Two concepts central to derivatives pricing are the replicating portfolio in the Black-Scholes model and the choice of numeraire in interest rate options. While at first glance they might seem unrelated, a deeper look unveils the inherent link between the two.
Black-Scholes and the Replicating Portfolio:
In the Black-Scholes framework, to price an option, one employs a replicating portfolio – a combination of the underlying asset and risk-free bonds that mimics the option's payouts. This portfolio eradicates the need to know the "actual" probability of the stock going up or down. Instead, one uses risk-neutral probabilities that transform the expected stock price growth into the risk-free rate. This approach enables the option price to be determined solely by arbitrage arguments, irrespective of any subjective probabilities or utility functions.
Numeraire in Interest Rate Options:
Switching gears to interest rate options, things are a bit more intricate. With multiple bonds having different maturities, we're left with a choice. Which bond (or interest rate) should we treat as the "baseline" or the standard of measurement? This is where the numeraire concept slides in. A numeraire is a chosen reference security whose price, by convention, is set to unity. The choice of numeraire provides a risk-neutral measure, similar to how the replicating portfolio does in the Black-Scholes model. The asset's price relative to this numeraire should be a martingale under this risk-neutral measure.
Consider a cap, which is essentially a series of caplets, each providing protection against a rise in interest rates over its respective period. For each caplet, when the reference rate (e.g., LIBOR) exceeds the cap's strike rate, the caplet pays the difference; otherwise, it's worthless.
In pricing a caplet (and by extension, a cap), a natural choice of numeraire is a zero-coupon bond maturing at the payment time of the caplet.
Choosing the zero-coupon bond as the numeraire transforms our complex world of multiple rates and maturities into a simpler one where the ratio of the caplet's price to the bond's price becomes a martingale. This martingale property, a core tenet in derivative pricing, ensures that the price process, when discounted using the chosen numeraire, has a zero expected drift.
Connecting the Dots:
At the heart of both these methodologies is the core principle of risk-neutral valuation. In the Black-Scholes world, the replicating portfolio simplifies complexities by ensuring the option and portfolio have the same future value, leading to risk-neutral pricing.
In interest rate options, the choice of numeraire plays a similar role.
By selecting an appropriate numeraire, we adjust the complex dynamics of interest rates into a simplified, risk-neutral world, making the pricing exercise tractable.