In finance and mathematical finance, a numéraire is a chosen unit of account or reference measure that allows for the pricing and valuation of financial instruments. It is used to express the relative value of different assets and cash flows. The concept of a numéraire is particularly useful in risk-neutral pricing and the evaluation of derivative securities.
Here's how a numéraire works:
1. Choice of Numéraire: choose a specific financial instrument or asset to serve as the numéraire: a risk-free bond, a stock, or a basket of goods.
2. Relative Valuation: Once you have chosen the numéraire, all other financial assets and cash flows are valued relative to this numéraire.
3. Pricing and Valuation: When pricing derivatives or assessing the value of different investments, you use the numéraire as the reference point.
The concept of changing the numéraire, often referred to as "change of numéraire," is a technique used in mathematical finance to simplify calculations or make them more tractable. It involves switching from one numéraire to another while ensuring that relative valuations remain unchanged.
1. Initial Numéraire: Start with an initial numéraire, which could be a risk-free bond.
2. Change of Numéraire: Select a new numéraire that may simplify calculations or modeling. This new numéraire could be a different financial asset or even a combination of assets.
3. Adjusting Values: Adjust the values of all other assets and cash flows to maintain their relative values in terms of the new numéraire.
4. Perform Calculations: you can perform your calculations using the new numéraire
Changing the numéraire can make complex financial models more manageable and is often used in techniques like the Girsanov theorem for stochastic processes or in risk-neutral pricing frameworks.
Suppose you are analyzing two stocks, A and B, and you want to value a derivative based on the prices of these stocks. To make the calculation more straightforward, you decide to change the numéraire from the risk-free bond to Stock A.
Initial Scenario:
- Numéraire: Risk-Free Bond
- Price of Stock A: $100
- Price of Stock B: $50
Now, you want to value a derivative that depends on both Stock A and Stock B. You can change the numéraire to Stock A by introducing a conversion factor. Let's say the conversion factor is 0.5, which means you'll value everything in terms of Stock A.
New Scenario:
- Numéraire: Stock A
- Price of Stock A (in terms of itself): 1
- Price of Stock B (in terms of Stock A): $50 / $100 = 0.5
- Derivative's Payoff: Some function F(S_A, S_B), where S_A is the price of Stock A and S_B is the price of Stock B
By changing the numéraire to Stock A, you've made Stock A the reference point for valuation. The price of Stock B is now expressed relative to Stock A, and you can value the derivative more conveniently.
#StochasticProcesses #GirsanovTheorem
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