Change of numeraire is a concept used in quantitative finance for pricing derivatives or fixed income securities. It involves changing the underlying asset used to value the instrument.
In traditional derivative pricing, the pricing is done using a single underlying asset, but in reality, the value of the asset can have high variability that may make the pricing process difficult. Therefore, changing the underlying asset can simplify the pricing process by reducing volatility or making it more manageable.
For example, suppose we have a European call option that expires in one year, with a strike price of $100, and the underlying asset is a stock. Let’s assume that the volatility of the stock is very high, and the market is not very stable. In this case, we can choose to change the underlying asset from the stock to a fixed income instrument, such as a zero-coupon bond. We could use the bond as the numeraire and value the option based on its payout relative to the bond.
By choosing a less volatile and more predictable asset as the pricing reference, we can produce a more stable valuation of the derivative. We can show how the option is valued relative to the bond, allowing for more transparent pricing through the expected values of the payoff due at different times.
Similarly, the concept of a change of numeraire can also be applied to fixed income securities. Fixed-income securities are typically priced using a default-free bond as the benchmark. In this context, the numeraire is the interest rate. However, when interest rates become volatile or behave in unexpected ways, this standard framework can also become problematic. To address this, alternative numerairess are used for the valuation of fixed-income securities. These numerairess may include inflation rates, commodity prices, or foreign exchange rates, among others.
Overall, changing the numeraire can help reduce complex pricing equations and make the valuation of derivatives or fixed income securities more manageable. It also leads to more consistent pricing by using a less volatile and more predictable asset.