Shreve I-1: Binomial asset pricing model

Wherein we give a slightly more intuitive version of the central replication derivation. Suppose we have a derivative security (which here really just means a random asset) worth $V_1(\omega_1)$ at time 1 and seek to determine its fair price at time 0, $V_0$. We will have $V_0=X_0$ where $X_0$ is an as-yet unknown amount of money that will be needed to replicate the security.

The security presumably depends, whether positively or negatively, on a stock valued at $S_t$ at time $t$. So to replicate the security we buy some as-yet unknown amount $\Delta_0$ of shares of the stock.

((The whole thing is easier in the case where interest rates are $r=0$. Say the security returns either 10 or 17 depending on whether the stock is at 5 or 2, respectively. Then we just seek to express the security as a linear function of the stock, i.e. we seek numbers $\Delta_0$ and $c$ such that
$$
\Delta_0(5\text{ or }2)+c = \text{10 or 17},
$$
and then value the security at $\Delta_0 S_0+c$.

If we find “risk-neutral” probabilities under which the stock has expected value equal to its current value (these must exist, i.e., we must have $dS_0<S_0<uS_0$, or else all money should be taken out of the money market and invested in the stock), then the price of the security is just the expected value of the security under these probabilities. [Proof: First, this is true if the security is just equal to the stock, by definition of risk-neutral probabilities. Then as we saw above a general security is a linear function of the stock, and expectations preserve linear combinations.] The main advantage of this is not that we believe in a risk-neutral world — we might as well have used a world where the risk premium $\mathbb E S_1/S_0$ is 10% or some other fixed easy-to-compute number (well, assuming we could be sure that $\tilde p$ and $\tilde q$ would exist in that case too!) (and, well, dividing 1 by is significantly easier than dividing by any other number) — but that once we have the risk-neutral probabilities we can calculate the prices of many securities as long as they are all based on the same underlying set of stocks.
))

This leaves us with the cash position, i.e. money on hand, of $X_0-\Delta_0 S_0$, which of course we invest in the money market, i.e., we let somebody (such as a bank) borrow the money in return for paying us interest.

At time 1 the value of our portfolio (of stock and money market accounts) is

$$
X_1(\omega_1) = \Delta_0 S_1(\omega_1) + (1+r)(X_0-\Delta_0 S_0)
$$
Here $\omega_1\in\{H,T\}$, so we actually have two equations in the two unknowns $\Delta_0$, $X_0$. We impose $V_1(\omega_1)=X_1(\omega_1)$ for both $\omega_1$, and we assume $V_1(\omega_1)$ is known for each $\omega_1$.

Rather than relying on magic intuition, we solve this system of equations, using the fact that the inverse of the matrix
$$
\left[\begin{array}{rr}
a & b \\
c & d \\
\end{array}\right]
$$
is
$$
\frac{1}{ad-bc}
\left[\begin{array}{rr}
d & -b \\
-c & a \\
\end{array}\right]
$$
Only then is it time to bring in the risk-neutral probabilities $\tilde p$ and $\tilde q$. Namely, we are curious whether there exist probabilities so that the expected value of the stock is just the return from the money market. Under real-world probabilities this should not happen, since it would make it needlessly risky to invest in stocks.

Then it turns out that, lo and behold,

the risk-neutral-expected return of our derivative security is equal to the money market return of… our sought-for time 0 price of the security.

So from these risk-neutral probabilities we can calculate the value $V_0$ of the security, using $(1+r)V_0 = \tilde{\mathbb{E}} V_1$.