A risk-neutral measure is a probability measure equivalent to the physical probability measure , meaning they agree on which events have zero probability. For assets without intermediate payments, prices expressed in the chosen numéraire must be martingales under . For a positive bank account and a stock without dividends, this requires to be a martingale; with dividends, use discounted dividend gains. In an unrestricted finite-state one-period financial market, existence of such a measure is equivalent to absence of arbitrage, by the fundamental theorem of asset pricing.
Physical probability describes the modeled frequency or belief about an up move. The risk-neutral probability is constrained by traded prices. In the worked binomial market, even if or . Calling risk-neutral does not assume that investors have no risk aversion; it changes the weights used for risk-neutral pricing.
Articles by others on the same topic
A risk-neutral measure is a concept used primarily in financial mathematics and quantitative finance, particularly in the context of pricing derivatives and financial instruments. It is a probability measure under which the present value of future cash flows can be calculated by discounting the expected payoffs at the risk-free rate, without needing to consider the risk preferences of investors. In a risk-neutral world, all investors are indifferent to risk, which means they require no additional return for taking on more risk.