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Checking Cap Volatility Surfaces by Stripping Caplet Prices

Article Quant Q&A · Author: Hasek
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Summary

The document considers how to assess a proposed interest-rate cap volatility surface for arbitrage. A cap consists of caplets, and the response recommends stripping caplet values from cap prices across maturities. In a simplified setting with a flat yield curve and a common strike, each incremental caplet price is obtained by subtracting the shorter-maturity cap price from the next longer-maturity cap price. A negative implied caplet price indicates an impossible option value and flags an inconsistency in the cap quotes.

The example shows a sequence of cap prices whose increments are nonnegative, then changes the final quote so that the last stripped caplet has negative value. The answer treats successful stripping as the condition for an arbitrage-free cap surface. Its example relies on simplifying assumptions, and the response asserts that caplet surfaces are arbitrage-free because the underlying rates differ across periods. It also distinguishes technical arbitrage from a possible relative-value opportunity due to irregular surface smoothness. Individual caplet tradability is raised in the question but not resolved in the answer.

Key ideas

  • Caps can be decomposed into individual caplets across successive accrual periods.
  • Strip caplet prices by taking differences between cap prices at adjacent maturities.
  • A negative stripped caplet price signals an arbitrage inconsistency in the cap quotes.
  • The numerical illustration assumes a flat yield curve and a shared strike.
  • The response does not fully address whether nontradable caplets change the practical arbitrage test.

Tags

Fixed incomeOptionsVolatilityArbitrageDerivatives pricing
Full text
# No-arbitrage conditions on a caps/floors volatility surface


# No-arbitrage conditions on a caps/floors volatility surface












Suppose that one has a caps/floors volatility surface and wants to check whether this surface admits arbitrage. What is the theoretical and practical way to do it?

Lets talk only about caps for simplicity, since a cap and a floor with the same strike and expiry have the same volatility (similarly to vanilla call and put options). An interest rate cap is a series of individual vanilla call options (caplets) on the interest rate. Given a flat cap volatility (the one that correctly reprices the cap as a sum of caplets with the flat volatility) one can derive spot volatilites of individual caplets via a procedure known as caplet volatility stripping. Therefore it is possible to build a caplet volatility surface from the given cap volatility surface, i.e. a volatility surface of vanilla European call options constituting caps.

Is it true that caps volatility surface is arbitrage-free if and only if the corresponding caplets volatility surface is arbitrage-free? Is it possible to check the absence of an arbitrage directly on caps without building the corresponding caplets surface? Note that we can't trade individual caplets constituting caps.

Any help, links to resources and thoughts on that matter will be greatly appreciated.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/68643

I would say that a cap volatility surface (meaning, a list of implied volatilities corresponding to various final maturities) is arbitrage free if and only if you can successfully build the corresponding caplet volatility surface.

Note that any caplet surface is arbitrage free because the underlying rates for each caplet are different. (Obviously, one would expect the caplet surface to be smooth, but even if it isn’t , it’s not technically an arbitrage. More of a relative value trading opportunity).

EDIT Suppose we have flat 1% yield curve for simplicity and then all caps and caplets are 1% strike. Sample cap prices: 6m 0.09 9m 0.18 1y 0.25 1y3m 0.38 Can be stripped into caplets by simple subtraction 3mx6m 0.09 6mx9m 0.09 9mx12m 0.07 12mx15m 0.13 But if the 1y3m cap price were 0.23 the stripping procedure would give 12mx15m -0.02 which is impossible as option prices are positive.

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