> For the complete documentation index, see [llms.txt](https://finwhite.gitbook.io/inflationcap/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://finwhite.gitbook.io/inflationcap/inflation-swap-cap-and-floor-model.md). # 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: --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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