Martingale pricing
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc. In contrast to the PDE approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using a Monte Carlo approach. As such, Martingale pricing is preferred when valuing high-dimensional contracts such as a basket of options. On the other hand, valuing American-style contracts  is troublesome and requires discretizing the problem (making it like a Bermudan option) and only in 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American options. Measure theory representation. Suppose the state of the market can be represented by the filtered probability space,formula_1. Let formula_2 be a stochastic price process on this space. One may price a derivative security, formula_3 under the philosophy of no arbitrage as, Where formula_4 is the risk-neutral measure. formula_5 is an formula_6-measurable (risk-free, possibly stochastic) interest rate process. This is accomplished through almost  sure replication of the derivative's time formula_7 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 formula_8 in continuous time, where he holds formula_9 shares of the underlying stock at each time formula_10, and formula_11 cash earning the risk-free rate formula_12. The portfolio obeys the stochastic differential equation formula_13 One will then attempt to apply Girsanov theorem by first computing formula_14; 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 formula_9 can be well-defined and constructed, then choosing formula_16 will result in formula_17, which immediately implies that this happens formula_18-almost surely as well, since the two measures are equivalent.