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.
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.
(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.
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.
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).
The owner can sell the option, or wait and hope the price changes.
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).
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.
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: 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.
When quantifying moneyness, it is computed as a single number with respect to spot (or forward) and strike, without specifying a reference option.
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).
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.
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.
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.
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.
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]
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.
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: The standardized moneyness is the average of these: and they are ordered as: differing only by a step of
The interpretation of these quantities is somewhat subtle, and consists of changing to a risk-neutral measure with specific choice of numéraire.
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−σ2/2, which is the reason for the correction factor.
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.