See Real options valuation#Technical considerations. (Another case where the modelling assumptions may depart from rational pricing is the valuation of employee stock options.) Risk neutral valuation Here the value of the option is calculated using the risk neutrality assumption. Under this assumption, the "expected value" (as opposed to "locked in" value) is discounted. The expected value is calculated using the intrinsic values from the later two nodes: "Option up" and "Option down", with u and d as price multipliers as above. These are then weighted by their respective probabilities: "probability" p of an up move in the underlying, and "probability" (1-p) of a down move. The expected value is then discounted at r, the risk-free rate. Solve for p under risk-neutrality, for no arbitrage to be possible in the share, today's price must represent its expected value discounted at the risk free rate (i.e., the share price is a Martingale): Solve for call value, using p for no arbitrage to be possible in the call, today's price must represent its expected value discounted at the risk free rate: The risk neutrality assumption Note that above, the risk neutral formula does not refer to the expected or forecast return of the underlying, nor its volatility p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices. Nevertheless, both arbitrage free pricing and risk neutral valuation deliver identical results. In fact, it can be shown that "delta hedging" and "risk-neutral valuation" use identical formulae expressed differently. Given this equivalence, it is valid to assume "risk neutrality" when pricing derivatives. A more formal relationship is described via the fundamental theorem of arbitrage-free pricing. Pricing shares The arbitrage pricing theory (APT), a general theory of asset pricing, has become influential in the pricing of shares. APT holds that the expected return of a financial asset can be modelled as a linear function of various macro-economic factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient: where is the risky asset's expected return, is the risk free rate, is the macroeconomic factor, is the sensitivity of the asset to factor , and is the risky asset's idiosyncratic random shock with mean zero. The model derived rate of return will then be used to price the asset correctly – the asset price should equal the expected end of period price discounted at the rate