Interest-Rate Trees, Forwards and Futures

In the stock trees so far the rate was a constant $r$ and the stock did all the moving. Now the rate moves. Toss a coin each period: $R_0$ is fixed today, and $R_n$ depends on the first $n$ tosses. One dollar put in the money market at time $n$ grows to $1 + R_n$ at time $n+1$. You know $R_n$ when you make the deposit, so the bank account is random over many periods but riskless over the next one.

The discount factor to time $m$ multiplies up the one-period rates along the path:

The last rate in the product, $R_{m-1}$, is set at time $m-1$, so $D_m$ depends only on the first $m-1$ tosses.

The tree is written directly under the risk-neutral measure: we choose the up and down probabilities and then define the time-0 price of any payment $X$ at time $m$ as $\tilde{\mathbb E}[D_m X]$. Every discounted price is then a martingale, so no trading strategy in the model can be an arbitrage. A model built this way, starting from the short rate, is a short-rate model. Ho-Lee and Black-Derman-Toy are discrete examples; Vasicek, CIR and Hull-White are the continuous-time ones.

Running example: $R_0 = 5\%$. After the first toss $R_1 = 7\%$ on heads or $3\%$ on tails, each with risk-neutral probability $\tfrac12$. Then