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Create `inflationcurve`

object for interest-rate curve from
dates and data

Build an `inflationcurve`

object using
`inflationcurve`

.

After creating a `inflationcurve`

object, you can use the associated
object function `indexvalues`

.

To price an `InflationBond`

,
`YearYearInflationSwap`

, or `ZeroCouponInflationSwap`

instrument, you must create an
`inflationcurve`

object and then create an `Inflation`

pricer
object.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available instruments, models, and pricing methods, see Choose Instruments, Models, and Pricers.

creates an `inflationcurve_obj`

= inflationcurve(`Dates`

,`InflationIndexValues`

)`inflationcurve`

object.

creates an `inflationcurve_obj`

= inflationcurve(___,`Name,Value`

)`inflationcurve`

object using name-value pairs
and any of the arguments in the previous syntax. For example,
```
myInflationCurve =
inflationcurve(InflationDates,InflationIndexValues,'Basis',4)
```

creates an `inflationcurve`

object. You can specify
multiple name-value pair arguments.

`indexvalues` | Calculate index values for `inflationcurve` object |

Build an inflation curve from a series of breakeven zero-coupon inflation swap (ZCIS) rates:

$$\begin{array}{l}I(0,{T}_{1Y})=I({T}_{0}){(}^{1}\\ I(0,{T}_{2Y})=I({T}_{0}){(}^{1}\\ I(0,{T}_{3Y})=I({T}_{0}){(}^{1}\\ \mathrm{...}\\ I(0,{T}_{i})=I({T}_{0}){(}^{1}\end{array}$$

where

$$I(0,{T}_{i})$$ is the breakeven inflation index reference number for maturity date

*T*_{i}.$$I({T}_{0})$$ is the base inflation index value for the starting date

*T*_{0}.$$b(0;{T}_{0},{T}_{i})$$ is the breakeven inflation rate for the ZCIS maturing on

*T*_{i}.

The ZCIS rates typically have maturities that increase in whole number of years. So the inflation curve is built on an annual basis. From the annual basis inflation curve, the annual unadjusted (that is, not seasonally adjusted) forward inflation rates are computed as follows:

$${f}_{i}=\frac{1}{({T}_{i}-{T}_{i-1})}\mathrm{log}\left(\frac{I(0,{T}_{i})}{I(0,{T}_{i-1})}\right)$$

The unadjusted forward inflation rates are used for interpolating and also for incorporating seasonality to the inflation curve.

For monthly periods that are not a whole number of years, seasonal adjustments can be made to reflect seasonal patterns of inflation within the year. These 12 monthly seasonal adjustments are annualized and they add up to zero to ensure that the cumulative seasonal adjustments are reset to zero every year.

$$\begin{array}{l}I(0,{T}_{i})=I({T}_{0})\mathrm{exp}\left({\displaystyle \underset{{T}_{0}}{\overset{{T}_{i}}{\int}}f(u)du)}\right)\mathrm{exp}\left({\displaystyle \underset{{T}_{0}}{\overset{{T}_{i}}{\int}}s(u)du)}\right)\\ I(0,{T}_{i})=I(0,{T}_{i-1})\mathrm{exp}(({T}_{i}-{T}_{i-1})({f}_{i}+{s}_{i}))\end{array}$$

where

$$I(0,{T}_{i})$$ is the breakeven inflation index reference number.

$$I(0,{T}_{i-1})$$ is the previous inflation reference number.

*f*_{i}is the annual unadjusted forward inflation rate.*s*_{i}is the annualized seasonal component for the period $$[{T}_{i-1},{T}_{i}]$$.

The first year seasonal adjustment may need special treatment, because typically, the breakeven inflation reference number of the first month is already known. If that is the case, the unadjusted forward inflation rate for the first year needs to be recomputed for the remaining 11 months.

[1] Brody, D. C., Crosby, J., and
Li, H. "Convexity Adjustments in Inflation-Linked Derivatives." *Risk
Magazine*. November 2008, pp. 124–129.

[2] Kerkhof, J. "Inflation
Derivatives Explained: Markets, Products, and Pricing." *Fixed Income
Quantitative Research*, Lehman Brothers, July 2005.

[3] Zhang, J. X. "Limited Price
Indexation (LPI) Swap Valuation Ideas." *Wilmott Magazine*. no. 57,
January 2012, pp. 58–69.