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JavaScript Finance: Calculate the theoretical price of an option using the Forward implied volatility

The Forward implied volatility is calculated using the following formula:

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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:

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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:

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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:

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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:

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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

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