# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: The Binomial Representation Theorem – III

BRP tells us that a previsible $\phi_k$ exists. Now the trick to use BRP to get a price for our claim $X$ is to do the following:

Let’s start at time $0$. Construct a portfolio $\Pi_0$:

\displaystyle \begin{aligned} \Pi_0 &= \phi_1 S_0 + \psi_1 B_0\\&=B_0(\phi_1 S_0 B^{-1}_0 + \psi_1)\\&=B_0(\phi_1 Z_0 + Y_0 - \phi_1 Z_0)\\&=B_0Y_0\\ &= Y_0 \\&= E^Q[B^{-1}_T | X]\end{aligned}

$; \psi_1 = Y_0 - \phi_1 B^{-1}_0 S_0 = Y_0 - \phi Z_0$

The value of this portfolio at time $1$, then, becomes:

$\Pi_1 = \phi_1 S_1 + \psi_1 B_1$

Substituting $\psi_1$ as above and rearranging gives:

\displaystyle \begin{aligned} \Pi_1 &= \phi_1 S_1 + B_1 Y_0 - B_1 \phi Z_0\\&=B_1(Y_o + \phi_1 Z_1 - \phi_1 Z_0)\\&=B_1(Y_0 + \phi_1 \Delta Z_1) \end{aligned}

Then, here we can exploit BRP to write $Y_1 = Y_0 + \phi_1 \Delta Z_1$, and this gives us:

$\Pi_1 = B_1(Y_0 + \phi_1 \Delta Z_1) = B_1Y_1$

Next, our strategy requires that we construct a porfolio $\Pi_1$:

\displaystyle \begin{aligned} \Pi_1 &= \phi_2 S_1 + \psi_2 B_1\\&=B_1(\phi_1 S_1 B^{-1}_0 + \psi_2)\\&=B_1(\phi_2 Z_1 + Y_1 - \phi_2 Z_1)\\&=B_1Y_1\end{aligned}

That is, it costs exactly the same as the value of the portfolio we ended up with at time $1$ when we created $\Pi_0$ at time $0$.

This can be carried on recursively to show that our original portfolio $\Pi_0$ is self-financing. At ‘maturity’ $T$, we end up with $B_T Y_T = B_T E^Q[B^{-1}_T X_T | F_T] = B_T B^{-1}_T X_T = X_T$, i.e. our original claim. That is, the price of claim $X_T$ should be exactly equal to the cost of the portfolio created originally (no money has come in or left the system) $E^Q[B^{-1}_T X] = Y_0 = \phi_1 S_0 + \psi_1 B_0$.

The moral of the story is that within a binomial two-asset setting there exists a self-financing strategy $(\phi_i, \psi_i)$ that duplicates the claim that we want to price. That is, the claim can be priced no matter what happens to the path of $S$ / filtration $F_i$ (ironically, then, the claim $X$ in the binomial world is redundant – as it can be reconstructed from the already existing $S$ and $B$).

I think it’s a good way to introduce the martingale representation theorem – but I personally would have preferred that it be done using the language of delta-hedging / risk-neutralily / no-arbitrage. In a way, I think it hampers BR in formally introducing the first fundamental theorem of asset pricing (they do point towards it “…as an afterthought”). Let’s conclude this post with the formal statement then:

First Fundamental Theorem of Asset Pricing

No-arbitrage implies that there exists a probability measure $Q$ under which the discounted price process $B^{-1}_i S_i$ is a martingale, and vice-versa. The measure $Q$, then, is called an equivalent martingale measure.