Let be a Brownian motion on a standard filtered probability space and let be the augmented 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
Consequently,
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 is 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 .
- Montin, Benoît. (2002) "Stochastic Processes Applied in Finance" [1]
- Elliott, Robert (1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 36, 213–226