A Payoff Inequality Linking Two Vanillas and a Cliquet

Summary

The note considers whether a short-term call and a cliquet can be worth at least as much as a longer-term call when their strikes are related. Its central result is a pointwise payoff inequality: for any values A and B, the positive part of their difference is at least the positive part of A minus the positive part of B. Applied to the option payoffs, this can establish a payoff dominance relation under the stated construction, which also implies a premium comparison when the instruments share compatible settlement and discounting assumptions.

The argument relies on convexity of the positive-part function rather than a correlation estimate or a pricing model. It is a concise payoff-level result, not a full treatment of every cliquet convention: exact payoff definitions, strike relationships, dates, and settlement terms must match the inequality being applied. The source provides no market data or numerical example, and the stated relationship alone does not establish equality or a universal price ordering for differently specified contracts.

Key ideas

• The positive-part function yields a pointwise inequality between a spread payoff and the corresponding vanilla payoffs.
• A payoff inequality can imply a price inequality when contracts have compatible terms and pricing assumptions.
• The result depends on mapping the vanilla payoffs correctly to the cliquet structure.
• Different conventions or settlement terms may prevent direct application of the inequality.

Tags

# is there some arbitrage relation one can show between a short term vanilla , a cliquet , and a long term vanilla?


# is there some arbitrage relation one can show between a short term vanilla , a cliquet , and a long term vanilla?












suppose all 3 are calls and have same strike moneyness.

The cliquet (V12) pays off S2-S1-K12

The short vanilla (V1) pays S1-K1

The long vanilla (V2) pays S2-K2.

All are floored at zero ofcourse.

K1,K2 , and K12 have some relationship eg K12 = K2-K1 with all of them having same moneyness

Is it possible to demonstrate that V1+V12 > V2 ? (talking about spot premiums here). (if it helps to assume zero interest rates we can do that).

I would think that essentially, in V2 your underlying and your strike is the sum of those of V1 and V12 , and so V2 is like a basket option , and we know that a basket is less than the sum of vanillas because correlation cannot be above 100%.

## Answer by Arshdeep (score 1, accepted)

https://quant.stackexchange.com/a/76109

$(A-B)+ >= (A)+ - (B)+$

where $A$ and $B$ are the vanilla option payoffs. This is a mathematical identity that only uses convexity of the $+$ function.

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