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How should you price Quanto option, where domestic = underlying?

Example: Let's say I want to price Quanto option in EUR on EUR/USD pair, but the option is settled in EUR.

Should I use normal Quanto option formula with $-1$ correlation and same sigma? (Due to the fact that the USD/EUR is $-1$ correlated to EUR/USD)

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  • $\begingroup$ What exactly is it that you price? A simple vanilla eurusd FX option in Euro is not a quanto, which would be of you settle in cad for example. $\endgroup$
    – AKdemy
    Commented Oct 4, 2022 at 16:20

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I suppose by the EUR/USD pair you mean the FX rate $X$ that is the price of one EUR in USD. We know that the arbitrage-free Black-Scholes model for an option on this is modelled by the GBM $$\tag{1} X_t=X_0\exp\Big(r_{USD}\,t-r_{EUR}\,t+\sigma\, W_t-\frac{\sigma^2\,t}{2}\Big)\,. $$ The non quanto payoff for a call is $$\tag{2} \operatorname{PlainVanilla}=\max(X_t-K,0)\,,\quad\quad\text{ in USD} $$ and the Black-Scholes price is -as we know- $$\tag{3} V_{\operatorname{PlainVanilla}}=X_0e^{-r_{EUR}\,t}\,\Phi(d_1)-e^{-r_{USD}\,t}\,K\Phi(d_2) \,,\quad\quad\text{ in USD} $$ where $$\tag{4} d_1=\frac{\log(X_0/K)+r_{USD}\,t-r_{EUR}\,t+\sigma^2\,t/2}{\sigma\sqrt{t}}\,,\quad d_2=d_1-\sigma\sqrt{t}\,. $$ If you settle the payoff (2) in EUR instead of USD it becomes a quanto and the price in USD becomes \begin{align}\tag{5} &e^{-r_{USD}\,t}\,\mathbb E_{\mathbb P}\Big[X_t\max(X_t-K,0)\Big]= X_0e^{-r_{EUR}\,t}\,\mathbb E_{\mathbb P}\Big[e^{\sigma W_t-\sigma^2 t/2}\max(X_t-K,0)\Big]\,. \end{align} By the Girsanov theorem $\widetilde{W}_t=W_t-\sigma\,t$ is a Brownian motion under the new measure $\mathbb Q$ that has Radon-Nikodym density $$\tag{6} \frac{d\mathbb Q}{d\mathbb P}=e^{\sigma W_t-\sigma^2 t/2}\,. $$ Therefore, (5) becomes $$\tag{7} X_0e^{-r_{EUR}\,t}\,\mathbb E_{\mathbb Q}\Big[\max(\widetilde{X}_t-K,0)\Big] $$ where $$\tag{8} \widetilde{X}_t=X_0\exp\Big(r_{USD}\,t-r_{EUR}\,t+\sigma\, \widetilde{W}_t\color{red}{+\sigma^2\,t}-\frac{\sigma^2\,t}{2}\Big)\,. $$ This leads to the call price $$\tag{9}\boxed{\quad V_{Quanto}=X_0^2\,e^{-r_{EUR}\,t\color{red}{\,+\,\sigma^2\,t}}\,\Phi(d_3)-e^{-r_{USD}\,t}\,K\,X_0\,\Phi(d_4)\quad} $$ where $$\tag{10} d_3=\frac{\log(X_0/K)+r_{USD}\,t-r_{EUR}\,t\color{red}{+\sigma^2\,t}+\sigma^2\,t/2}{\sigma\sqrt{t}}\,,\quad d_4=d_3-\sigma\sqrt{t}\,. $$

  • Note that the new terms $\color{red}{+\sigma^2\,t}$ have an enormous impact on the vega of that quanto option.
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Pricing of a Foreign Exchange Vanilla Option I think this post can help you. You don't need to understand how bloomberg calculator is worked. Just look at it as information for option pricing. Edit. If you will open this post, you can see that the main problem of this post is how to price option on USDCAD settled in USD. Also I quote E. Reiner "Quanto Mechanics":
"In a global equity market it is possible to link foreign stock and currency exposures in a variety of interesting ways: investors may choose to combine their investments in foreign equities with differing degrees of protection against adverse moves in exchange rates, equity prices, or combinations thereof. Four scenarios, in roughly increasing order of complexity, and the pay-offs that match them are:

  1. An investor wants to participate in gains in a foreign equity, desires protection against losses in that equity, but is unconcerned about the translation risk arising from a potential drop in the exchange rate. Such an investor might desire the pay-off of a foreign equity call struck in foreign currency: $C_1 = X^* \max[S'^* - K', 0]$, where $S'^*$ is the equity price in its own currency after time $t$ and $K'$ is a foreign currency amount. In this formula, $X^*$ appears in front of the maximum function, indicating that the final pay-off must be converted into domestic currency.
  2. An investor wishes to receive any positive returns from the foreign market, but wants to be certain that those returns are meaningful when translated back into his own currency. For him, it is the product of the foreign asset price and the exchange rate at expiry that is important, and he might be interested in a pay-off like that of a foreign equity call struck in domestic currency: $C_2^* = \max[S'^*X^* - K, 0]$, where K is now a domestic currency amount and X* multiplies S'* only, representing tran slation of the foreign equity value into domestic terms... And etc... "

Look at the first point. It's your case: $C = USDEUR\cdot \max[EURUSD - K]$. And it also discussed in Pricing of a Foreign Exchange Vanilla Option in the latest answer from jherek. Either I'm wrong or no one read further than the name.

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  • $\begingroup$ Aside from the fact that you do need to understand what something is doing, the link you posted is about pricing FX vanillas, not Quanto options. $\endgroup$
    – user34971
    Commented Oct 4, 2022 at 12:54
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    $\begingroup$ @Frido Rolloos, the way the question is worded now eould not be a quanto option imho. $\endgroup$
    – AKdemy
    Commented Oct 4, 2022 at 16:23
  • $\begingroup$ @AKdemy Yes, a quanto with underlying USDEUR is not a quanto. My first comment was regarding the first answer which has now been deleted. $\endgroup$
    – user34971
    Commented Oct 4, 2022 at 16:29
  • $\begingroup$ +1 for your expanded answer. I indeed did not carefully enough read the OPs question. $\endgroup$
    – user34971
    Commented Oct 4, 2022 at 16:41
  • $\begingroup$ My answer to the question in the link provided above contains links to other questions that I answered (this is a reoccurring question) where FX vanilla options are priced (Julia code, which is usually easy to read) in all directions and notations. $\endgroup$
    – AKdemy
    Commented Oct 4, 2022 at 18:53

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