Editing Moneyness (section)

==Definition==

===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.

===Conventions===
When quantifying moneyness, it is computed as a single number with respect to spot (or forward) and strike, without specifying a reference option. There are thus two conventions, depending on direction: call moneyness, where moneyness increases if spot increases relative to strike, and put moneyness, where moneyness increases if spot decreases relative to strike. These can be switched by changing sign, possibly with a shift or scale factor (e.g., the probability that a put with strike ''K'' expires ITM is one minus the probability that a call with strike ''K'' expires ITM, as these are complementary events). Switching spot and strike also switches these conventions, and spot and strike are often complementary in formulas for moneyness, but need not be. Which convention is used depends on the purpose. The sequel uses ''call'' moneyness – as spot increases, moneyness increases – and is the same direction as using call Delta as moneyness.

While moneyness is a function of both spot and strike, usually one of these is fixed, and the other varies. Given a specific option, the strike is fixed, and different spots yield the moneyness of that option at different market prices; this is useful in option pricing and understanding the [[Black–Scholes formula]]. Conversely, given market data at a given point in time, the spot is fixed at the current market price, while different options have different strikes, and hence different moneyness; this is useful in constructing an [[implied volatility surface]], or more simply plotting a [[volatility smile]].<ref name="neftci">{{Harv|Neftçi|2008|loc=11.2 How Can We Define Moneyness? |pp=[https://books.google.com/books?id=IOEyEY3TM9AC&pg=PA458 458–460]}}</ref>

===Simple examples===
This section outlines moneyness measures from simple but less useful to more complex but more useful.<ref name="hafner85">{{Harv|Häfner|2004|loc=Section 5.3.1, Choice of Moneyness Measure, pp. [https://books.google.com/books?id=67OE8cKXWBQC&pg=PA85 85–87]}}</ref> Simpler measures of moneyness can be computed immediately from observable market data without any theoretical assumptions, while more complex measures use the implied volatility, and thus the Black–Scholes model.

The simplest (put) moneyness is '''fixed-strike moneyness''',<ref name="hafner42" /> where ''M''=''K,'' and the simplest call moneyness is '''fixed-spot moneyness''', where ''M''=''S.'' These are also known as '''absolute moneyness''', and correspond to not changing coordinates, instead using the raw prices as measures of moneyness; the corresponding volatility surface, with coordinates ''K'' and ''T'' (tenor) is the ''absolute volatility surface''. The simplest non-trivial moneyness is the ratio of these, either ''S''/''K'' or its reciprocal ''K''/''S,'' which is known as the (spot) '''simple moneyness''',<ref name="hafner85" /> with analogous forward simple moneyness. Conventionally the fixed quantity is in the denominator, while the variable quantity is in the numerator, so ''S''/''K'' for a single option and varying spots, and ''K''/''S'' for different options at a given spot, such as when constructing a volatility surface. A volatility surface using coordinates a non-trivial moneyness ''M'' and time to expiry ''τ'' is called the ''relative volatility surface'' (with respect to the moneyness ''M'').

While the spot is often used by traders, the forward is preferred in theory, as it has better properties,<ref name="hafner85" /><ref>{{Harv|Natenberg|1994|loc=pp. 106–110}}</ref> thus ''F''/''K'' will be used in the sequel. In practice, for low interest rates and short tenors, spot versus forward makes little difference.<ref name="hafner42" />

In (call) simple moneyness, ATM corresponds to moneyness of 1, while ITM corresponds to greater than 1, and OTM corresponds to less than 1, with equivalent levels of ITM/OTM corresponding to reciprocals. This is linearized by taking the log, yielding the '''log simple moneyness''' <math>\ln\left(F/K\right).</math> In the log simple moneyness, ATM corresponds to 0, while ITM is positive and OTM is negative, and corresponding levels of ITM/OTM corresponding to switching sign. Note that once logs are taken, moneyness in terms of forward or spot differ by an additive factor (log of discount factor), as <math>\ln\left(F/K\right) = \ln(S/K)+rT.</math>

The above measures are independent of time, but for a given simple moneyness, options near expiry and far from expiry behave differently, as options far from expiry have more time for the underlying to change. Accordingly, one may incorporate time to maturity ''τ'' into moneyness. Since dispersion of Brownian motion is proportional to the square root of time, one may divide the log simple moneyness by this factor, yielding:<ref>{{Harv|Natenberg|1994}}</ref> <math>\ln\left(F/K\right) \Big/ \sqrt{\tau}.</math> This effectively normalizes for time to expiry – with this measure of moneyness, volatility smiles are largely independent of time to expiry.<ref name="hafner85" />

This measure does not account for the volatility ''σ'' of the underlying asset. Unlike previous inputs, volatility is not directly observable from market data, but must instead be computed in some model, primarily using ATM implied volatility in the Black–Scholes model. Dispersion is proportional to volatility, so [[standardizing]] by volatility yields:<ref>{{Harv|Tompkins|1994}}, who uses spot rather than forward.</ref>
:<math>
m = \frac{\ln\left(F/K\right)}{\sigma\sqrt{\tau}}.
</math>
This is known as the '''standardized moneyness''' (forward), and measures moneyness in standard deviation units.

In words, the standardized moneyness is the number of [[standard deviations]] the current forward price is above the strike price. Thus the moneyness is zero when the forward price of the underlying equals the [[strike price]], when the option is ''at-the-money-forward''. Standardized moneyness is measured in standard deviations from this point, with a positive value meaning an in-the-money call option and a negative value meaning an out-of-the-money call option (with signs reversed for a put option).

===Black–Scholes formula auxiliary variables===
The standardized moneyness is closely related to the auxiliary variables in the Black–Scholes formula, namely the terms ''d''<sub>+</sub> = ''d''<sub>1</sub> and ''d''<sub>&minus;</sub> = ''d''<sub>2</sub>, which are defined as:
:<math>d_\pm = \frac{\ln\left(F/K\right) \pm (\sigma^2/2) \tau}{\sigma\sqrt{\tau}}.</math>
The standardized moneyness is the average of these:
:<math>m = \frac{\ln(F/K)}{\sigma\sqrt{\tau}} = \tfrac{1}{2}\left(d_- + d_+\right),</math>
and they are ordered as:
:<math>d_- < m < d_+,</math>
differing only by a step of <math>\sigma\sqrt{\tau}/2</math> in each case. This is often small, so the quantities are often confused or conflated, though they have distinct interpretations.

As these are all in units of standard deviations, it makes sense to convert these to percentages, by evaluating the [[standard normal]] [[cumulative distribution function]] ''N'' for these values. The [[Black–Scholes#Interpretation|interpretation]] of these quantities is somewhat subtle, and consists of changing to a [[risk-neutral measure]] with specific choice of [[numéraire]]. In brief, these are interpreted (for a call option) as:
* ''N''(''d''<sub>&minus;</sub>) is the (Future Value) price of a [[binary call option]], or the risk-neutral likelihood that the option will expire ITM, with numéraire cash (the risk-free asset);
* ''N''(''m'') is the percentage corresponding to standardized moneyness;
* ''N''(''d''<sub>+</sub>) is the [[Delta (finance)|Delta]], or the risk-neutral likelihood that the option will expire ITM, with numéraire asset.
These have the same ordering, as ''N'' is monotonic (since it is a CDF):
:<math>N(d_-) < N(m) < N(d_+) = \Delta.</math>

Of these, ''N''(''d''<sub>&minus;</sub>) is the (risk-neutral) "likelihood of expiring in the money", and thus the theoretically correct '''percent moneyness''', with ''d''<sub>&minus;</sub> the correct moneyness. The percent moneyness is the implied probability that the derivative will expire in the money, in the risk-neutral measure. Thus a moneyness of 0 yields a 50% probability of expiring ITM, while a moneyness of 1 yields an approximately 84% probability of expiring ITM.

This corresponds to the asset following [[geometric Brownian motion]] with drift ''r,'' the risk-free rate, and diffusion ''σ,'' the implied volatility. Drift is the mean, with the corresponding median (50th [[percentile]]) being ''r''&minus;''σ''<sup>2</sup>/2, which is the reason for the correction factor. Note that this is the ''implied'' probability, ''not'' the real-world probability.

The other quantities – (percent) standardized moneyness and Delta – are not identical to the actual percent moneyness, but in many practical cases these are quite close (unless volatility is high or time to expiry is long), and Delta is commonly used by traders as a measure of (percent) moneyness.<ref name="hafner42" /> Delta is more than moneyness, with the (percent) standardized moneyness in between. Thus a 25 Delta call option has less than 25% moneyness, usually slightly less, and a 50 Delta "ATM" call option has less than 50% moneyness; these discrepancies can be observed in prices of binary options and [[vertical spread]]s. Note that for puts, Delta is negative, and thus negative Delta is used – more uniformly, absolute value of Delta is used for call/put moneyness.

The meaning of the factor of (''σ''<sup>2</sup>/2)''τ'' is relatively subtle. For ''d''<sub>&minus;</sub> and ''m'' this corresponds to the difference between the median and mean (respectively) of [[geometric Brownian motion]] (the [[log-normal distribution]]), and is the same correction factor in [[Itō's lemma#Geometric Brownian motion|Itō's lemma for geometric Brownian motion]]. The interpretation of ''d''<sub>+</sub>, as used in Delta, is subtler, and can be interpreted most elegantly as change of numéraire. In more elementary terms, the probability that the option expires in the money and the value of the underlying at exercise are not independent – the higher the price of the underlying, the more likely it is to expire in the money ''and'' the higher the value at exercise, hence why Delta is higher than moneyness.