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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?

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  • $\begingroup$ I don't understand what you're trying to do sorry. $S_t X_t$ is not an asset but rather the value of a foreign asset denominated in domestic currency units. What are you trying to achieve, make this value insensitive to fluctuations of the spot exchange rate? Or rather, form a self-financing strategy which consists in going "long" $S_t X_t$ and short something else so that in total, you are insensitive to $X_t$ ? $\endgroup$ – Quantuple Jun 13 '16 at 10:41
  • $\begingroup$ For now what you did boils down to saying: I've calculated the volatility of $S_T X_T$ and its not the same as what the theoretical formula predicts. Regardless of whatever hedge you set up. $\endgroup$ – Quantuple Jun 13 '16 at 10:42
  • $\begingroup$ Well the payoff I have provided is the value of the foreign asset denominated in domestic currency units (yes) that is true + the value of all the FX hedges hedged with FX forwards (you can see this through the summation). So I guess what I'm trying to do is the hedge the foreign asset using FX forwards, and I'm now trying to hedge continiously - Maybe my title of "quanto" is not quite accurate. $\endgroup$ – jim Jun 13 '16 at 10:48
  • $\begingroup$ Why do you call that a payoff? Do you mean a terminal wealth and in that case what is your initial investment and what is the strategy? I don't understand what you do... I mean If you take $N\rightarrow\infty$ by AOA you have $F^X(t_{i-1},t_i) = X_{t_i}$ (forward exchange rate over very short horizon = spot exchange rate), so the summation disappears and you are left with $S_T X_T$... in what way is that a hedge of $S_T X_T$ ?! $\endgroup$ – Quantuple Jun 13 '16 at 10:53
  • $\begingroup$ I'm saying that my formula is a payoff because well the first part, $S_T X_T$ is the value of the foreign asset in domestic units at maturity. This I think we both agree with and there is no hedging involved. At every index $i$ of the summation, I buy $S_{t_{i-1}}$ units FX forward that mature at time $t_i$ and valued at time $t_{i-1}$. Obviously in terms of payoff, if the FX rate falls at time $t_i$, then I will make a profit and vice versa. Hence I need to subtract $S_{t_{i-1}}$ of the FX rate $X_{t_i}$ $\endgroup$ – jim Jun 13 '16 at 10:59
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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.

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  • $\begingroup$ Yes I agree that this is true. But I'm more concerned about what happens if I hedge continiously. Would the asset-currency correlation/FX volatility still have an effect? $\endgroup$ – jim Jun 13 '16 at 7:40
  • $\begingroup$ Are you referring to longer horizon here by stating '' if I hedge continiously"? $\endgroup$ – Manish Jun 13 '16 at 7:48
  • $\begingroup$ No I mean like rather than hedging say monthly, I hedge continuously. I know it's not possible in reality, but theoretically you can $\endgroup$ – jim Jun 13 '16 at 8:19
  • $\begingroup$ OK. So theoretically, if you hedge continuously for shorter period, hedging reduces return volatility. However for longer horizons, foreign asset class display greater return volatility when hedged and this also results changes in minimum variance hedge ratios. $\endgroup$ – Manish Jun 13 '16 at 9:29
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[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:

  1. By construction, the price of a quanto option does not depend on the foreign exchange rate
  2. 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.
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  • $\begingroup$ So does this mean that say I FX hedge a foreign asset monthly and arrive at a specific volatility, say $\sigma_A$, does that mean that if I continuously hedge this asset, then $\sigma_{ASSET NEW} = \sigma_{ASSET}$ $\endgroup$ – jim Jun 13 '16 at 9:13
  • $\begingroup$ Did you read the paper I mentioned in my answer? $\endgroup$ – Quantuple Jun 13 '16 at 9:25
  • $\begingroup$ Yes and it says as such. My approach was slightly different. I did the following: The payoff in investing/hedging in such an asset is given by $S_T X_T + \sum_{i=1}^N S_{t_{i-1}}(F_{t_{i-1},t_i}^X-X_{t_i})$, where N is the number of hedges such that $t_N = T$. I then used moment matching to determine the corresponding volatility. I got an answer, but the problem is, if i let the increments between the hedges tend towards zero, I don't seem to get $\sigma_{ASSET}$. This was the rationale behind my question. Is this an appropriate way to approach this? $\endgroup$ – jim Jun 13 '16 at 9:38
  • $\begingroup$ I'm sorry but I really don't understand your question. First in your OP, you talk about drift-adjustment while you simply describe the variance. Then you're dealing with quantos but rather deal with $S_t X_t$ i.e. the value of the foreign asset in domestic units. Could you please clarify everything and re-phrase your question? $\endgroup$ – Quantuple Jun 13 '16 at 9:54

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