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HOW TO PRICE YEARONYEAR INFLATION SWAPS AND OPTIONS

In the inflation market we witness an interesting peculiarity: whilst the most traded type of inflation swap is (as now) the Zero Coupon type, inflation caps and floors are instead based on a Year on Year coupon type. This implies that the underlying of caps/floors (the floating leg of a year on year swap) is largely not observable in the market place and must instead be built synthetically.

SETTINGS:

Our {SWIL } function allows to generate Year on Year quotes by customizing the following

inputs: A) Inflation Zero Coupon curves can be either obtained from a contributor (namely a market maker, an interdealer broker or a Bloomberg composite price) or be user inputted;

B) Inflation Volatility. Inflation volatility is fully customizable, the market currently quotes caps and floors premiums (basis points upfront) which we convert into volatility surfaces. Volatility has a twofold usage; it is an input to calculate the convexity adjustments necessary to convert zero coupon quotes into year on year quotes (therefore providing the underlying for both year on year swaps and caps/floors) and also to appraise the time value of inflation caps/floors.

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Inflation volatility can be expressed either as shifted lognormal or normal, also allowing for negative strike prices.

C) Once defined the settings for zero coupon curves and volatility, the Details page of SWIL allows to choose the settings for the generation of the convexity adjustment vector.

Users can decide whether or not:

to apply the convexity adjustment for valuation of all swaps and options on inflation; to extract the adjustment from market prices or input it;

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to apply a Put Call Parity model or a JY Model. Put Call Parity: This method is model independent and based on the assumption that the contributor selected in the Volatility tab of the setting function SWIL provides cap and floor quotes with at least one common strike. The YoY convexity correction is then calculated by applying the put call parity to the quoted prices: the difference between a cap, with given strike K and maturity T, and a floor, with the same strike and maturity, is a YoY swap with maturity T and with fixed rate K. JY Model: This method calibrates the functional form of convexity correction derived under the model of Jarrow and Yildirim (Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model. Journal of Financial and Quantitative Analysis 38(2), 409 430. 2003). The JY model is a three factor model where nominal and real instantaneous short rates are assumed to follow HJM Gaussian processes and the CPI evolves according to a lognormal process. The JY model has the advantage of implying closed form formulas for the convexity correction and for caps and floors.

D) Once all the above settings have been chosen, we create vectors of bid and ask rates for year on year swaps

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PRICING:

After determining with accuracy the vector of Year on Year swap rates, we can proceed to price derivatives based upon it. Our {SWPM } function allows to structure, price, save and re evaluate the following inflation

instruments: ZC Inflation Swaps (please consult the ZC help document for more information) versus fixed or float; YoY Inflation Swaps versus fixed or float; YoY Caps/Floors.

YoY SWAPS:

The traditional Year on Year structure will merge a YoY inflation leg with a Fixed leg; the swap coupon will be solved for that level that makes the swap par at inception.

Some useful explanation: The lag period refers to the difference between the CPI date and the inflation leg's effective or maturity date. Since most CPIs only provide monthly data, interpolation method is the method by which to determine the settlement date pattern if the deal starts or ends in the middle of one month.

SWPM supports the following interpolation methods:

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MIR Monthly Interpolation. The settlement is always set on the 1st of the start and end month, which means CPI values would be the direct monthly value for these particular months. For example, if the lag period is three months, the coupon rate for one period is: (Monthly CPI on 3 month before next Payment date / Monthly CPI on 3 month before previous payment date 1) x 100%.

DIR Daily Interpolation. The settlement dates are the original dates that can be in the middle of a month. CPI values are the daily weighted average of that month and next month's CPI values. The calculation that appears in the definition of MIR above displays further information.

The leverage represents the magnitude of the reflection on CPI movements on the inflation leg coupons (a value of 1.000 means that a 2% increase in CPI in a year will generate a 2% coupon on the inflation leg; a value of 2.000 would mean that a 2% increase in CPI in a year will generate a 3% coupon on the inflation leg. Etc..)

The spread is a facultative fixed coupon to add to the CPI appreciation within the inflation leg. Whilst cashflows are discounted at libor, they are generated using all the inflation inputs set in SWIL, namely ZC curves, Inflation volatility surface, convexity adjustment method (put call parity or JY) and seasonality. The Curves page in SWPM summarizes some of this information:

Additional information is displayed (and can be further customized) in the Leg Detail page, where the deal can also be customized with an amortizing schedule.

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YoY CAPS/FLOORS:

For inflation caps/floors most of the explanation of settings and the fields available for inflation

swaps (leverage, lag, interpolation, convexity adjustment, etc) hold. Here we will provide more information on the specific features of these options.

Caps default to a 5 year caplets strip (obviously the expiry is customizable) and the caps strike is set at the ATM level for the specific currency and expiry.

The implied volatility used to price the option is derived from the Bloomberg inflation volatility surface (from the SWIL settings function) and is always expressed in SWPM as shifted lognormal volatility.

The yield value represents the premium translated into running yield, based on the deal frequency and day count bases.

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SHORTCUTS:

The pricing functionality can be accessed with shortcuts; a list:

{SWPM ILFX } Zero coupon inflation swap versus fixed

{SWPM ILFL } Zero coupon inflation swap versus float

{SWPM ILFXYOY } Year on Year inflation swap versus fixed

{SWPM ILFLYOY } Year on Year inflation swap versus float

{SWPM ILCAP } Year on Year inflation caps

{SWPM ILFLR } Year on Year inflation floors