Inflation Index Volatility in the Jarrow–Yildirim Model

Summary

The document presents the Jarrow–Yildirim model’s risk-neutral dynamics for the nominal short rate, real short rate, and inflation index. The index’s proportional change has a drift given by the nominal rate minus the real rate, alongside a stochastic term scaled by the parameter σI. The rate processes are mean reverting, and their drift functions are intended to match the term structure; the Brownian motions are correlated.

The author asks why the index needs its own volatility parameter if inflation is determined by the nominal-real rate difference. This frames a useful modeling distinction: that rate difference specifies the index’s drift, while σI represents random variation around that drift. The document itself supplies the equations and defines the parameters but contains no answer, derivation, calibration example, or evidence about how σI should be estimated. It therefore introduces the model structure and the conceptual question without resolving practical modeling choices.

Key ideas

• The model specifies the inflation index’s drift as the difference between nominal and real short rates.
• The parameter σI scales a separate stochastic component in the index dynamics.
• The nominal and real rates follow mean-reverting processes with drift functions used to match the term structure.
• Correlations among the Brownian motions affect how the modeled rates and index move together.
• The document poses the interpretation question but does not provide a derivation or calibration method.

Tags

# Jarrow-Yildirim $\sigma_I$


# Jarrow-Yildirim $\sigma_I$












Under the Jarrow-Yildirim model, the nominal short rate $r_n$, the real rate $r_r$ and index $I$ are modelled according to the following stochastic differential equations under the Martingale measure $\mathbb{Q}$:

\begin{eqnarray*} d r_n (t) & = & [\theta_n (t) - a_n r_n (t)] d t + \sigma_n d W_n (t)\\ d r_r (t) & = & [\theta_r (t) - \sigma_r \sigma_i \rho_{r, I} - a_r r_r (t)] d t + \sigma_r d W_r (t)\\ \frac{d I (t)}{I (t)} & = & [r_n (t) - r_r (t)] d t + \sigma_I d W_I (t) \end{eqnarray*}

where $W_n$, $W_r$ and $W_I$ are correlated Brownian motions, $\rho_{r, I}$ is the correlation between $W_r$ and $W_I$, $\theta_n$ and $\theta_r$ are drift functions used to match the term structure, $a_n$ and $a_r$, are constant mean reversion speeds and $\sigma_n$, $\sigma_r$ and $\sigma_I$ are constant volatility parameters.

What is the purpose of the $\sigma_I$ parameter? I am confused since my naiive intuition tells me that an instantaneous percentage increment of the inflation index should be fully determined by the inflation short rate: $r_n (t) - r_r (t)$. I.e.

\begin{eqnarray*} \frac{d I (t)}{I (t)} & = & [r_n (t) - r_r (t)] d t \end{eqnarray*}

Any pointers as to where my logic is failing would be appreciated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.