Quanto Option Delta Hedging Across Equity and FX Exposures

# Delta hedging of a quanto call option


# Delta hedging of a quanto call option












When dynamically hedging a quanto call option buying/selling delta units of the foreign underlying under the domestic risk neutral measure, does it make sense to use the domestic money market as cash account only? Or does one to consider the foreign money market as well? why? Assuming delta hedging is done "continuously" with this approach and the delta used is the delta of the analytical BS quanto formula.

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/81109

Let's call call the price of this option $V(S)$ (in units of domestic currency).

Then $\Delta=\frac{\partial V(s)}{\partial s}$.

We need to buy enough unity of $S$ to neutralise it. But, since $S$ is denominated in a foreign currency, the real value of one $S$ for me is $S \cdot X$ where $X$ is the exchange rate (the right way round of course, depends on your conventions).

So, in order to neutralise my $\Delta$, I need to buy $\frac{\Delta}{X}$ units of $S$.

But this gives me exposure to something new $X$, which I will have to neutralise too, buying / selling the right amount of the exchange rate.

So, when you do all the maths correctly, you will find your portfolio contains: the call option, some shares, some FX.

Of course, all of that should in theory be re-hedged continuously.

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/81111

More generally, no matter what products, suppose that your accounting is denominated in the local currency, and that the value of your portfolio may depend on:

- the price of the underlying expressed in foreign currency

- the foreign/domestic currency exchange rate

- the domestic interest rate, having a term structure

- the foreign interest rate, including the cross-currency basis, having a term structure

- possibly other inputs, such as implied volatilities

You should monitor the deltas (sensitivities to the inputs, denominated in the domestic currency), and when they exceed your risk appetite / limits, re-hedge them. You can use a monte-carlo simulation to estimate how often you might need to re-hedge based on some volatility assumptions.

You should also monitor the pairwise cross gammas between all the inputs and also time. These cross-gammas and the passage of time should explain the reasons for the changing deltas. While articulating a risk appetite / limits for cross gammas would be unusual, it is common industry practice to check that the cross-gammas explain most ly the changes in the deltas, similar to P&L attribution. As a simple example, if your accounting is in USD, and you own an ADR of a EUR stock, and the EUR price moves 10%, then this causes the FX delta to also move 10%. If the FX rate moves 1%, then this causes the (USD denominated) stock price delta to move 1%. If an option has 1 day less left to expiry, then this theta drives down not only the value of the option, but also the deltas. Having an automated process check that these all add up is a pretty standard part of ongoing performance monitoring for pricing models.

So, if you have more exposure to the foreign interest rate (including cross-currency basis), how should you hedge it? Practically, limiting yourself to instruments that also have the cross-currency basis, like fx forwards and cross-currency swaps, may be simpler than rates instruments that do not include a cross-currency basis, like bonds or same-currency interest rate swaps, because once you create a separate exposure to the cross-currency basis, you need to ponder your risk appetite for that. You can hedge the fx exposure with foreign currency cash, i.e. not hedge the foreign interest rate exposure, if your risk appetite lets you - sorry for any tautology.