Risk-Neutral Measures
A risk neutral measure is a probability measure used in mathematical finance to aid in pricing derivatives and other financial assets. Risk neutral measures give investors a mathematical interpretation of the overall market’s risk averseness to a particular asset, which must be taken into account in order to estimate the correct price for that asset.A risk neutral measure is also known as an equilibrium measure or equivalent martingale measure.
Risk-Neutral Measure
Definition
The risk-neutral measure is a probability measure used in mathematical finance for pricing derivatives and other financial assets. It provides investors with a mathematical explanation of the market's overall risk aversion towards a specific asset, which must be considered when estimating the correct price of that asset. The risk-neutral measure is also known as the equilibrium measure or equivalent martingale measure.
Origin
The concept of the risk-neutral measure originated in the 1970s and matured with the development of financial mathematics and financial engineering. In 1973, Fischer Black and Myron Scholes introduced the famous Black-Scholes option pricing model, which uses the risk-neutral measure as its core for pricing options.
Categories and Characteristics
The risk-neutral measure can be classified into the following types:
- Real-world probability measure: This is the probability measure observed by investors in the real world, reflecting the actual risk preferences of the market.
- Risk-neutral probability measure: This is the probability measure used in theoretical models, assuming that investors are neither risk-averse nor risk-seeking.
Characteristics:
- Simplified calculations: Using the risk-neutral measure can simplify the process of pricing derivatives.
- Theoretical foundation: Provides a solid theoretical foundation for financial models.
- Market consistency: Reflects the overall market attitude towards risk.
Specific Cases
Case 1: Option Pricing
In option pricing, the risk-neutral measure is used to calculate the theoretical price of an option. Suppose a stock's current price is $100, with a volatility of 20% over the next year, and a risk-free interest rate of 5%. Using the Black-Scholes model, the option's price can be calculated under the risk-neutral measure.
Case 2: Bond Pricing
In the bond market, the risk-neutral measure is also used to price complex bond products. For example, a company issues a convertible bond that investors can convert into company stock at a future date. Using the risk-neutral measure, the fair value of this convertible bond can be calculated.
Common Questions
Question 1: What is the difference between the risk-neutral measure and the real-world probability measure?
The risk-neutral measure assumes that investors are neither risk-averse nor risk-seeking, while the real-world probability measure reflects the actual risk preferences of the market.
Question 2: Why use the risk-neutral measure?
Using the risk-neutral measure can simplify the process of pricing derivatives and provides a solid theoretical foundation for financial models.
