Calculate Implied Volatility with the Bisection Method

Get an Excel spreadsheet & VBA to calculate implied volatility with the bisection method. Discover how numerical bisection works, its advantages and disadvantages

The implied volatility of an option is the volatility that matches its current market price (given the governing stock price, strike price, time to maturity, risk-free rate and dividend yield). Analysts compare implied and historical volatility to judge whether the option is fairly priced.

Quantitative analysts use several numerical methods to calculate implied volatility, including Newton-Raphson iteration and numerical bisection. The latter is easy to implement, and, unlike the Newton-Raphson approach, does not need numerical derivatives in its calculation.

Let’s explore the bisection algorithm in greater detail.

Step-by-Step Guide to Numerical Bisection

Step 1
If you want to calculate the implied volatility of an option, conceptually we want to find the root of this equation

f(v) = BlackScholesCall(S, X, T, r, d, v) – Price

i.e. the value of v for which f(v) = 0.

Start by picking upper and lower bounds of the volatility (volLower and volUpper) such that f(volLower) and f(volUpper) are opposite in sign. This means that the root (i.e. the implied volatility) lies between volLower and volUpper

Step 2
Calculate a volatility that lies halfway between volUpper and volLower, i.e. volMid = (volLower + volUpper)/2

Step 3
If f(volMid) = 0 (or is less than a threshold value, called epsilonABS in the VBA), then you’ve found your implied volatility. You can stop now!

Step 4
Evaluate f(volLower) and f(volMid). If these two are opposite in sign, i.e. if f(volLower) × f(volMid) < 0, then the root lies between volLower and volMid.

If, however, f(volLower) × f(volMid) > 0, then the root lies between volMid and volUpper.

Step 4
If f(volLower) × f(volMid) < 0, then let volUpper = volMid, and go back to Step 2. But if f(volLower) × f(volMid) > 0, then let volLower = volMid, and go back to Step 2.

Bisection is as simple as that, and the VBA code below implements this algorithm. It’s a robust method to calculate implied volatility. But when using bisection for other applications, you could need to account for several complications in the way your function behaves between the upper and lower bounds.

For example, a function may have two or more roots between the upper and lower search bounds.

Or there may be no roots between your upper and lower search bounds

Advantages and Disadvantages of the Bisection Method

The bisection method benefits from several advantages

1. The method always converges when the function changes sign – there’s no way for the method to diverge
2. The maximum error will always be the difference between the upper and lower bounds of the volatility. By increasing the number of iterations, you can control the error; each iteration halves the error

VBA and Excel Spreadsheet

The VBA includes two functions

• BlackScholesCall() calculates the price of a call option
• ImpliedVolatility() calculates implied volatility. It uses the bisection method to calculate volatility using BlackScholesCall()

   Function BlackScholesCall( _
    ByVal S As Double, _
    ByVal X As Double, _
    ByVal T As Double, _
    ByVal r As Double, _
    ByVal d As Double, _
    ByVal v As Double) As Double
    Dim d1 As Double
    Dim d2 As Double
    d1 = (Log(S / X) + (r - d + v ^ 2 / 2) * T) / v / Sqr(T)
    d2 = d1 - v * Sqr(T)
    BlackScholesCall = Exp(-d * T) * S * Application.NormSDist(d1) - X * Exp(-r * T) * Application.NormSDist(d2)
  End Function
 
  Function ImpliedVolatility( _
    ByVal S As Double, _
    ByVal X As Double, _
    ByVal T As Double, _
    ByVal r As Double, _
    ByVal d As Double, _
    ByVal Price As Double) As Double
 
    Dim epsilonABS As Double
    Dim epsilonSTEP As Double
    Dim volMid As Double
    Dim niter As Integer
    Dim volLower As Double
    Dim volUpper As Double
 
    epsilonABS = 0.0000001
    epsilonSTEP = 0.0000001
    niter = 0
    volLower = 0.001
    volUpper = 1
 
    Do While volUpper - volLower >= epsilonSTEP Or Abs(BlackScholesCall(S, X, T, r, d, volLower) - Price) >= epsilonABS And epsilonABS <= Abs(BlackScholesCall(S, X, T, r, d, volUpper) - Price) >= epsilonABS
      volMid = (volLower + volUpper) / 2
      If Abs(BlackScholesCall(S, X, T, r, d, volMid) - Price) <= epsilonABS Then
        Exit Do
      ElseIf ((BlackScholesCall(S, X, T, r, d, volLower) - Price) * (BlackScholesCall(S, X, T, r, d, volMid) - Price) < 0) Then
        volUpper = volMid
      Else
        volLower = volMid
      End If
      niter = niter + 1
    Loop
 
    ImpliedVolatility = volLower
 
  End Function

This spreadsheet uses the VBA given above.

Download Excel Spreadsheet to Calculate Implied Volatility with the Bisection Method