A convertible bond denominated in USD is issued by an Indian company (with equity traded in INR). The bond will be repaid in USD and if converted into equity in the company, the conversion price will be based on a pre-determined fixed USD/INR rate. When valuing the option embedded in the bond, should we use the USD risk-free rate or INR risk-free rate?
Use the USD rate. The actual tricky bit is in the volatility. Normally for a cross currency bond you would use the volatility of the foreign shares as denominated in US currency. However, the fixed FX rate in this case means that the correct volatility to use is the volatility in INR.
The answer is, that it does not matter. Choose one currency as the numeraire, and stick to it. This is because of the foreign exchange interest rate carry arbitrage relationship.
If that relationship doesn't hold, skip the bond and lock-in the arbitrage on the interest rate differential embedded in the USD/INR exchange rate.