FX Hedging, Quanto Payoffs, and Currency Exposure in Volatility

# Quanto derivatives and FX risk management


# Quanto derivatives and FX risk management












Let us assume that we have a foreign asset with volatility $\sigma_{ASSET}$. Now, I know that when pricing this under the foreign measure, I need to do a drift adjustment, namely $\sigma_{ASSET NEW}^2 = \sigma_{ASSET}^2 + \sigma_{FX}^2 +2\rho\sigma_{ASSET}\sigma_{FX}$.

On the other hand, I know that this can be FX hedged. My question is, what happens to this volatility when I hedge continuously?

My thinking is that anything to do with FX should be removed, but I can't justfiy this.

Edit: I'm further clarifying what I am after. In trying to find the volatility of a hedged asset (hedged against FX), I determine the volatility by looking at the payoff in domestic units. That is, $S_T X_T + \sum_{i=1}^N S_{t_{i-1}}(F_{t_{i-1},t_i}^X-X_{t_i})$, where $N$ is the number of hedges such that $t_N = T$, and $T$ is the maturity.

I come up with a volatility using moment matching (which looks really messy so I won't post here unless required). Now I am looking at what happens as the increments between the hedges gets smaller. That is, $N\rightarrow \infty$ and $t_i-t_{i-1}\rightarrow 0$. What I think may happen is as described above, where anything to do with the FX is altogther gone from the volatility. However, that doesn't seem to be the case with the volatility I have. Is my approach invalid?

## Answer by user18663 (score 0)

https://quant.stackexchange.com/a/27590

When you are hedging through FX then there are two factors influencing the impact that currency will have on the portfolio: the volatility of currency relative to that of the underlying asset and the interaction between currency and the underlying asset. The larger the volatility ratio ( volatility foreign currency/volatility asset ), the greater the impact of the foreign-currency exposure on the portfolio’s volatility.The lower the volatility ratio, the more important the asset–currency correlation will be in determining the portfolio risk outcome. It is the net effect of the two influences that determines whether total portfolio risk is increased or decreased by hedging the foreign-currency exposure.

## Answer by Quantuple (score 0)

https://quant.stackexchange.com/a/27591

[Answer]

Well yes, this comes from the interpretation of an option price as the initial endowment of a perfect replicating portfolio. Here the replication consists in holding a self-financing portfolio of shares + foreign/domestic bonds where one dynamically re-balances the delta continuously (assuming all usual assumptions hold: i.e. no transaction consts etc.).

> I recommend you read section 3.2 of this document. Hedging of quanto options is specifically discussed in section 3.2.4. The interpretation of the hedging strategy p.33 is what matters to you.

[Some details]

Consider an equity underlying $S_t$ denominated in a foreign currency.

Consider the FOR/DOM exchange rate $X_t$, which is the price at time $t$ of one unit of foreign currency expressed in domestic currency. The DOM/FOR exchange rate at $t$ is obviously given by $Y_t=(X_t)^{-1}$.

Assume that both $S_t$ and $Y_t$ are lognormal under the foreign risk-neutral measure $\mathbb{Q}^f$, with a linear correlation $\rho$ between their driving Brownian motions $$ S_t^f \sim GBM(r^f, \sigma_{ASSET}) \iff S_t^f \sim N(\ln(F(0,t))-\frac{1}{2}\sigma_{ASSET}^2 t, \sigma_{ASSET}^2 t)$$ $$ Y_t \sim GBM(r^f - r^d, \sigma) \iff Y_t \sim N(\ln(F^Y(0,t))-\frac{1}{2}\sigma^2 t, \sigma^2 t)$$ where $F(0,t)$ denotes the equity forward in the foreign risk-neutral measure, and $F^Y(0,t)$ the forward DOM/FOR exchange rate.

From the above, the price of the equity underlying expressed in domestic currency units, i.e. $S_t X_t$, is indeed a log-normally distributed random variable with variance: $$ (\sigma_{ASSET}^2 + \sigma^2 + 2\rho \sigma_{ASSET} \sigma)t $$ this is because $S_t X_t$ is the product of 2 correlated log-normal variables (i.e. the log of $S_t X_t$ is a sum of two correlated normal variables. To see that $X_t$ is indeed lognormally distributed you may want to apply Itô's lemma to $X_t=1/Y_t$).

Now, although this is true, I don't think it will be useful for what you are trying to prove, especially since you are dealing with quanto options with payoff functions: $$ \phi(S_T) = f(S_T), \text{e.g. } X^{quanto} (S_T-K)^+ $$ where $X^{quanto}$ is a constant (typically $X^{quanto} = 1$) and not compo options with payoff functions: $$ \phi(S_T,X_T) = f(S_T X_T), \text{e.g. } (S_T X_T - K)^+ $$

When you look at a quanto option there are 2 things that matter:

- By construction, the price of a quanto option does not depend on the foreign exchange rate

- The delta of a quanto option naturally embeds the FX risk. This is because, although your option pays in the DOM currency (hence a delta in DOM units), you hedge by buying shares of the foreign underlying (hence in FOR units). Therefore, although you have a constant notional in DOM units, your delta notional in FOR units constantly changes due to the fluctuations of the exchange rate.