Martingale pricing

Suppose the state of the market can be represented by the filtered probability space,${\displaystyle (\Omega ,({\mathcal {F}}_{t})_{t\in [0,T]},{\tilde {\mathbb {P} }})}$. 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,

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

Where ${\displaystyle {\tilde {\mathbb {P} }}}$ is the risk-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 through almost 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 ${\displaystyle \Delta (t)}$ shares of the underlying stock at each time ${\displaystyle t}$, and ${\displaystyle X(t)-\Delta (t)S(t)}$ cash earning the risk-free rate ${\displaystyle r(t)}$. The portfolio obeys the stochastic differential equation

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

One will then attempt to apply Girsanov theorem by first computing ${\displaystyle {\frac {d{\tilde {\mathbb {P} }}}{d\mathbb {P} }}}$; 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 ${\displaystyle \Delta (t)}$ 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)