Editing Moneyness (section)

===Moneyness function===
Intuitively speaking, moneyness and time to expiry form a two-dimensional [[coordinate system]] for valuing options (either in currency (dollar) value or in implied volatility), and changing from spot (or forward, or strike) to moneyness is a [[change of variables]]. Thus a '''moneyness function''' is a function ''M'' with input the spot price (or forward, or strike) and output a real number, which is called the '''moneyness'''. The condition of being a change of variables is that this function is monotone (either increasing for all inputs, or decreasing for all inputs), and the function can depend on the other parameters of the [[Black–Scholes model]], notably time to expiry, interest rates, and [[implied volatility]] (concretely the ATM implied volatility), yielding a function:
:<math>M(S, K, \tau, r, \sigma),</math>
where ''S'' is the spot price of the underlying, ''K'' is the strike price, ''τ'' is the time to expiry, ''r'' is the [[risk-free rate]], and ''σ'' is the implied volatility. The forward price ''F'' can be computed from the spot price ''S'' and the risk-free rate ''r.'' All of these are observables except for the implied volatility, which can computed from the observable price using the Black–Scholes formula.

In order for this function to reflect moneyness – i.e., for moneyness to increase as spot and strike move relative to each other – it must be [[Monotonic function|monotone]] in both spot ''S'' and in strike ''K'' (equivalently forward ''F,'' which is monotone in ''S''), with at least one of these strictly monotone, and have opposite direction: either increasing in ''S'' and decreasing in ''K'' (call moneyness) or decreasing in ''S'' and increasing in ''K'' (put moneyness). Somewhat different formalizations are possible.<ref name="hafner42">{{Harv|Häfner|2004|loc=Definition 3.12, p. [https://books.google.com/books?id=67OE8cKXWBQC&pg=PA42 42]}}</ref> Further axioms may also be added to define a "valid" moneyness.

This definition is abstract and notationally heavy; in practice relatively simple and concrete moneyness functions are used, and arguments to the function are suppressed for clarity.