# Pricing inflation lags

I've been looking into a short piece of maths I found on pricing inflation with payment delays, and was hoping someone could confirm whether my understanding was correct or if the maths isn't quite right to begin with. The setup: suppose, for example, you have an inflation-linked cash flow that captures the level of inflation between 2014-15, but this cash flow isn't paid until a year later in 2016.

Then the value of this cashflow is not necessarily just the expected 2014 $\rightarrow$ 2015 inflation discounted back from 2016, because you may have some correlation between interest rates and inflation. In essence, $\mathbb{E}(\text{I}_{2015} \cdot \text{DF}_{2016}) \ne \mathbb{E}(\text{I}_{2015}) \cdot \mathbb{E} (\text{DF}_{2016})$ if your discount factor and inflation are correlated.

This is not surprising - if inflation and rates are +100% correlated then an increase in inflation is partially "hedged" by rates rising, and conversely given -100% correlation a rise in inflation will be further compounded by lower discount rates. In either case, the expected value of your cashflow is nonlinear vs inflation, and you would expect it to be some function of correlation.

My aim is to understand the magnitude of the "convexity adjustment" $\Delta$, where $\mathbb{E}(\text{I}_{2015}) \cdot \text{DF}_{2016} = \Delta \cdot (\mathbb{E}(\text{I}_{2015}) \cdot \mathbb{E} (\text{DF}_{2016}))$. The simplistic maths I found to estimate this adjustment is below (verbatim). I am aware it's relatively crude but it doesn't need to be perfect for my purposes - I just want something in the right ballpark, better than making no adjustment at all. There are more sophisticated discussions of the maths such as this paper but I'd like to understand the simplified calculation first if possible.

Let $I_T =$ inflation fixing at time $T$, but paid on lagged date $T_L = T + D$. Write your discount factor from time al$a$ to $b$, as seen at time $c$, as $\delta_c(a,b)$. Then a payment of $I_T$ at time $T_L$ is equivalent to $I_T \cdot \delta_T(T,T_L)$ at time $T$.

Letting $r$ be the short-term interest rate observed at $T$, we can say $\delta_T (T,T_L) = e^{-r D}$.

Suppose now $I_T$ is lognormally distributed, variance $\sigma_I^2 T$, and $r$ is normally distributed with variance $\sigma_r^2 T$, and correlation is $\rho$. Then we have

$\mathbb{E}(I_T \cdot \delta_T(T,T_L)) = \exp{[-\rho \cdot \sigma_I \cdot \sigma_r \cdot D \cdot T]} \cdot \mathbb{E}(I_T) \cdot \delta_0(T,T_L) \,\,\,\, (*)$

and the first term is your convexity adjustment $\Delta$.

My first question is: is all of that maths valid, if oversimplified? I'm a little suspicious of saying $\delta_{T}\,(T,T_L) = e^{-r D}$ for a "short term rate" $r$ if $D$ is a long delay, e.g. 5 years. It seems like by simply using $r \times D$, you're inconsistently treating $r$ like a fixed number over the period $T \to T + D$. Or is the key point that once you reach time $T$, $r$ is observed and can be treated as "fixed"?

Assuming the maths is ok, is $(*)$ just a consequence of the fact that for correlated normal R.V.s $X,\, Y$ we have $\mathbb{E}(e^X e^{-Y}) = \mathbb{E}(e^{X-Y})$ and $X-Y \sim N(\mu_X - \mu_Y, \sigma_X^2 + \sigma_Y^2 -2\rho \sigma_X \sigma_Y)$? If I understand correctly you can just use that relation and substitute for $X$ and $Y$ in the above.

Lastly, in this context would $\sigma_I$ and $\sigma_r$ both be normal vols? That's how I read the maths, but the numerical example I have suggests that $\sigma_I$ should be more like a lognormal order of magnitude (e.g. nearer to 10% than 1%). Thanks for your help.