The Forward implied volatility is calculated using the following formula:
1 | FIV = sqrt((ln(F/K) + (r + (sigma^2)/2) * T) / T) |
Where F
is the forward price, K
is the strike price, r
is the risk-free interest rate, sigma
is the volatility of the underlying asset, and T
is the time to expiration.
To calculate the theoretical price of an option using the Forward implied volatility, we can use the Black-Scholes formula:
1 2 | Call = F * N(d1) - K * e^(-r * T) * N(d2) Put = K * e^(-r * T) * N(-d2) - F * N(-d1) |
Where N(x)
is the cumulative normal distribution function, and d1
and d2
are calculated as follows:
1 2 | d1 = (ln(F/K) + (r + (sigma^2)/2) * T) / (sigma * sqrt(T)) d2 = d1 - sigma * sqrt(T) |
Here is an example of a function that calculates the theoretical price of a European call option using the Forward implied volatility in JavaScript:
1 2 3 4 5 6 7 8 | function callOptionPrice(F, K, r, sigma, T) { // Calculate d1 and d2 var d1 = (Math.log(F / K) + (r + (sigma * sigma) / 2) * T) / (sigma * Math.sqrt(T)); var d2 = d1 - sigma * Math.sqrt(T); // Calculate the theoretical price using the Black-Scholes formula var price = F * normalCDF(d1) - K * Math.exp(-r * T) * normalCDF(d2); return price; } |
Note that this function assumes that the normalCDF(x)
function is defined and returns the cumulative normal distribution function for a given value of x
. This function can be implemented using the error function, as shown in the following example:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | function normalCDF(x) { return (1 / 2) * (1 + erf(x / Math.sqrt(2))); } function erf(x) { // Approximate the error function using a Taylor series expansion var a1 = 0.254829592; var a2 = -0.284496736; var a3 = 1.421413741; var a4 = -1.453152027; var a5 = 1.061405429; var p = 0.3275911; var sign = 1; if (x < 0) { sign = -1; } x = Math.abs(x); var t = 1 / (1 + p * x); var y = 1 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x); return sign * y; } |
Note that this implementation of the error