Quanto Forward Returns and Foreign Exchange Exposure

# Quanto Total Return of a Foreign Asset into Domestic


# Quanto Total Return of a Foreign Asset into Domestic












Say we have a product that pays the following at expiry $T$:

$$\text{Payoff}_{in\ USD} = \text{Notional}_{in\ USD} \cdot \frac{DAXLevel_{in\ EUR}\ at\ t=T}{DAXLevel_{in\ EUR}\ at\ t=0}$$ i.e. it simply pays the total DAX return in USD (as opposed to EUR, which is the currency in which the DAX is quoted). So it gives USD-based investors exposure to the EUR-denominated index DAX without actually having to invest in EUR. That is, FX risk is seemingly nullified (or is it).

Now my question is, how would one replicate such a payoff? After playing around with this for a while (however, without actually finding a working replicating strategy), I also get the impression that FX risk is not zero (i.e. we have non-zero EURUSD fx delta). The reason being that the replicating portfolio will invest in

- The DAX itself (in EUR)

- A cash account (in EUR)

- A cash account (in USD)

i.e. there will have to be conversion of USD into EUR and back.

Would anybody be able to point me in the right direction?

## Answer by Gordon (score 5, accepted)

https://quant.stackexchange.com/a/28407

To see the exposure to FX risk and the difficulty for hedging, we assume constant interest rates and constant volatilities. Let $r_d$ and $r_f$ denote respectively the interest rates for USD and EUR. Moreover, let $X_t$ be the exchange rate at time $t$ from one unit USD to units of EUR. Finally, let $S_t$ be the price level of DAX at time $t$. We assume that, under the EUR risk-neutral measure $Q_f$, \begin{align*} \frac{dX_t}{X_t} &= (r_f-r_d) dt + \sigma_X dW_t\\ \frac{dS_t}{S_t} &= r_f dt + \sigma_S\big( \rho dW_t + \sqrt{1-\rho^2} dB_t\big), \end{align*} where $\{W_t, t \ge 0\}$ and $\{B_t, t \ge 0\}$ are two independent standard Brownian motions, and $\rho$ is the correlation.

Let $Q_d$ be the USD risk-neutral measure. Note that \begin{align*} \frac{dQ_d}{dQ_f}\big|_t = \frac{e^{r_dt} X_t}{e^{r_f t}X_0}. \end{align*} Therefore, \begin{align*} e^{-r_dT} E_{Q_d} \left(\frac{S_T}{S_0} \right) &=e^{-r_dT} E_{Q_f} \left(\frac{S_T}{S_0} \frac{e^{r_dT} X_T}{e^{r_f T}X_0}\right)\\ &= e^{-r_fT} E_{Q_f} \left(\frac{S_T X_T}{S_0X_0}\right) \tag{1}\\ &= e^{(r_f-r_d)T}E_{Q_f}\left(e^{-\frac{1}{2}\sigma_S^2 T +\sigma_T (\rho W_T + \sqrt{1-\rho^2} B_T) -\frac{1}{2}\sigma_X^2 T + \sigma_X W_T)} \right)\\ &=e^{(r_f-r_d)T}e^{\rho\sigma_S\sigma_X T}. \tag{2} \end{align*} Here, $\rho\sigma_S\sigma_X T$ is the quanto adjustment.

From $(1)$, we note that the payoff $S_T/S_0$ of the quanto forward, in USD, is equivalent to the payoff $S_TX_T/(S_0X_0)$ in EUR. Therefore, a quanto forward is exposed to FX risk. We also note that, though the FX rate does not explicitly appear in valuation formula $(2)$, both the FX risk factors $\rho$ and $\sigma_X$ are presented. Moreover, Formula $(2)$ is based on the constant volatility assumption, while in the local volatility framework, the volatility $\sigma_X$ will depend on the spot FX rate $X_0$. Regarding replication, we basically need to replicate the payoff $S_TX_T/(S_0X_0)$ in EUR, which does not appear an easy exercise.

EDIT: There is an interesting discussion of a similar product in Section 12.4.5 of the book Financial Risk Management.