Pricing Self-Quanto FX Options When Settlement Uses the Underlying Currency

# Quanto options, domestic = underlying


# Quanto options, domestic = underlying












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)

## Answer by Kurt G. (score 3)

https://quant.stackexchange.com/a/73679

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 $$\tag{9}\boxed{\quad V_{Quanto}=X_0^2\,e^{-2\,r_{EUR}\,t\,+\,r_{USD}\,t\color{red}{\,+\,\sigma^2\,t}}\;\Phi(d_3)-e^{-r_{EUR}\,t}\,K\,X_0\,\Phi(d_4)\quad} $$ 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.

## Answer by K. Roman (score 1)

https://quant.stackexchange.com/a/73394

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:

- 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.

- 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.

## Answer by user35980 (score 1)

https://quant.stackexchange.com/a/81599

Bit late to the game here, but I think the original question is referring to what's known as a self-quanto FX option i.e. an option where the payoff is not in the domestic currency (as is the case for vanillas) but rather the foreign currency (assuming FOR-DOM currency convention).

Crucially, this conversion of the payoff to the foreign currency is done at some fixed FX rate set in the trade description, but is usually the option strike $K$. Note that this is very different to converting a vanilla payoff (originally in domestic terms) into a foreign amount at the spot rate at maturity $S_T$ (which is a variable). Fixing the conversion rate introduces non-linearity in the payoff function, as explained below.

For a vanilla call (FOR-call, DOM-put) with nominal $N_{FOR}$ (in units of the foreign currency) and spot rate $S_T$, the payoff in domestic currency is $$ \text{payoff}_{DOM}=N_{FOR}\cdot max \left( S_T-K,0\right)$$ while for a self-quanto it is $$ \text{payoff}_{DOM}=N_{FOR}\cdot max \left( \frac{S_T-K}{K},0\right)S_T.$$ It follows that self-quanto calls are worth more than vanillas (and vice versa for puts) and have a monotonically increasing convex payout function (concave for puts). Thus being short one of these calls in a blow-up scenario can be a very unpleasant experience.

A long self-quanto call can be replicated as a long vanilla call plus a strip of $n$ weighted nominal off-strike ($K+n\epsilon$) vanilla calls (while a long self-quanto put is a long vanilla put minus a strip of weighted nominal off-strike ($K-n\epsilon$) puts), for $n\in \mathbb{N}$ and some small $\epsilon$ chosen to generate the strip strikes. This is reminiscent of replication approaches in CMS convexity adjustments in the rates world (only there we are linearizing a non-linear payoff, here we're going the other way!).

The replication argument illustrates that:

- the top-side vega profile for self-quanto calls is materially accentuated (while puts have shorter downside vega), as compared to vanillas

- skew/flys have a significant impact in the valuation of FX self-quantos.

See Giles Jewitt's excellent "FX Derivatives Trader School", Wiley 2015