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

1. The DAX itself (in EUR)
2. A cash account (in EUR)
3. 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?

• A hint: To replicate you will have to buy DAX (clearly) , but this will create an unwanted exposure to EUR, you will then have to find a way to offset (eliminate) this. How to do that? You need a second leg: you must borrow EUR (i.e. hold a negative amount of EUR which precisely offsets the above). So the replication has two parts, which need to be adjusted daily. Jul 30, 2016 at 18:59
• @AlexC Thank you for this. I will have a look at the details. Another thing, when I purely look at the discounted PV of the Payoff function above, I don't see how a change in EURUSD Spot would effect the PV at all. As in exp(-rT)*E[NotionalDAX_T/DAX_0] seems utterly independent of EURUSD. Jul 31, 2016 at 12:10

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.

• Hey Gordon from a dimensional standpoint it seems odd that the S0 and X0 are present in the denominator
– dm63
Aug 1, 2016 at 0:47
• @Gordon Thank you for this. Just two small questions: 1) would you be able to provide some more background on the exact derivation of the Radon-Nikodym derivative? 2) in your final formula i do not see any dependence on $X_0$, so how exactly does the PV depend on the fx rate? Aug 1, 2016 at 8:07
• @Tom: revised. The Radon-Nikodym derivative is the ratio of the respective numeraire, converted to the same currency and normalized with respective the corresponding initial values; see Stochastic Calculus for Finance for more details. Since we have the dynamics in the foreign (i.e., EUR) currency, we need to change the measure in order to compute the expectation. Aug 1, 2016 at 14:23
• I think it is correct that there is no fx risk. If you write this contract, your position is short foreign equities offset by long the euro. Hence you are flat.
– dm63
Aug 1, 2016 at 17:48
• There is an interesting discussion of a similar question in Section 12.4.5 of the book Financial Risk Management. Aug 3, 2016 at 17:52