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.

  • $\begingroup$ 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? $\endgroup$
    – KevinT
    May 25 at 16:50

The correlation comes into the replication (and thus hedging) of a quanto and not explicitly in the final payoff. In a sense you are trying to hedge a linear payoff with a linear hedging instrument (exchange rate) and a non-linear hedging instrument (foreign security converted into local currency) and the correct hedge ratio depends on the correlation.

  • $\begingroup$ Does this correlation still interact if we consider options paying relative performance of an index : (Final value-Initial Value)/Initial Value ? $\endgroup$
    – Delepine
    May 25 at 11:39
  • $\begingroup$ Yes, if the accounting currency is not the natural currency of the payout (ie a quanto) then you have this effect. Your hedge instrument is still non-linear, being the product of the exchange rate and the index level. $\endgroup$
    – river_rat
    May 25 at 13:14

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