Question

Derive a formula for the volga of a European call option on a no-dividend underlying asset in the Black-Scholes model.

Answer 1

The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

The price of a corresponding put option based on put-call parity is:

For both, as above:

is the cumulative distribution function of the standard normal distribution

is the time to maturity (expressed in years)

is the spot price of the underlying asset

is the strike price

is the risk-free rate (annual rate, expressed in terms of continuous compounding)

is the volatility of returns of the underlying asset

Alternative formulation[edit]

Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula):

The auxiliary variables are:

is the time to expiry (remaining time, backward time)

is the discount factor

is the forward price of the underlying asset, and {\displaystyle S=DF}

with d₊ = d₁ and d₋ = d₂ to clarify notation.

Given put-call parity, which is expressed in these terms as:

the price of a put option is:

Interpretation[edit]

The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the (and a fortiori ) terms, particularly   and why there are two different terms.^[14]

The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options but are easier to analyze.

Thus the formula:

breaks up as:

where is the present value of an asset-or-nothing call and is the present value of a cash-or-nothing call. The D factor is for discounting, because the expiration date is in the future, and removing it changes present value to future value (value at expiry). Thus is the future value of an asset-or-nothing call and is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.

The naive, and not quite correct, interpretation of these terms is that is the probability of the option expiring in the money , times the value of the underlying at expiry F, while is the probability of the option expiring in the money times the value of the cash at expiry K. This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for asset or it is not), but the probabilities and are not equal. In fact, can be interpreted as measures of moneyness (in standard deviations) and as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option,, is correct, as the value of the cash is independent of movements of the underlying, and thus can be interpreted as a simple product of "probability times value", while the {\displaystyle N(d_{+})F} is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.^[14] More precisely, the value of the asset at expiry is variable in terms of cash but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.

If one uses spot S instead of forward F, in instead of the term there is which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of d₋ for moneyness rather than the standardized moneyness – in other words, the reason for the factor – is due to the difference between the median and mean of the log-normal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing N(d₊) by N(d₋) in the formula yields a negative value for out-of-the-money call options.^[14]:6

In detail, the terms are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.^[14] The risk-neutral probability density for the stock price is

where is defined as above.

Specifically, is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. , however, does not lend itself to a simple probability interpretation. is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.^[15] For related discussion – and graphical representation – see section "Interpretation" under Datar–Mathews method for real option valuation.

The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure-theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.