Deriving Zero-Coupon Bond Price Dynamics from a Stochastic Yield
Summary
The document derives the price process of a zero-coupon discount bond from a stochastic process for its continuously compounded yield to maturity. It represents the bond price as the exponential of negative yield times remaining maturity, then applies Itô’s lemma to account for changes in yield, time to maturity, and yield variance.
The resulting bond-price dynamics have a state-dependent drift and volatility: both depend on the current yield and time remaining, with the yield’s mean-reverting drift contributing to the bond’s drift. This is a symbolic derivation rather than an empirical result. It assumes the stated yield process and the zero-coupon bond pricing relation; it does not address calibration, risk-neutral valuation, or whether the assumed yield dynamics are appropriate in a particular market.
Key ideas
- A zero-coupon bond paying one unit at maturity is priced as the exponential of negative yield times remaining maturity.
- Applying Itô’s lemma converts the yield process into a bond-price process.
- The bond’s instantaneous volatility depends on current yield and time remaining until maturity.
- The bond-price drift includes both the yield’s mean-reverting component and a variance correction.
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Full text
# Bond price and its process
# Bond price and its process
Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process
$dx=a(x_0-x)dt + sxdz$
where $a, x_0$ and $s$ are positive constants and $dz$ is the wiener process. What is the process followed by a bond price?
Solution: $dS=\mu Sdt+\sigma S dz$
where S is the bond price and $\mu$ and $\sigma$ are expected instantaneous return and instantaneous volatility respectively. Yield to maturity is the total return anticipated on a bond if the bond is held until the end of its lifetime.
## Answer by LocalVolatility (score 3, accepted)
https://quant.stackexchange.com/a/31660
If I understand your question correctly, then the zero-coupon bond price for the maturity $T$ is given by
\begin{equation} B_t = e^{-x_t (T - t)}, \end{equation}
where $t \in [0, T]$ and $x_t$ is the per annum yield-to-maturity. Note that you didn't make the definition of $x$ fully clear in your question.
To get the dynamics of $B$, you just apply the Ito formula to the function $f(t, x) = e^{-x (T - t)}$ with
\begin{equation} f_t(t, x) = x f(t, x), \quad f_x(t, x) = -(T - t) f(t, x), \quad f_{xx}(t, x) = (T - t)^2 f(t, x). \end{equation}
Then
\begin{eqnarray} \mathrm{d}B_t & = & \mathrm{d}f \left( t, x_t \right)\\ & = & x_t B_t \mathrm{d}t - (T - t) B_t \mathrm{d}x_t + \frac{1}{2} (T - t)^2 B_t \mathrm{d} \langle x \rangle_t\\ & = & \left( x_t + \frac{1}{2} (T - t)^2 s^2 x_t^2 \right) B_t \mathrm{d}t - (T - t) B_t \mathrm{d}x_t\\ & = & \underbrace{\left( x_t + \frac{1}{2} (T - t)^2 s^2 x_t^2 - (T - t) a \left( x_0 - x_t \right) \right)}_{\mu \left( t, x_t \right)} B_t \mathrm{d}t + \underbrace{(t - T) s x_t}_{\sigma \left( t, x_t \right)} \mathrm{B}_t \mathrm{d}z. \end{eqnarray}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.