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.