Martingale representation theorem
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using Malliavin calculus.
Statement of the theorem
Let be a Brownian motion on a standard filtered probability space and let be the augmentation of the filtration generated by . If X is a square integrable random variable measurable with respect to , then there exists a predictable process C which is adapted with respect to , such that
Application in finance
The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that is a Q-martingale process, whose volatility is always non-zero. Then, if is any other Q-martingale, there exists an -previsible process , unique up to sets of measure 0, such that with probability one, and N can be written as:
The replicating strategy is defined to be:
- hold units of the stock at the time t, and
- hold units of the bond.
where is the stock price discounted by the bond price to time and is the expected payoff of the option at time .
At the expiration day T, the value of the portfolio is:
and it's easy to check that the strategy is self-financing: the change in the value of the portfolio only depends on the change of the asset prices .
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