Reciprocal FX Options and Their Equivalent Payoffs

Summary

The document examines how to relate options written on reciprocal currency pairs. It asks whether the delta of a USDJPY put corresponds to the delta of a JPYUSD call at the reciprocal strike and with the same maturity. The answer derives the terminal payoff of each contract and expresses the JPYUSD call payoff in terms of the USDJPY exchange rate.

The payoff transformation shows that the reciprocal-pair call is proportional to a put on the original pair, with the factor reflecting the unit and currency conversion. This establishes a payoff equivalence after conversion, rather than asserting that the quoted deltas are numerically identical under every convention. The note does not work through delta definitions, premium-adjusted versus unadjusted conventions, or the effects of quoting and discounting currencies. Those details matter when translating a payoff relationship into a practical FX delta conversion.

Key ideas

• A call on a reciprocal FX rate can be rewritten as a payoff on the original exchange rate.
• At reciprocal strikes, the transformed payoff corresponds to a put on the original pair.
• The payoff equivalence includes a scale factor from converting contract units and currencies.
• Delta conversion still depends on quotation and delta conventions that the document does not analyze.

Tags

# Option Delta Conversion for currency pairs


# Option Delta Conversion for currency pairs












Assuming I have a USDJPY put option at strike 100 (1USD = 100Yen) and the delta is D1. What is the delta of a corresponding JPYUSD call option at strike 0.01 (1Yen = 0.01USD) with the same maturity? Would it be related to D1?

Thank you very much!

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/30577

Since I'm not a FX guru I need to prove the comment of @alex c

A USDJPY 100 call on one dollar has payoff:

max(0, FX-100) Yen

where FX=USDJPY at maturity. A JPYUSD 0.01 call on one Yen has payoff

max(0, 1/FX - 0.01) Dollars = 0.01/FX * max(0,100 - FX) Dollars = 0.01 * max(0,100 - FX) Yen

which is the same as a USDJPY 100 put on 0.01 dollars, as suggested.

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