# 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.

• Isn’t it because the nominal-real interest rate differential gives the expected inflation, not the actual realized inflation. The latter has some randomness on top. – dm63 Mar 24 at 7:37
• Thanks dm63. I think that you are right. Many thanks – Nick Deguillaume Mar 24 at 8:40