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A debt security that doesn't pay interest (a coupon) but is traded at a deep discount, rendering profit at maturity when the bond is redeemed for its full face value.

A zero coupon bond with maturity date $T$, also called a $T$-bond, is a contract which guarantees the holder $1$ dollar to be paid on the date $T$. The convention that the payment at the time of maturity, known as the principal value or face value, equals one is made for computational convenience.The price at time t of a bond with maturity date T is denoted by $P(t, T)$ and it is assumed

  • There exists a (frictionless) market for $T$-bonds for every $T > 0$.
  • The relation $P(t, t) = 1$ holds for all $t$.
  • For each fixed t, the bond price $P(t, T)$ is differentiable w.r.t. time of maturity $T$.

The relation $P(t, t) = 1$ above is necessary in order to avoid arbitrage.

Risk neutral valuation

Zero Coupon Bond prices are given by the formula $$P(t,T)=\mathbb{E}^Q\left[\exp\left(\int_{t}^{T}r_s ds\right)|\mathcal{F}_t\right]$$