Numerical Methods for Calculating Implied Volatility from Call Prices
Summary
The document explains how to infer implied volatility from a call option’s market price. It gives an example with a stock price, strike, option price, time to expiry, and interest rate, for which the stated volatility is 30%. The central point is that implied volatility is obtained by finding the volatility input that makes a pricing model reproduce the observed option price.
For Black–Scholes implied volatility, the pricing equation has no closed-form solution for volatility, so a numerical root finder is needed. The responses name bisection, Newton–Raphson, the secant method, and Brent’s method. The choice involves tradeoffs such as speed and numerical stability, and the method must be implemented around a selected pricing model; other models would produce different model-implied volatilities. The discussion does not provide a complete implementation or compare the methods on the example, and its reference to a 30% result is not accompanied by calculation details.
Key ideas
- Implied volatility is found by matching a model option price to the observed market price.
- Black–Scholes implied volatility has no analytical closed-form solution.
- Root-finding approaches include bisection, Newton–Raphson, secant, and Brent methods.
- Method selection involves numerical stability and speed, while the pricing model defines which implied volatility is being calculated.
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Full text
# Calculate volatility from call option price # Calculate volatility from call option price Given call option price, what is the simplest formula to get the volatility value ? Test Data: ``` Stock price : $60 Option strike: $65 Call option price: $1.766 Duration (in year) : 0.25 (equivalent to 63 trading days) Interest rate: 0.25% ``` Result: ``` Volatility: 30% ``` I am especially looking for a formula that can be programmed into C#. ## Answer by delta hedge (score 4, accepted) https://quant.stackexchange.com/a/10635 Implied volatility cannot be calculated analytically with a closed formula. Instead, you have to approximate it numerically. There are multiple methods to compute IV on an option: Bi-section method Newton-Raphson method Secant method A quick google search came up with the following code for C++ using bi-section and newton methods: Implied volatility calculations in C++ ## Answer by Probilitator (score 1) https://quant.stackexchange.com/a/10639 Theory: First of all you must decide which implied volatility you want. Probably you are looking for the Black and Scholes implied vol. (but one could also caclute Heston implied vols etc) If we are in a B&S setting one desires to retrieve the implied vol by solving the B&S pricing equation for $\sigma$. Unfortunately there is no analytical solution and you will have to use a numerical root finder. Delta hedge has already provided some frequently used algorithms. I will also mention Brent's Method here. You will have to decide which one you like best. In general there are two main criteria you should consider - numerical stability - speed (for more see here) Implementation in C# - for most algorithms you will find pseudocode online which can be easily translated into C# (see e.g. the one for Newton's Method). Every root finder can be coded up in C#. If you are in a hurry just google "root finder c#" and this will give you some already implemented methods.
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