Valuing a Cross-Currency Relative-Performance Knock-In Call

Summary

The question concerns a one-year call on a euro-denominated stock with a euro strike, subject to a knock-in condition based on relative performance. The option pays only when the euro stock finishes above its strike and its growth over the year exceeds the growth of a USD-denominated stock. The questioner proposes simulating correlated geometric Brownian motions and asks what further modeling details matter.

The answer gives a direct valuation route under a simplifying assumption: model the joint terminal distribution of the two stocks and integrate the option payoff over the region where the euro stock clears its strike and outperforms the dollar stock in percentage growth. It does not provide parameter choices, a complete payoff formula, or simulation details. It also explicitly sets aside volatility skew, so the suggested method is a simplified starting point rather than a full market-calibrated valuation.

Key ideas

• The payoff requires both the euro stock to exceed its strike and to outperform the dollar stock in growth.
• A joint terminal distribution can represent the dependence between the two stock prices.
• Valuation can integrate payoffs over the region satisfying both conditions.
• The suggested simple calculation does not explicitly model volatility skew.

Tags

# Simulating assets of different currencies


# Simulating assets of different currencies












I have a situation as follows:

- One year call option on a Euro stock with a Euro denominated strike.

- 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?

## Answer by ZRH (score 1)

https://quant.stackexchange.com/a/44049

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 $p_{EUR}(T)/p_{EUR}(0)>p_{USD}(T)/p_{USD}(0)$

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