Choosing an Initial Guess for Newton Implied Volatility

Summary

The document addresses how to choose a starting volatility when using Newton’s method to infer the implied volatility of a European vanilla call. The answer explains that a call’s theoretical price rises monotonically with volatility, so the price error as a function of volatility has a single crossing under ordinary conditions. It argues that many positive starting values can converge quickly and that the supplied starting formula is a convenient choice associated with setting d-minus to zero.

A second starting formula can instead set d-plus to zero, and the answer says other values may also work. This is a qualitative explanation rather than a convergence proof or a comparison across market conditions. It does not discuss safeguards for edge cases, such as very low vega, invalid option prices, or iterations that leave the positive-volatility domain, so practical implementations may need additional controls.

Key ideas

• A European call’s Black-Scholes price increases with implied volatility.
• Newton iteration seeks the volatility at which model price matches the observed option price.
• The stated initial guess has an interpretation through the d-minus term, while another can be chosen through d-plus.
• The answer gives no general convergence guarantee or implementation safeguards for difficult cases.

Tags

# Newton's Algorithm for Implied Volatility


# Newton's Algorithm for Implied Volatility












I was studying the implied volatility for European Vanilla Call option. My notes said that we can apply Newton's algorithm to calculate implied volatility numerically. I understand how the algorithm works and the updating part is straightforward. However, I am confused by the initial guess of $\sigma$: $$ \sigma_0 = \sqrt{\frac{2\log(S_te^{r(T-t)}/K)}{T}}. $$ I don't understand why I have to choose the initial guess like this. Does a random guessed number affect the convergence of the algorithm?

## Answer by StackG (score 8)

https://quant.stackexchange.com/a/58636

For a vanilla call option, the price of the option increases monotonically with implied volatility. For functions like this, newton's method works really nicely, and it's not very sensitive to the choice of starting parameter

I've borrowed an image from this webpage, detailing the technique:

If you think of the red line as being the price of your option minus the observed market price against the implied vol, you'll see that no matter which initial guess you choose (as long as it's above $0$), you'll home in to the true value very quickly.

Given this, your initial guess above corresponds to $d_- = 0$, but you could just as easily choose an initial guess that makes $d_+ = 0$: \begin{align} \sigma_0 = \sqrt{{\frac {2 \log{{\frac K {S_t e^{r(T-t)}}}}} {T}}} \end{align} or almost any other value...

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.