Martingale pricing is a pricing approach based on the notions of
credit derivatives
, etc.
In contrast to the
Measure theory representation
Suppose the state of the market can be represented by the
filtered probability space
,
. Let
be a stochastic price process on this space. One may price a derivative security,
under the philosophy of no arbitrage as,
Where is the risk-neutral measure.
- is an -measurable (risk-free, possibly stochastic) interest rate process.
This is accomplished through almost sure replication of the derivative's time payoff using only underlying securities, and the risk-free money market (MMA). These underlyings have prices that are observable and known.
Specifically, one constructs a portfolio process in continuous time, where he holds shares of the underlying stock at each time , and cash earning the risk-free rate . The portfolio obeys the stochastic differential equation
One will then attempt to apply Girsanov theorem by first computing ; that is, the
Radon–Nikodym derivative
with respect to the observed market probability distribution. This ensures that the discounted replicating portfolio process is a Martingale under risk neutral conditions.
If such a process can be well-defined and constructed, then choosing will result in , which immediately implies that this happens -almost surely as well, since the two measures are equivalent.
See also
References
- from the original on 2009-10-16. Retrieved October 8, 2011.