Common Quanto adjustment

In this paper (Quanto Skew, by Jackel), on second page, the author states:

a) Determine an effective volatility coeffcient $\hat \sigma$ exactly as in (2) using the forward as if the option was entirely domestic. Subsequently, price the quanto option with Black's formula replacing the forward $F$ by $F^{'}$ as given in (1).

What does the author mean with "using the forward as if the option was entirely domestic"?

I understand that the idea is to calibrate a model (let's use the Heston model for example, as I know it) against market data in order to get an "effective volatility" and then apply standard Black-Scholes equation using the modified Forward and this "effective volatility".

But which market data do I use? The underlying asset is quoted in FOR currency. Do I get all the foreign vanilla options on this asset to calibrate the model? And then how do I derive the "effective volatility" from the calibrated model?

EDIT - Question 2 On page 4 the author says that the DFAQ case is the applciation of equation 6. But on page 2 he had said that it is the application o Black's formula, using the effective volatility and the forward $F^{'}$. Even more confusing is that also on page 4 he says that for the DFAQ case one should take $F=S_0$ and not $F=F^{'}$ as expected. Could you please clarify what the author is stating?

1 Answer

First of all, quanto options (options denominated in FOR currency but whose value we wish to determine is in DOM currency) are mainly traded over-the-counter, hence their prices are not likely available, as opposed to non-quanto options (options denominated in DOM currency with payoff also in DOM currency) for wich we have some quotes. The idea of the methodology presented by Jackel is to use the later to determine the price of the former.

Regarding your questions:

What does the author mean with "using the forward as if the option was entirely domestic"?

The author means to derive/calibrate an implied volatility while forgetting about the quanto feature (i.e. you only focus on the skew of the domestic currency, DOM, and not of the skew of the FX rate DOM/FOR).

But which market data do I use? The underlying asset is quoted in FOR currency. Do I get all the foreign vanilla options on this asset to calibrate the model? And then how do I derive the "efficient volatility" from the calibrated model?

For the implied volatility calibration, just use options data on options written in DOM currency (European or American options without quanto feature, e.g. call & put options written on the S&P 500 (in USD) with a payoff in USD and quoted in USD) as they will give you the skew you are interested in (i.e. DOM currency). Using such data, derive your "efficient volatility" using usual models (Heston, CEV, ...).

Once you have your "efficient volatility" $\hat{ \sigma}_{DOM}$ smile, you can extract the at-the-money volatility (or more generally, the volatility corresponding to the moneyness of the option you want to price) and your quanto option price will be computed as:

$$C_{quanto}(F',K,\hat{ \sigma}_{DOM},r,T)$$

where $C_{quanto}$ is the Black 76' formula for pricing an option with strike $K$, interest rate $r$ and time-to-maturity $T$. $F'$ is the so-called effective forward capturing the quanto feature derived from equation (1):

$$F' = F \exp (\hat{c}) = F \exp \left(\hat{ \sigma}_{DOM} \cdot \rho \cdot \hat{\sigma}_{FOR} \cdot T\right)$$

The FX volatility $\hat{ \sigma}_{FOR}$ is determined similarly to $\hat{ \sigma}_{DOM}$ using available options from the market written on the DOM/FOR FX rate. Again, the author suggests to extract the at-the-money volatility. Of course, it is not likely that you find an "exact" at-the-money" option in the market as the underlying price moves constantly.

Note: In my answer, I suggested that you calibrate a smile before extracting the ATM volatility. Of course it is your choice to assume which volatility model to use: you can assume a flat volatility taken from the available option which is the closest to the ATM region. I suggested a calibration to generalize the methodology to non-ATM options by building a smile and then extract the volatility at the desired moneyness level.

• JejeBelfort, what do you mean with "options written in DOM currency", what is the undeluing of these such options? The asset referred in the paper is quoted in FOR currency, hence there is no option in the domestic market for this asset.Moreover, why is it necessary to calibrate a model (say, Heston) to market data in order only to get the ATM volatility. The ATM IMPLIED vol can be implied from standard black-scholes formula using the option market price. I do not understand why to calibrate a model only in oredr to get something that we already have at hand. Could you add more details? – John Jul 13 '17 at 12:44
• @John Sure, will do – JejeBelfort Jul 13 '17 at 13:02
• JejeBelfort, thank you so much. It is much better now. I have another simple doubt: what do you mean with "or more generally, the volatility corresponding to the moneyness of the option you want to price". Does it mean that if I want to price a Quanto option deep out of the money, then I should get, from my calibrated model, an "effective volatility" deep out of the money as well? Or I should always get the ATM vol? Thank you. – John Jul 13 '17 at 20:40
• JejeBelfort, I added another doubt in an EDIT, could you please take a look? – John Jul 13 '17 at 21:56
• JejeBelfort, I accepted your answer and gave one point. Thank you for your attention. – John Jul 15 '17 at 0:04