Correlation effect in Quanto options

My question will probably be stupid but here it is. I try to understand the effect of the correlation between exchange rate and underlying in a quanto option. And to have a non-precise understanding of this effect, I will consider a simple binomial tree. Suppose I have one underlying valuing 100 \$& current exchange rate 1€=1\$. The quanto option pays at maturity max(S-100,0) paid in €. I consider now two extreme cases (correlation=+/- 1):

• At maturity, S=200\$, 1€=2\$ or S=50\$, 1€=0.5\$
• At maturity, S=50 \$, 1€=2\$ or S=200\$, 1€=0.5 \$

In both cases, the final payoff will be 50€=0.5 * 100€+0.5 * 0€ , whatever the correlation between underlying and exchange rate is. Therefore, the current value of the option would be the NPV in € of 100 € and is independent of the correlation between exchange rate and underlying.

Where is the error in this simulation?

PS: By the way, we can use multistep binomial trees. The evolution of the underlying does not depend on the exchange rate.

• I might be misreading your tree data, however, I don't even see the terminal payoffs being equal for your two corner cases. The OTM case is clear, 0$= 0 always. The ITM case for a correlation of +1 would pay max[200-100, 0]$ times 0.5 for every 1$, so 50. For a correlation of -1, you'd have max[200-100,0]$ times 2 for every 1$, so 200? May 25 '21 at 16:50 • @KevinT that's because by definition in quanto options, the exchange rate for the final payoff is fixed at the beginning of the contract. In this case it is 1€ = 1$. Oct 26 '21 at 4:26