Modelling the yield curve

In our previous blog post on the OLOs of belgium Debt, we introduced the yield curve which dictates the prices for bonds with different maturities. More precisely the yield curve dictates how a euro paid at horizon \(\tau\) is worth today. This is what we call the discount factor \(D(\tau)\). As the yield curve moves over time due to factors described in our previous blog, we need a formal theory that enables us to model these shifts in the yield curve. We do this not by modelling the yield curve directly as this would be a complicated endeavour, but rather we model the short rate \(r\) and assume the no-arbitrage system holds. we can affectively reconstruct the yield curve by modelling the short rate \(r\) as we can derive the discounting to time \(\tau\) by compounding the short rate:

\[D = e^{-\int_t^Tr(s) ds }\]

But we do not know how the short rate \(r\) will behave in the future so we therefore need to model the price of a bond as the average over all possible paths:

\[P(t, T) = \mathbb{E}[ e^{-\int_t^Tr(s) ds}]\]

Also, the stochastic process can vary depending on the approach. In this blog post we will focus on the Vacisek stochastic differential equation (SDE):

\[dr = \kappa( \theta - r(t)) dt + \sigma dW\]

There is a lot to unpack here and let us begin with the two time varying blocks called the noise \(\sigma dW\) and the drift \(\kappa (\theta - r(t))\). The drift is multiplied with \(dt\) which is the infinitesimal slice of clock time. it's deterministic. \(\kappa\) is the mean reversion speed, \(\theta\) is the long run level, \(r(t)\) is the state of the current short rate \(r\). The noise has a volatilty term \(\sigma\) and an infinitesimal increment \(dW\) of a Wiener process (Brownian motion) \(W(t)\) in it.
This discretizes in:

\[\begin{aligned} \Delta r = \underbrace{\kappa(\theta - r)\,\Delta t}_{\text{drift } \propto\,\Delta t, } + \underbrace{\sigma\sqrt{\Delta t}\,Z}_{\text{noise }\propto\,\sqrt{\Delta t}} \end{aligned}\]

\(\sigma\) is the volatility. 1

Footnotes

1

Brownian motion is continuous but nowhere differentiable. There's no smooth function hiding inside \(dW\); it's an irreducibly separate object, which is exactly why stochastic calculus exists as its own thing.The deterministic piece scales like \(\Delta t\); the random piece scales like \(\sqrt{\Delta t}\)​. Those are not the same, and the gap explodes as you zoom in. Take \(\Delta t=0.01\): the drift scale is \(0.01\), the noise scale is \(\sqrt{0.01}=0.1\) — the randomness is ten times larger. Shrink the step further and the ratio gets worse. So on very short timescales an SDE path is essentially all noise; the drift is invisible. Zoom out instead, and drift accumulates linearly in time while noise only accumulates as the square root — so over long horizons the drift eventually wins. That crossover between \(\Delta t\) and \(\sqrt{\Delta t}\) is the entire personality of a stochastic process.