Girsanov's theorem and the risk neutral measure

In this section we generalize Girsanov's theorem for the continuous time case, and derive the risk neutral measure. The methodology is very important, because

1. It allows one to compute derivative prices under assumptions which are more general than the BS ones.

2. It allows one to compute prices for derivatives that are path dependent, like Asian or lookback options.

3. The option price is expressed as an expectation under the risk neutral probability measure; evaluating such an expectation is usually easier than solving a second degree PDE with highly nonlinear terminal conditions. Even if the expectation cannot be solved analytically, one can utilize Monte Carlo methods.

The Girsanov theorem is as follows:

Theorem 42 (Girsanov-Martin-Cameron)   Suppose that we are given a probability space $\left( \Omega ,\mathscr{F}%% ,P\right)$, where $P$ is the market probability measure. Let $B\left( t\right)$ be a Brownian motion on that measure, and let $\mathscr{F}\left( t\right)$ be the filtration generated by this Brownian motion. In addition consider a process $\theta \left( t\right)$ adapted to $\mathscr{F}\left( t\right)$. Define the diffusion

$$B^{Q}\left( t\right) = \int_{0}^{t}\theta \left( s\right) ds+B\left( t\right) dB^{Q}\left( t\right) =\theta \left( t\right) dt+dB\left( t\right)$$

$$M\left( t\right) = \exp \left\{ -\int_{0}^{t}\theta \left( s\right) dB\left( s\right) -\frac{1}{2}\int_{0}^{t}\theta ^{2}\left( s\right) ds\right\}$$

In addition define a probability measure by

$$Q\left( F\right) =\int_{F}M\left( T\right) dP,\forall F\in \mathscr{F}$$.

Then, under $Q$ the process $B^{Q}\left( t\right)$ is a Brownian motion.

We can make the following observations about $M\left( t\right)$ and the measure $Q$

• $M\left( t\right)$ is a martingale under $P$. Specifically, Ito's formula gives that $dM\left( t\right) =-\theta \left( t\right) M\left( t\right) dB\left( t\right)$;

• $Q$ is an equivalent [to $P$] probability measure. In particular $%% Q\left( F\right) \geq 0,\forall F\in \mathscr{F}$, $\left( P\left( F\right) \Leftrightarrow Q\left( F\right) \right)$, and $Q\left( \Omega \right) =\int_{\Omega }M\left( T\right) dP=\mathbf{E}_{P}M\left( T\right) =1$;

• The expectations under $Q$ are given by $\mathbf{E}_{Q}X=\mathbf{E}%% _{P}\left[ M\left( T\right) X\right]$;

• If $\theta \left( t\right) =\theta$ [constant], then the distribution of $B^{Q}\left( t\right)$ under $P$ is normal with $%% B^{Q}\left( t\right) \sim \mathcal{N}_{P}\left( \theta T,T\right)$, while $%% B^{Q}\left( t\right) \sim \mathcal{N}_{Q}\left( 0,T\right)$ [therefore $%% B\left( t\right) \sim \mathcal{N}_{P}\left( 0,T\right)$, and $B\left( t\right) \sim \mathcal{N}_{Q}\left( -\theta T,T\right)$]^7.2

• The means change as we noted above, but the variances don't. This can be seen by observing that
  

  $$dB^{Q}\left( t\right) dB^{Q}\left( t\right) = \left[ \theta \left( t\right) dt+dB\left( t\right) \right] \left[ \theta \left( t\right) dt+dB\left( t\right) \right]$$

  $$= dB\left( t\right) dB\left( t\right)$$

  $$= dt.$$

  
  

• If $X$ is $\mathscr{F}\left( t\right)$-measurable [like a derivative price for example] since $M\left( t\right)$ is a martingale we have
  

  $$\mathbf{E}_{Q}\left[ X\vert\mathscr{F}\left( 0\right) \right] = \mathbf{E}_{P}%% \left[ M\left( T\right) X\vert\mathscr{F}\left( 0\right) \right]$$

  $$= \mathbf{E}\left[ \mathbf{E}\left[ M\left( T\right) X\vert\mathscr{F}\left( t\right) \right] \vert\mathscr{F}\left( 0\right) \right]$$

  $$= \mathbf{E}\left[ X\mathbf{E}\left[ M\left( T\right) \vert\mathscr{F}\left( t\right) \right] \vert\mathscr{F}\left( 0\right) \right]$$

  $$= \mathbf{E}\left[ XM\left( t\right) \vert\mathscr{F}\left( 0\right) \right] .$$

  
  

As usually, we will define as an arbitrage opportunity a portfolio $%% \mathbf{H}=\left\{ H\left( t\right) \right\} _{t=0}^{T}$ that satisfies

$$X\left( 0\right) =0$$, $$P\left[ X\left( T\right) \geq 0\right] =1$$, and $$P\left[ X\left( T\right) >0\right] >0$$.

We define a risk neutral probability measure as one that makes all discounted asset prices to form martingales, and we try to drive this measure. The stock price diffusion is written as

$$\frac{dS\left( t\right) }{S\left( t\right) } = \mu dt+\sigma dB\left( t\right)$$

$$= rdt+\sigma \left[ dB\left( t\right) +\frac{\mu -r}{\sigma }dt\right]$$

$$= rdt+\sigma dB^{Q}\left( t\right)$$

We can easily observe that in order to switch from the market to the risk neutral probability measure, we want to use Girsanov's theorem with $M\left( t\right) =\exp \left\{ -\int_{0}^{t}\theta dB\left( s\right) -\frac{1}{2}%% \int_{0}^{t}\theta ^{2}ds\right\}$, $\theta =\frac{\mu -r}{\sigma }$, the price of [volatility] risk.