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Risk-neutral probabilities are defined as the specific set of probabilities chosen precisely to make the discounted price process of any asset a martingale. They are not derived from historical data or investor beliefs, but from the mathematical requirement that discounted prices behave like fair games.

Conditions: Operating within a risk-neutral valuation framework.; Prices are discounted at the risk-free rate.

Risk-neutral probabilities are defined as the specific set of probabilities chosen precisely to make the discounted price process of any asset a martingale. They are not derived from historical data or investor beliefs, but from the mathematical requirement that discounted prices behave like fair games.

Conditions: Operating within a risk-neutral valuation framework.; Prices are discounted at the risk-free rate.

A sequence of random variables $M_0, M_1, \ldots, M_N$ is a martingale if it is adapted and satisfies the condition $M_n = \mathbb{E}_n[M_{n+1}]$ for all $n = 0, 1, \ldots, N - 1$. This means the expected value of the next variable, given all prior information, is exactly equal to the current variable's value.

Conditions: The sequence must be adapted, meaning $M_n$ depends only on the first $n$ events.; The equality must hold for all time steps from $0$ to $N-1$.

A sequence of random variables $M_0, M_1, \ldots, M_N$ is a martingale if it is adapted and satisfies the condition $M_n = \mathbb{E}_n[M_{n+1}]$ for all $n = 0, 1, \ldots, N - 1$. This means the expected value of the next variable, given all prior information, is exactly equal to the current variable's value.

Conditions: The sequence must be adapted, meaning $M_n$ depends only on the first $n$ events.; The equality must hold for all time steps from $0$ to $N-1$.