LocalVariance

Local variance is a concept used in mathematical finance, particularly in the context of option pricing and volatility modeling. It is related to the local volatility, which is a function that represents the instantaneous volatility of a stock or other financial instrument at a given point in time and price.

The risk-neutral expectation is a measure used to calculate the expected value of a random variable under the assumption of risk-neutrality. This means that investors in this hypothetical scenario do not require a premium for bearing risk, and the value of assets is based solely on their expected future cash flows, discounted at the risk-free rate.

The local variance is the risk-neutral expectation of the local variance of a financial instrument, conditional upon the final stock price being equal to the strike price. This can be expressed mathematically as:

$$E[LV(S, t) | S(T) = K]$$

where:

• $E$ denotes the risk-neutral expectation operator,
• $LV(S, t)$ represents the local variance of the stock price S at time t,
• $S(T)$ is the stock price at the expiration time T, and
• $K$ is the strike price of the option.

This expectation provides information about the volatility dynamics of the underlying asset at the time when the option is exercised. It can be used to better understand the pricing of options and other derivatives, as well as to manage risk in a portfolio that includes these financial instruments.