Why Year-on-Year Inflation Options Do Not Map Directly to Zero-Coupon Options

Summary

The document asks whether a year-on-year inflation option payoff, and its vega exposure, can be represented using zero-coupon inflation options. The response cautions that there is no direct payoff mapping. Converting zero-coupon inflation quantities into year-on-year quantities requires a convexity adjustment to the CPI forward ratio and CPI correlations to translate volatilities.

With flat volatility across strikes, implied correlations could in principle be inferred from zero-coupon and year-on-year implied volatilities. In practice, volatility smiles complicate this approach because it requires choosing how year-on-year strikes correspond to zero-coupon strikes. The discussion offers conceptual guidance rather than a full pricing derivation or a worked example, and it does not provide a reliable direct method for netting the vegas of the two option types.

Key ideas

• Year-on-year inflation option payoffs cannot be readily replicated by zero-coupon option payoffs.
• Conversion requires a convexity adjustment to the CPI forward ratio.
• CPI correlations are needed to relate zero-coupon volatility to year-on-year volatility.
• Flat strike volatility can permit implied correlation estimates in principle.
• Volatility smiles make the mapping dependent on how strikes are matched.

Tags

# how to simplify Inflation year-on-year option to Zero-coupon option


# how to simplify Inflation year-on-year option to Zero-coupon option












Belgrade 2004 paper basically proposes that inflation year-on-year volatilities (and hence yoy options) are basically the spread vols between the Zero-coupon vols from (t0 to T) minus the zero-coupon vols from (t0 to T-1).

Is it possible to express the payoff and hence a year-on-year option unto zero-coupon options? I am trying to map the yoy vega's of my options so that they can be netted off against the vega's of the zero-coupon options.

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/45165

You can't readily map YY options payoffs into ZC options payoffs.

To go from ZC to YY requires:

- a convexity adjustment for transforming the CPI forwards ratio into a YY forward

- CPI correlations for transforming the CPI volatilities into a YY volatility

With flat (in strike) volatilities you can in principle compute CPI implied correlations from ZC implied volatilities and YY implied volatilities. However since there is a smile for both volatilities things would be more complicated as you would need to arbitrarily map YY strikes into ZC strikes.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.