## Remarks on Quanto Options in Section 17.4.3

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### Remarks on Quanto Options in Section 17.4.3

The option pay-off in eq. (17.5) is denominated in EUR. The payoff is a function of the FX spot price F(T) at time T, which I assume is denominated in EUR per USD, plus the share & strike prices, S(T) & K, both denominated in USD. Note that the Max(.) operator appears to be missing in eq. (17.5).

The terminal value of the payoff is a function of F(T), the terminal FX spot price, therefore in the strictest sense this option is not a Quanto, since for this to be case the FX price featuring in the payoff would be agreed (fixed) at deal inception at time 'zero'.

Putting this issue aside, the specification of the pay-off influences the choice of numeraire, & hence probability measure, that we use to calculate the option value. We have:

V_{EUR}(T) = F(T).Max[S(T)-K ; 0]

and in the simplest case we may use the EUR Money Market Account (MMA) as the numeraire. We could have also written the terminal payoff as:

V_{USD}(T) = Max[S(T)-K ; 0]

where: V_{USD}(T) = V_{EUR}(T) / F(T), and for t <=T we have:

V_{USD}(t) = V_{EUR}(t) / F(t)

Now writing the pay-off as V_{USD}(T) we might choose the USD MMA as the numeraire. When we write the stochastic price processes in eq. (17.3) we must do so wrt a probability measure.

If we choose the USD MMA as the numeraire then we have:

dS/S = r_f.dt + sig_s.dW_1(t)

and

dF/F = (r_f - r_d).dt + sig_F.dW_2(t)

where dW_1(t) and dW_2(t) are Brownian motion increments wrt to the probability measure induced by the choice of the USD MMA as the numeraire. Here F would now have units of USD per EUR (the market convention).

If we choose the EUR MMA as the numeraire then we have:

dS/S = (rho.sig_s.sig_F + r_f).dt + sig_s.dZ_1(t)

and

dF/F = (r_d - r_f).dt - sig_F.dZ_2(t)

where dZ_1(t) and dZ_2(t) are Brownian motion increments wrt to the probability measure induced by the choice of the EUR MMA as numeraire, and rho is the instantaneous correlation between the Brownian motion increments dZ. Note that the change of measure has introduced a drift adjustment for the process S. Here the process F also has a drift adjustment due to the measure change (c.f Siegels Paradox ). F would have units of EUR per USD.

Now using MC (or analytic results) one can calculate either V_{USD}(t) or V_{EUR}(t) but one must use the correct price processes for the probability measure chosen. Note that using the USD MMA as numeraire the option becomes trivial to value since it will simply be the price of a USD denominated call option on the share V_{USD}(t), multiplied by the spot FX price F(t) in units of EUR per USD.

I may have misunderstood the example in the text, apologies if I have done so.
rdoubble

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