The Black-Scholes formula III

The martingale representation theorem allows us to price derivative securities when Girsanov's theorem is not sufficient, usually being used when one considers incomplete markets where the processes exhibit jumps, switches, etc. The idea behind this approach is based on two assumptions: that the marginal rate of substitution [MRS] has the same source of randomness as the stock price, namely $B\left( t\right)$; and that the detrended percentage change of the MRS forms a martingale [the latter is there in order to ensure that the MRS is always nonnegative].

Then the martingale representation theorem states that there exists an $%% \mathscr{F}\left( t\right)$-measurable process $\theta \left( t\right)$ such that

$$\frac{d\mathcal{M}\left( t\right) }{\mathcal{M}\left( t\right) }-\alpha \left( t\right) dt = \theta \left( t\right) dB\left( t\right)$$

$$\mathcal{M}\left( t\right) = \mathcal{M}\left( 0\right) \exp \left\{ \left[ \alpha \left( t\ri... ...{2}\left( t\right) \right] dt+\theta \left( t\right) dB\left( t\right) \right\}$$

Now, there is one restriction that has to be satisfied. If the interest rate is $r\left( t\right)$, then $\frac{dB\left( t\right) }{B\left( t\right) }%% =-rdt$

$$-r\left( t\right) dt=\mathbf{E}\frac{d\mathcal{M}\left( t\right) }{\mathcal{M}\left( t\right) }=\alpha \left( t\right) dt$$,

which sets the drift of the MRS equal to the negative risk free rate of return. In addition

$$S\left( 0\right) =\mathbf{E}\left[ \frac{\mathcal{M}\left( t\righ... ...0\right) }S\left( t\right) \right] ,\forall t\in \left[ 0,T\right] \text{, or}$$

$$1=\mathbf{E}\left[ \exp \left\{ \int_{0}^{t}\left[ \mu \left( s\r... ...ft( s\right) +\sigma \left( s\right) \right] dB\left( s\right) \right\} \right]$$   ,

$$0=\int_{0}^{t}\left[ \mu \left( s\right) -r\left( s\right) -\frac... ...%% \left[ \theta \left( s\right) +\sigma \left( s\right) \right] ^{2}\right] dt$$,

$$0=\int_{0}^{t}\left[ \mu \left( s\right) -r\left( s\right) +\theta \left( s\right) \sigma \left( s\right) \right] dt$$.

Since the equality holds $\forall t\in \left[ 0,T\right]$ the solution is obviously

$$\theta \left( t\right) =\frac{\mu \left( t\right) -r\left( t\right) }{\sigma \left( t\right) }\text{,}$$

which is the price of risk that was considered in the previous approach. Then, the derivative prices that pay $g\left[ S\left( T\right) \right]$ on maturity are just

$$C=\mathbf{E}\left[ \frac{\mathcal{M}\left( T\right) }{\mathcal{M}\left( 0\right) }g\left[ S\left( T\right) \right] \right] \text{,}$$

which is just the ones obtained through Girsanov's theorem approach, since

$$\frac{\mathcal{M}\left( T\right) }{\mathcal{M}\left( 0\right) }=\exp \left\{ -\int_{0}^{t}r\left( s\right) ds\right\} M\left( T\right)$$   .