The Bank Account, Short Rate, and the Fundamental Pricing Equation

Every interest rate model is a machine for converting a specification of the short rate into bond prices. The short rate $r(t)$ is the instantaneous continuously compounded rate at which the money-market account grows at time $t$. It is the most local object in the term structure: a single number that cannot be observed directly but determines, via a risk-neutral expectation, every zero-coupon bond price in the market.

This note establishes the three building blocks of no-arbitrage pricing in interest rate models:

1. The money-market account $B(t) = e^{\int_0^t r(s)\,ds}$ — the risk-free savings account that earns the short rate continuously.
2. The stochastic discount factor $D(t,T) = B(t)/B(T)$ — the random variable that discounts a payoff at $T$ back to $t$.
3. The fundamental pricing equation — the statement that every zero-coupon bond price is a risk-neutral conditional expectation of the stochastic discount factor.

• define the money-market account, state its SDE, and write it as a stochastic exponential;
• define the short rate as the instantaneous spot rate and relate it to the bank account;
• state the stochastic discount factor $D(t,T)$ and interpret it as the random price of $1 at time$T$viewed from time$t$;
• state the fundamental pricing equation and explain why the expectation is taken under $\mathbb{Q}$, not $\mathbb{P}$;
• verify that the formula is consistent with the zero-coupon bond conventions (P(T,T)=1, deterministic short rate).

1. 1. The money-market account
2. 1. The stochastic discount factor
3. 1. The fundamental pricing equation
4. 1. More general claims
5. 1. Term structure from the short rate