Implied Vol Newton Step

Problem

Take one Newton step toward the implied volatility of a European call.

Compute the Black-Scholes model price at the current sigma, the vega S * phi(d1) * sqrt(T), and return sigma - (model - price) / vega. Return sigma unchanged when vega is zero.

Hint

Vega is largest near the money, which makes the Newton step well behaved there.

Test cases

case 1

in: [0.2,10.45,100,100,0.05,1]

expected: 0.19998444804269158

case 2

in: [0.25,5,100,100,0.03,0.5]

expected: 0.1506720363707626

case 3

in: [0.3,15,100,90,0.05,1]

expected: 0.15283331733069402

Starter codePython

def implied_vol_newton_step(sigma, price, S, K, r, T):
    # Your code here
    pass

3 test cases · run your code against them in the editor