Deriving Foreign-Domestic Symmetry for FX Call Options

Summary

The response derives a pricing relationship between a call option on an exchange rate and a put option on the inverse exchange rate. It defines the rate as domestic currency per unit of foreign currency, then rewrites the call payoff using the reciprocal exchange rate and reciprocal strike.

The derivation changes from the domestic risk-neutral measure to the foreign risk-neutral measure using the Radon–Nikodym density. Substituting the transformed payoff and measure-change factor yields the symmetry: the domestic-currency call price equals the initial exchange rate times the original strike times the corresponding foreign-currency put price at reciprocal strike. This is a concise mathematical argument, not a broad treatment of FX option conventions or assumptions. Correct interpretation depends on the stated currency and exchange-rate quotation convention and on the money-market numéraires used for each measure.

Key ideas

• A call payoff on an exchange rate can be rewritten as a scaled put payoff on the inverse rate.
• The reciprocal transformation maps the original strike to its inverse.
• Changing between domestic and foreign risk-neutral measures introduces a density involving the exchange rate and money-market accounts.
• The resulting call-put symmetry scales the inverse-rate put price by the initial exchange rate and original strike.
• The identity depends on consistent currency quotation and numéraire conventions.

Tags

# Foreign-Domestic Symmetry


# Foreign-Domestic Symmetry












I try to find the proof of the theorem of Foreign-Domestic Symmetry for FX options, but I did not found, Some paper or book, where it is explained?

Theorem 2.1 in this paper:

https://arxiv.org/pdf/1912.01387.pdf

Thx.

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/54737

Let $S_t$ be the exchange rate from one unit foreign currency FGN to units of domestic currency DOM. Note that, for maturity $T$ and strike $K$, \begin{align*} \max(S_T-K, \, 0) = S_TK\max\Big(\frac{1}{K}-\frac{1}{S_T}, \, 0\Big). \end{align*}

Moreover, let

- $B^d_t$ and $B^f_t$ be the respective domestic and foreign money-market account values at time $t$,

- $P^d$ and $P^f$ be the respective domestic and foreign risk-neutral measures,

- $E_d$ and $E_f$ be the respective expectations corresponding to $P^d$ and $P^f$.

Then, for $t\ge 0$, \begin{align*} \frac{dP^d}{dP^f}\big|_t &= \frac{S_0 B^d_t}{S_t B^f_t}. \end{align*} Therefore, \begin{align*} C_{DOM/FGN}(0, T, K) &= E_d\left(\frac{\max(S_T-K, \, 0)}{B^d_T} \right)\\ &=E_f\left(\frac{dP^d}{dP^f}\big|_T\frac{\max(S_T-K, \, 0)}{B^d_T} \right)\\ &=E_f\left(\frac{S_0 B^d_T}{S_T B^f_T}\,\frac{S_TK\max\big(\frac{1}{K}-\frac{1}{S_T}, 0\big)}{B^d_T} \right)\\ &=S_0 KE_f\left(\frac{\max\big(\frac{1}{K}-\frac{1}{S_T}, \, 0\big)}{B^f_T} \right)\\ &=S_0 K P_{FGN/DOM}\Big(0, T, \frac{1}{K}\Big). \end{align*}

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