In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, moneyness is the relative position of the current
price
A price is the (usually not negative) quantity of payment or compensation given by one party to another in return for goods or services. In some situations, the price of production has a different name. If the product is a "good" in the c ...
(or future price) of an underlying
asset
In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value ...
(e.g., a
stock
In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...
) with respect to the
strike price
In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set b ...
of a
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
, most commonly a
call option
In finance, a call option, often simply labeled a "call", is a contract between the buyer and the seller of the call option to exchange a security at a set price. The buyer of the call option has the right, but not the obligation, to buy an ...
or a
put option
In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a s ...
. Moneyness is firstly a three-fold classification:
* If the derivative would have positive
intrinsic value if it were to
expire
Expire was an American hardcore punk band from Milwaukee, Wisconsin, United States, active from 2009 to 2017. They were signed to Bridge 9 Records.
History
Expire was formed in 2009. They released three full-length albums, three EPs and one spl ...
today, it is said to be in the money;
* If the derivative would be worthless if expiring with the underlying at its current price, it is said to be out of the money;
* And if the current underlying price and strike price are equal, the derivative is said to be at the money.
There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc.
This rough classification can be quantified by various
definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definit ...
to express the moneyness as a number, measuring how far the asset is in the money or out of the money with respect to the strike – or, conversely, how far a strike is in or out of the money with respect to the spot (or forward) price of the asset. This quantified notion of moneyness is most importantly used in defining the ''relative''
volatility surface
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
: the implied
volatility in terms of moneyness, rather than absolute price. The most basic of these measures is simple moneyness, which is the ratio of spot (or forward) to strike, or the reciprocal, depending on convention. A particularly important measure of moneyness is the likelihood that the derivative will expire in the money, in the
risk-neutral measure
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price und ...
. It can be measured in percentage
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of expiring in the money, which is the forward value of a
binary call option
A binary option is a financial exotic option in which the payoff is either some fixed monetary amount or nothing at all.Breeden, D. T., & Litzenberger, R. H. (1978). "Prices of state-contingent claims implicit in option prices". ''Journal of Busi ...
with the given strike, and is equal to the auxiliary ''N''(''d''
2) term in the
Black–Scholes formula. This can also be measured in
standard deviations
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
, measuring how far above or below the strike price the current price is, in terms of volatility; this quantity is given by ''d''
2. (Standard deviations refer to the price fluctuations of the underlying instrument, not of the option itself.) Another measure closely related to moneyness is the
Delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, at a river mouth
* D (NATO phonetic alphabet: "Delta")
* Delta Air Lines, US
* Delta variant of SARS-CoV-2 that causes COVID-19
Delta may also re ...
of a call or put option. There are other proxies for moneyness, with convention depending on market.
Example
Suppose the current stock price of
IBM is $100. A
call
Call or Calls may refer to:
Arts, entertainment, and media Games
* Call, a type of betting in poker
* Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage
Music and dance
* Call (band), from Lahore, Paki ...
or
put option
In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a s ...
with a strike of $100 is at-the-money. A
call
Call or Calls may refer to:
Arts, entertainment, and media Games
* Call, a type of betting in poker
* Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage
Music and dance
* Call (band), from Lahore, Paki ...
with a strike of $80 is in-the-money (100 − 80 = 20 > 0). A
put option
In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a s ...
with a strike at $80 is out-of-the-money (80 − 100 = −20 < 0). Conversely, a call option with a $120 strike is out-of-the-money and a put option with a $120 strike is in-the-money.
The above is a traditional way of defining ITM, OTM and ATM, but some new authors find the comparison of strike price with current market price meaningless and recommend the use of Forward Reference Rate instead of Current Market Price. For example, a put option will be in the money if the strike price of the option is greater than the Forward Reference Rate.
Intrinsic value and time value
The intrinsic value (or "monetary value") of an option is its value assuming it were exercised immediately. Thus if the current (
spot
Spot or SPOT may refer to:
Places
* Spot, North Carolina, a community in the United States
* The Spot, New South Wales, a locality in Sydney, Australia
* South Pole Traverse, sometimes called the South Pole Overland Traverse
People
* Spot (produ ...
) price of the underlying security (or commodity etc.) is above the agreed (
strike
Strike may refer to:
People
*Strike (surname)
Physical confrontation or removal
*Strike (attack), attack with an inanimate object or a part of the human body intended to cause harm
*Airstrike, military strike by air forces on either a suspected ...
) price, a
call
Call or Calls may refer to:
Arts, entertainment, and media Games
* Call, a type of betting in poker
* Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage
Music and dance
* Call (band), from Lahore, Paki ...
has positive intrinsic value (and is called "in the money"), while a
put has zero intrinsic value (and is "out of the money").
The
time value of an option is the total value of the option, less the intrinsic value. It partly arises from the uncertainty of future price movements of the underlying. A component of the time value also arises from the unwinding of the discount rate between now and the expiry date. In the case of a European option, the option cannot be exercised before the expiry date, so it is possible for the time value to be negative; for an American option if the time value is ever negative, you exercise it (ignoring special circumstances such as the security going ex dividend): this yields a
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
.
Moneyness terms
At the money
An
option is at the money (ATM) if the
strike price
In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set b ...
is the same as the current spot price of the underlying security. An at-the-money option has no intrinsic value, only time value.
[At the Money Definition](_blank)
, Cash Bauer 2012
For example, with an "at the money" call stock option, the current share price and strike price are the same. Exercising the option will not earn the seller a profit, but any move upward in stock price will give the option value.
Since an option will rarely be exactly at the money, except for when it is written (when one may buy or sell an ATM option), one may speak informally of an option being near the money or close to the money.
[Near The Money]
, Investopedia Similarly, given standardized options (at a fixed set of strikes, say every $1), one can speak of which one is nearest the money; "near the money" may narrowly refer specifically to the nearest the money strike. Conversely, one may speak informally of an option being far from the money.
In the money
An in the money (ITM) option has positive intrinsic value as well as time value. A call option is in the money when the strike price is below the spot price. A put option is in the money when the strike price is above the spot price.
With an "in the money" call stock option, the current share price is greater than the strike price so exercising the option will give the owner of that option a profit. That will be equal to the market price of the share, minus the option strike price, times the number of shares granted by the option (minus any commission).
Out of the money
An out of the money (OTM) option has no intrinsic value. A call option is out of the money when the strike price is above the spot price of the underlying security. A put option is out of the money when the strike price is below the spot price.
With an "out of the money" call stock option, the current share price is less than the strike price so there is no reason to exercise the option. The owner can sell the option, or wait and hope the price changes.
Spot versus forward
Assets can have a forward price (a price for delivery in future) as well as a spot price. One can also talk about moneyness with respect to the forward price: thus one talks about ATMF, "ATM Forward", and so forth. For instance, if the spot price for USD/JPY is 120, and the forward price one year hence is 110, then a call struck at 110 is ATMF but not ATM.
Use
Buying an ITM option is effectively lending money in the amount of the intrinsic value. Further, an ITM call can be replicated by entering a forward and buying an OTM put (and conversely). Consequently, ATM and OTM options are the main traded ones.
Definition
Moneyness function
Intuitively speaking, moneyness and time to expiry form a two-dimensional
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
. 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
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Blac ...
, notably time to expiry, interest rates, and
implied volatility In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equa ...
(concretely the ATM implied volatility), yielding a function:
:
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
The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations.
Since the risk-free ra ...
, 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
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
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.
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
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
, or more simply plotting a
volatility smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
.
Simple examples
This section outlines moneyness measures from simple but less useful to more complex but more useful.
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,
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,
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,
thus ''F''/''K'' will be used in the sequel. In practice, for low interest rates and short tenors, spot versus forward makes little difference.
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
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
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:
This effectively normalizes for time to expiry – with this measure of moneyness, volatility smiles are largely independent of time to expiry.
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
In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...
by volatility yields:
[, who uses spot rather than forward.]
:
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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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
In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set b ...
, 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''
+ = ''d''
1 and ''d''
− = ''d''
2, which are defined as:
:
The standardized moneyness is the average of these:
:
and they are ordered as:
:
differing only by a step of
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
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
''N'' for these values. The
interpretation of these quantities is somewhat subtle, and consists of changing to a
risk-neutral measure
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price und ...
with specific choice of
numéraire
The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, acting ...
. In brief, these are interpreted (for a call option) as:
* ''N''(''d''
−) is the (Future Value) price of a
binary call option
A binary option is a financial exotic option in which the payoff is either some fixed monetary amount or nothing at all.Breeden, D. T., & Litzenberger, R. H. (1978). "Prices of state-contingent claims implicit in option prices". ''Journal of Busi ...
, 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''
+) is the
Delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, at a river mouth
* D (NATO phonetic alphabet: "Delta")
* Delta Air Lines, US
* Delta variant of SARS-CoV-2 that causes COVID-19
Delta may also re ...
, 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):
:
Of these, ''N''(''d''
−) is the (risk-neutral) "likelihood of expiring in the money", and thus the theoretically correct percent moneyness, with ''d''
− 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
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...
with drift ''r,'' the risk-free rate, and diffusion ''σ,'' the implied volatility. Drift is the mean, with the corresponding median (50th
percentile
In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage falls ...
) being ''r''−''σ''
2/2, 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.
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
In options trading, a vertical spread is an options strategy involving buying and selling of multiple options of the same underlying security, same expiration date, but at different strike prices. They can be created with either all calls or all p ...
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''
2/2)''τ'' is relatively subtle. For ''d''
− and ''m'' this corresponds to the difference between the median and mean (respectively) of
geometric Brownian motion
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...
(the
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
), and is the same correction factor in
Itō's lemma for geometric Brownian motion. The interpretation of ''d''
+, 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.
References
*
*
*
*
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{{Derivatives market
Options (finance)
Derivatives (finance)