# Moneyness

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In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a three-fold classification: if the derivative would have positive intrinsic value if it were to expire today, it is said to be in the money; if it would be worthless if expiring at the current price it is said to be out of the money, and if the current price and strike price are equal, it 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.

Finance is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

In finance, the strike price of an option is the fixed price at which the owner of the option can buy, or sell, the underlying security or commodity. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium.

In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying." Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation or getting access to otherwise hard-to-trade assets or markets. Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the New York Stock Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges. Derivatives are one of the three main categories of financial instruments, the other two being stocks and debt. The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed a century ago, are a more recent historical example.

## Contents

This rough classification can be quantified by various definitions 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: 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. It can be measured in percentage probability of expiring in the money, which is the forward value of a binary call option with the given strike, and is equal to the auxiliary N(d2) term in the Black–Scholes formula. This can also be measured in standard deviations, measuring how far above or below the strike price the current price is, in terms of volatility; this quantity is given by d2. (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 of a call or put option. There are other proxies for moneyness, with convention depending on market. [1]

In finance, the expiration date of an option contract is the last date on which the holder of the option may exercise it according to its terms. In the case of options with "automatic exercise" the net value of the option is credited to the long and debited to the short position holders.

In mathematical finance, a risk-neutral measure is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free.

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

## Example

Suppose the current stock price of IBM is $100. A call or put option with a strike of$100 is at-the-money. A call with a strike of $80 is in-the-money (100 80 = 20 > 0). A put option 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. Though the above is a traditional way of calculating ITM, OTM and ATM, 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, the option will be In The Money if Strike Price of Buy PUT on underlying is greater than the Forward Reference Rate. [2]

International Business Machines Corporation (IBM) is an American multinational information technology company headquartered in Armonk, New York, with operations in over 170 countries. The company began in 1911, founded in Endicott, New York, as the Computing-Tabulating-Recording Company (CTR) and was renamed "International Business Machines" in 1924.

A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation, to buy an agreed quantity of a particular commodity or financial instrument from the seller of the option at a certain time for a certain price. The seller is obligated to sell the commodity or financial instrument to the buyer if the buyer so decides. The buyer pays a fee for this right. The term "call" comes from the fact that the owner has the right to "call the stock away" from the seller.

In finance, a put or put option is a stock market device which gives the owner the right, but not the obligation, to sell an asset, at a specified price, by a predetermined date to a given party. The purchase of a put option is interpreted as a negative sentiment about the future value of the underlying stock. The term "put" comes from the fact that the owner has the right to "put up for sale" the stock or index.

## 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) price of the underlying security (or commodity etc.) is above the agreed (strike) price, a call 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 finance, the time value (TV) of an option is the premium a rational investor would pay over its current exercise value, based on the probability it will increase in value before expiry. For an American option this value is always greater than zero in a fair market, thus an option is always worth more than its current exercise value.. As an option can be thought of as 'price insurance', TV can be thought of as the risk premium the option seller charges the buyer—the higher the expected risk, the higher the premium. Conversely, TV can be thought of as the price an investor is willing to pay for potential upside.

## Moneyness terms

### At the money

An option is at the money (ATM) if the strike price is the same as the current spot price of the underlying security. An at-the-money option has no intrinsic value, only time value. [3]

In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed.

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. [4] 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 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:

${\displaystyle M(S,K,\tau ,r,\sigma ),}$

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 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. [5] 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. [1]

### Simple examples

This section outlines moneyness measures from simple but less useful to more complex but more useful. [6] 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, [5] 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, [6] 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, [6] [7] thus F/K will be used in the sequel. In practice, for low interest rates and short tenors, spot versus forward makes little difference. [5]

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${\displaystyle \ln \left(F/K\right).}$ 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 ${\displaystyle \ln \left(F/K\right)=\ln(S/K)+rT.}$

The above measures are independent of time, but for a given simple moneyness, options near expiry and far for 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: [8] ${\displaystyle \ln \left(F/K\right){\Big /}{\sqrt {\tau }}.}$ This effectively normalizes for time to expiry – with this measure of moneyness, volatility smiles are largely independent of time to expiry. [6]

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: [9]

${\displaystyle m={\frac {\ln \left(F/K\right)}{\sigma {\sqrt {\tau }}}}.}$

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+ = d1 and d = d2, which are defined as:

${\displaystyle d_{\pm }={\frac {\ln \left(F/K\right)\pm (\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}.}$

The standardized moneyness is the average of these:

${\displaystyle m={\frac {\ln(F/K)}{\sigma {\sqrt {\tau }}}}={\tfrac {1}{2}}\left(d_{-}+d_{+}\right),}$

and they are ordered as:

${\displaystyle d_{-}

differing only by a step of ${\displaystyle \sigma {\sqrt {\tau }}/2}$ 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 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) 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+) is the 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):

${\displaystyle N(d_{-})

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 with drift r, the risk-free rate, and diffusion σ, the implied volatility. Drift is the mean, with the corresponding median (50th percentile) being rσ2/2, 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. [5] 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 spreads. 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 (σ2/2)τ is relatively subtle. For d 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 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.

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## References

1. ( Neftci 2008 , 11.2 How Can We Define Moneyness? pp. 458–460 )
2. Chugh, Aman (2013). Financial Derivatives- The Currency and Rates Factor (First ed.). New Delhi: Dorling Kindersly (India) Pvt Ltd, licensees of Pearson Education in South Asia. p. 60. ISBN   978-81-317-7433-5 . Retrieved 18 August 2014.
3. At the Money Definition Archived 2012-06-16 at the Wayback Machine ., Cash Bauer 2012
4. "Near The Money", Investopedia
5. ( Häfner 2004 , Definition 3.12, p. 42 )
6. ( Häfner 2004 , Section 5.3.1, Choice of Moneyness Measure, pp. 85–87 )
7. ( Natenberg 1994 , pp. 106–110)
8. ( Tompkins 1994 ), who uses spot rather than forward.

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