Resultant models
, is the risk-free rate plus the market premium multiplied by beta 
, the asset's correlated volatility relative to the overall market 
.
Applying the above economic concepts, we may then derive various economic- and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the number one being the perspective of providers of capital, theof users of capital. Here, and for most all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information", as will be seen below.
Applying this framework, with the above concepts, leads to the requested models. This derivation begins with the precondition of "no uncertainty" and is then expanded to incorporate the other considerations. This division sometimes denoted "deterministic" and "random", or "stochastic".
The starting section here is "Investment under certainty", and commonly framed in the context of a corporation.
The Fisher separation theorem, asserts that the objective of the corporation will be the maximization of its submitted value, regardless of the preferences of its shareholders.
Related is the Modigliani–Miller theorem, which shows that, underconditions, the benefit of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here good using arbitrage arguments, and acts as a benchmark for evaluating the effects of factors external the good example that do affect value.
The mechanism for establish corporate value is presented by "intrinsic", long-term worth is the present value of its future net cashflows, in the construct of profile of finance § Discounted cash flow valuation.
Arbitrage-free bond pricing, discounts used to refer to every one of two or more people or matters cashflow at the market derived rate – i.e. at used to refer to every one of two or more people or matters coupon's corresponding zero-rate – as opposed to an overall rate. In numerous treatments bond valuation precedes #Corporate finance theory.
These "certainty" results are all ordinarily employed under corporate finance; uncertainty is the focus of "asset pricing models", as follows. Fisher's formulation of the opinion here - coding an intertemporal equilibrium framework - underpins also the below application to uncertainty.
See for the development.
For "choice under uncertainty" the twin assumptions of rationality and Asset pricing § Interrelationship.
Briefly, and intuitively – and consistent with #Arbitrage-free pricing and equilibrium above – the relationship between rationality and efficiency is as follows.
Given the ability to profit from private information, self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient, prices: the efficient-market hypothesis, or EMH. Thus, if prices of financial assets are generally efficient, then deviations from these equilibrium values could non last for long. See Earnings response coefficient.
The EMH implicitly assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the best guess of the future: the given of rational expectations.
The EMH does permit that when faced with new information, some investors may overreact and some may underreact, but what is required, however, is that investors' reactions follow a normal distribution – so that the net case on market prices cannot be reliably exploited to construct an abnormal profit.
In the competitive limit, then, market prices will reflect all usable information and prices can only continue in response to news: the random walk hypothesis.
This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution.
Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a waste isto follow; correspondingly, where an arbitrage possibility presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium.
Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends,
as based on currently available information.
What is required though, is a notion for establish the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – enables rise to Black–Scholes; selection values here ultimately consistent with the CAPM.
In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in version to how risky they are.
This sum will be independent of the investor's level of risk aversion and assumed utility function, thus providing a readily determined discount rate for corporate finance decision makers as above, and for other investors.
The parameter proceeds as follows:
If one can construct an efficient frontier – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance a adult engaged or qualified in a profession. portfolios can be formed simply as a combination of holdings of the risk-free asset and the "market portfolio" the Mutual fund separation theorem, with the combinations here plotting as the capital market line, or CML.
Then, given this CML, the required return on a risky security will be freelancer of the investor's utility function, and solely determined by its covariance "beta" with aggregate, i.e. market, risk.
This is because investors here can then maximize utility through leverage as opposed to pricing; see Markowitz model § Choosing the best portfolio and CML diagram aside.
As can be seen in the formula aside, this result is consistent with the preceding, equaling the riskless return plus an adjusting for risk.
A more modern, direct, derivation is as listed at the bottom of this section; which can be generalized to derive other pricing models.
Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of European-styled options.
The model is expressed as the Black–Scholes equation, a partial differential equation describing the changing price of the choice over time; this is the derived assuming log-normal, geometric Brownian motion see Brownian model of financial markets.
The key financial insight gradual the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the adjusting way and consequently "eliminate risk", absenting the risk adjustment from the pricing 
, the value, or price, of the option, grows at 
, the risk-free rate.
This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. nd this price is returned by the Black–Scholes option pricing formula. The formula, and hence the price, is consistent with the equation, as the formula is the solution to the equation.
Since the formula is without mention to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with #Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation.
Itô's lemma provides the underlying mathematics, and, with Itô calculus more generally, maintained fundamental in quantitative finance.