
, which, as was seen above, is also a
-martingale. Therefore,
according to Theorem A.5 there exists using the martingale representation
a process
adapted on
with
-almost sure,
so that for all
it holds that:
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The result obtained from the last theorem was already formulated in Chapter 6 as equation (6.24).
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