Price Index

The Price Index can be monthly or quarterly. It is a flat-right interpolated curve $I(t)$ that contains historical values. The curve is defined on a discrete set of dates $\mathcal T$ where for $\tau\in\mathcal T, I(\tau)$ is the published price index for period start date $\tau$. If $\tau_0=$max$\mathcal T$ is the latest historical start period, then for $\tau>\tau_0$,

$$I(\tau)=I(\tau_0)F(\tau-\tau_0)$$

Note that no seasonal factor adjustments are made (but can be included here). Also note that Last Period Start should be set to $\tau_0$.

If $p(\tau)$ is the publication date for period starting $\tau$ (typically $p(\tau)\gt\tau$) and $s(t)$ is the greatest $\tau$ for a given date $t$ for which $p(\tau) \le t$, then $p(s(t))\le t$ and $s(p(\tau))=\tau$. The value of the Index at $t$ is $I(t)=I(s(t))$.

Price Index References

If an inflation contract needs the value of a price index sampled at time $T$, the sampled value is called a reference value $I_R(T)$. Generally, $I_R(T)$ will be a function of several previously published price index values.

For deal valuation, the price index reference convention needs to be known. The following conventions have been implemented:

• IndexReference$\mathscr l$M (for $\mathscr l$=2,3) gives the published price index on the first day of the month that is $\mathscr l$ calendar months prior to the month of $T$:

$$I_R(T)=I((T-\mathscr {l} m)^{(1)})$$

where $T^{(i)}$ denotes the $i^{th}$ day in month $T$ and $T-\mathscr {l} m$ is result of subtracting $\mathscr l$ calendar months from T.

• IndexReferenceInterpolated$\mathscr l$M (for $\mathscr l$=3 or 4) gives the following interpolation of $I((T-\mathscr {l} m)^{(1)})$ and $I((T-(\mathscr {l}-1) m)^{(1)})$:

$$I_R(T)=I((T-\mathscr {l} m)^{(1)})+\Biggl( \frac{T-T^{(1)}}{(T^{(1)}+1m)-T^{(1)}} \Biggl)\Big( I((T-(\mathscr {l}-1) m)^{(1)})-I((T-\mathscr {l}m)^{(1)})\Big)$$ .

Inflation Rates

Inflation rate price factors are similar to interest rate price factors but have an associated price index factor.

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GBMPriceIndexModel

The model is specified as follows:

$$F(t) = exp \Big( (\mu-\frac{\sigma^2}{2})t + \sigma W(t) \Big)$$

Where:

• $\mu$ is the drift of the price index
• $\sigma$ is the volatility of the price index
• $W(t)$ is a standard Wiener Process under the real-world measure

Note that the simulation of this model is identical to plain Geometric Brownian Motion with the exception of modifying the scenario grid to coincide with allowable publication dates obtained by the corresponding Price Index