# Simulating assets of different currencies

I have a situation as follows:

1. One year call option on a Euro stock with a Euro denominated strike.
2. Knock in feature as follows - The option can only pay out if the growth in the Euro stock over the year exceeds the growth in a USD denominated stock over that period.

How would I go about valuing this option? My instinct is to simulate the Euro and USD stocks using correlated GBM, but I am wondering if there are any intricacies that I am missing in doing this?

## 1 Answer

For a simple calculation, where you do not model skew explicitly, it would suffice to write down the joint density of the EUR and USD stock, and to integrate over the area, where $$p_{EUR}>K_{EUR}$$ and $$p_{EUR}(T)/p_{EUR}(0)>p_{USD}(T)/p_{USD}(0)$$

• Since the payoff is in EUR, do not forget to make a quanto adjustment to get the dynamics of $p_{\text{USD}}$ under the EUR risk neutral measure. – Antoine Conze Feb 14 at 9:50
• As I understood it, the payoff was in EUR on the EUR stock, only conditional on the USD stock having higher return – ZRH Feb 14 at 9:52
• because of the condition on the USD stock you need the USD stock under $Q_T^{\text{EUR}}$ the EUR risk neutral measure: $\text{npv} = D^{\text{EUR}}(T)E^{Q_T^{\text{EUR}}}\left[\text{payoff}(p_{\text{EUR}}(T), p_{\text{USD}}(T)) \right]$ – Antoine Conze Feb 14 at 10:08