Explain the concept of "change of numeraire" and its application in the pricing of financial derivatives under different measures.?

The concept of a "change of numeraire" refers to the technique of changing the unit of account or the reference asset to value financial derivatives. The motivation behind this technique is to simplify the valuation of these derivatives under different measures, often making the price easier to compute.

To explain change of numeraire, let’s first look at the concept of measures in the context of derivative pricing. A measure is a probability distribution that defines the weight of all possible outcomes in a stochastic model. In the realm of financial mathematics, the most important measures are the historical measure, the real-world probability measure (also denoted as P), and the risk-neutral measure (denoted as Q). The historical measure represents the real-world probabilities of different scenarios, while the risk-neutral measure ensures that asset prices under this measure have no-arbitrage opportunities. Often, financial derivatives are priced under the risk-neutral measure.

Now, let’s consider a financial market with multiple assets, denoted as S₀(t), S₁(t), …, S_n(t). In this context, a numeraire is an asset, say N(t), among those available in the market, that we use as the reference or the unit of account to value other assets. The numeraire itself is a stochastic process that evolves over time.

When we change the reference asset or numeraire, we effectively switch from one probability measure, Q₁, to another measure, Q₂, such that, in both measures, the price process of a traded asset (in terms of numeraire) should be a martingale. This implies that by finding the measure Q₂, we can then find the price of the asset and the derivative whose payoff is based on this asset.

Mathematically, the change of numeraire technique involves finding a process Λ(t) such that the likelihood ratio satisfies:

$$\frac{dQ_2}{dQ_1} \equiv \Lambda(T) = \frac{N_1(T)S^*_2(T)}{N_2(T)S^*_1(T)}$$

Here, S^* denotes the discounted price process, and N₁(t) and N₂(t) are the two numeraires corresponding to measures Q₁ and Q₂, respectively.

Let’s consider an example to illustrate the change of numeraire technique. Suppose we have a zero-coupon bond B(t, T), maturing at time T, with a value B(T, T) = 1. Traditionally, we price this bond using the money market account as the numeraire, N(t) = B(t, 0), but another numeraire can be chosen.

One popular choice is to use another zero-coupon bond, say N(t) = B(t, M), maturing at time M with a value B(M, M) = 1. The change of numeraire technique states that, given the measure Q^B under the traditional numeraire, there exists a measure Q^N = Q^B(·,M) under the alternative numeraire N(t) = B(t, M), such that:

$$\frac{dQ^N}{dQ^B} \equiv \Lambda(T) = \frac{B(T,0)B(T,M)}{B(T,0)B(T,M)}$$

In this case, since the zero-coupon bond B(t, T) is priced in terms of the bond B(t, M), the likelihood ratio simplifies to:

$$\Lambda(T) = \frac{B(T,M)}{B(0,M)}$$

Thus, by finding the new measure Q^N under the alternative numeraire N(t) = B(t, M), we can then price the zero-coupon bond B(t, T).

In conclusion, the change of numeraire technique is a powerful tool in quant finance for simplifying the pricing of financial derivatives under different measures. By choosing an appropriate numeraire and its corresponding measure, we can exploit the martingale property of asset prices to find the fair price of the derivative.