The Replicating Portfolio in Simple Terms

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.

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.

Replicating Portfolio in Black-Scholes :

$V(t)=\mathrm{\Delta }(t)S(t)+B(t)$

Here, $V(t)$ is the value of the portfolio, $\mathrm{\Delta }(t)$ represents the amount of the underlying asset held, $S(t)$ is the price of the underlying asset, and $B(t)$ is the amount invested in risk-free bonds.

The goal is to find $\mathrm{\Delta }(t)$ such that the portfolio replicates the option's payoff. This leads to the Black-Scholes partial differential equation:

$$\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0$$

Solving this PDE gives the Black-Scholes formula for option prices, effectively allowing valuation in a risk-neutral framework.

Choice of 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.

Numeraire for Caplet Pricing :

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.

$P(t,T)=e^{-r(T-t)}$

Here, $P(t,T)$ is the price of a zero-coupon bond maturing at time $T$, and $r$ is the continuously compounded risk-free rate.

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:

$\frac{C(t)}{P(t,T)}$

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.

Link Between Replicating Portfolios and Numeraire :

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.

The replicating portfolio technique allows us to hedge and price options by dynamically adjusting positions in the underlying and risk-free assets. Meanwhile, the numeraire technique simplifies multi-dimensional problems in interest rate models by scaling prices relative to a risk-free benchmark, ensuring martingale properties under the selected measure.

Both methods demonstrate the elegance of mathematical finance in reducing complexity, ensuring arbitrage-free pricing, and enabling effective risk management.