Choosing the Rate and Volatility for a Cross-Currency Convertible Bond

Summary

The document considers an embedded equity conversion option in a bond issued by an Indian company. The bond is denominated and repaid in US dollars, while the underlying shares trade in Indian rupees; the conversion price uses a fixed USD/INR rate. One response recommends using the USD risk-free rate and the shares’ INR volatility, reasoning that the fixed conversion rate affects which equity volatility is relevant.

A second response says the choice of currency numeraire should not affect the valuation when foreign-exchange interest-rate carry relationships hold. It also points to a potential arbitrage if those relationships fail. These are brief answers rather than a derivation or a worked valuation, and the document does not reconcile the apparent difference in emphasis. Applying the guidance requires modeling the currency and conversion terms consistently.

Key ideas

• The example combines a USD-denominated bond with equity traded in INR and a fixed exchange rate for conversion.
• One answer recommends the USD risk-free rate and INR share volatility for the embedded option.
• Another answer says either currency numeraire can be used if foreign-exchange carry relationships are respected.
• The discussion is conceptual and does not provide a full derivation or numerical example.

Tags

# Which risk-free rate to use to price a bond issued in one currency but convertible into equity in another?


# Which risk-free rate to use to price a bond issued in one currency but convertible into equity in another?












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?

## Answer by Brian B (score 3)

https://quant.stackexchange.com/a/609

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.

## Answer by glyphard (score 2)

https://quant.stackexchange.com/a/606

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.