# Inflation Swap, Cap and Floor Model

 

Pricing Inflation Swap, Cap and Floor

 

A model is presented for pricing swaps, caps, and floors on inflation index returns. To capture general term structures of interest rates and index volatilities, the model requires time-averaged forward rate, and volatility inputs.

 

For a series of reset times

      ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png) 

 

we consider swap-type payments of the following form

 

·         At maturity time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png)                                         (Zero swap)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)               (Zero cap)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png)               (Zero floor)

 

·         At each reset time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png)                                       (Reset swap)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png)             (Reset cap)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png)             (Reset floor)

 

where

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) denotes the level of the inflation index at time ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image022.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image024.png)  denotes the reset cap (floor) return strike.

 

We assume that the level of the inflation index satisfies a domestic risk-neutral SDE of the form

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image026.png)                                  (1)

 

where

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png) is the deterministic domestic continuously compounded risk-free,

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image030.png) is a deterministic continuous dividend yield parameter,

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image032.png) is a deterministic volatility parameter,

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image034.png) is a standard Brownian motion.

 

Given SDE (1) for index levels, we may model

                             ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image036.png)                     (2)

 

where

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image040.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image042.png) is a standard Brownian motion.

 

The model

                             ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image044.png)                                  (3)

 

where

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image038.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image047.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image049.png) is a standard Brownian motion.

 

 

We note that, in marginal distribution, Equations 2 and 3 are equivalent.

###  

### In the inflation index market, forward levels ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image051.png) for a series of “key maturities” ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image053.png) are traded.  In particular, forward levels of the form

 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image055.png)

 

where

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image057.png) is the domestic discount factor, at time 0, to time![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image059.png)

·         ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image028.png) is the domestic continuously compounded risk-free rate.

 

 

Given the assumptions described in Sections 3.1 and 3.2 above, we have

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image062.png)

 

and

![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image064.png)             (4)

 

for all ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image066.png)

 

References:

<https://finpricing.com/lib/FiBond.html>

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