Martingale pricing is a pricing approach based on the notions ofmartingale andrisk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety ofderivatives contracts, e.g.options,futures,interest rate derivatives,credit derivatives, etc.

In contrast to thePDE approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using aMonte Carlo approach. As such, martingale pricing is preferred when valuing high-dimensional contracts such as a basket of options. On the other hand, valuingAmerican-style contracts is troublesome and requires discretizing the problem (making it like aBermudan option) and only in 2001F. A. Longstaff andE. S. Schwartz developed a practical Monte Carlo method for pricing American options.^[1]

Suppose the state of the market can be represented by thefiltered probability space,${\displaystyle (\mathrm{\Omega },({\mathcal{F}}_t{)}_{t\in [0,T]},\tilde{\mathbb{P}})}$. Let${\displaystyle \{S(t){\}}_{t\in [0,T]}}$ be a stochastic price process on this space. One may price a derivative security,${\displaystyle V(t,S(t))}$ under the philosophy of no arbitrage as,

${\displaystyle D(t)V(t,S(t))=\tilde{\mathbb{E}}[D(T)V(T,S(T))|{\mathcal{F}}_t],dD(t)=-r(t)D(t)dt}$

Where${\displaystyle \tilde{\mathbb{P}}}$ is therisk-neutral measure.

${\displaystyle (r(t){)}_{t\in [0,T]}}$ is an${\displaystyle {\mathcal{F}}_t}$-measurable (risk-free, possibly stochastic) interest rate process.

This is accomplished throughalmost sure replication of the derivative's time${\displaystyle T}$ 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${\displaystyle \{X(t){\}}_{t\in [0,T]}}$ in continuous time, where he holds${\displaystyle \mathrm{\Delta }(t)}$ shares of the underlying stock at each time${\displaystyle t}$, and${\displaystyle X(t)-\mathrm{\Delta }(t)S(t)}$ cash earning the risk-free rate${\displaystyle r(t)}$. The portfolio obeys the stochastic differential equation

${\displaystyle dX(t)=\mathrm{\Delta }(t)dS(t)+r(t)(X(t)-\mathrm{\Delta }(t)S(t))dt}$

One will then attempt to applyGirsanov theorem by first computing${\displaystyle \frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}}$; that is, theRadon–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${\displaystyle \mathrm{\Delta }(t)}$ can be well-defined and constructed, then choosing${\displaystyle V(0,S(0))=X(0)}$ will result in${\displaystyle \tilde{\mathbb{P}}[X(T)=V(T)]=1}$, which immediately implies that this happens${\displaystyle \mathbb{P}}$-almost surely as well, since the two measures are equivalent.

• Brownian model of financial markets
• Martingale (probability theory)

• Finance theories
• Mathematical finance
• Pricing
• Financial models