The Forward implied volatility is calculated using the following formula:
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:
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:
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:
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:
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