Implied Volatility, Interpolation, and Realized Volatility
Summary
The document separates three tasks that are easy to conflate when constructing an option volatility term structure. Implied volatility is inferred by finding the volatility input at which an option-pricing model, such as Black–Scholes, matches the traded option price. Newton–Raphson is one numerical method for finding that input by iteratively reducing the pricing error; it is not itself a method for filling missing maturities or strikes.
Interpolation comes afterward, using implied volatilities calculated from traded options to estimate values where quotes are unavailable. The response notes that the resulting surface should avoid arbitrage, but it does not specify an interpolation method or arbitrage checks. It contrasts model-dependent implied volatility with actual volatility, which cannot be known in advance and is estimated statistically, for example from return variability. The explanation is introductory: it does not detail local volatility models, estimation windows, or how realized volatility depends on the sampling choices.
Key ideas
- Newton–Raphson can iteratively find the volatility input that aligns a model price with a traded option price.
- Implied volatility is the model-dependent volatility that matches an observed option price.
- Interpolation estimates volatility values between quoted options after their implied volatilities are calculated.
- An interpolated volatility surface should be constructed to avoid arbitrage.
- Actual or realized volatility is estimated from returns rather than directly known in advance.
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Full text
# Implied Volatility vs Actual Volatility Calculation # Implied Volatility vs Actual Volatility Calculation To build a term structure I need different volatilities; as I don't get them at every strike, I use interpolation technique to calculate the rest and plot. This is how I calculate the implied vols. How is the Newton-Raphson method used to calculate the implied vols? Is it another way to calculate like I calculate with interpolation techniques? Is my understanding correct? Now what ae actual vols and how are they calculated? Edit: Basically I want to understand how the Newton-Raphson method is used to calculate to implied volatility and what is the difference between the Newton-Raphson method and interpolation methods? What is the difference between implied volatility and actual/local volatility? ## Answer by Dhruv Mahajan (score 0, accepted) https://quant.stackexchange.com/a/57879 Newton-Raphson is not an implied volatility calculation method, it's just a way to minimize (above a certain threshold) the difference between traded options prices and BS prices, the volatility at which this minimization happens is called implied volatility. You cannot have said options for all maturities trading in the market at the time of calculation, so you need to create an interpolation curve. E.g you may have an option with 1m to maturity and 3m to maturity but not 2m, you'll calculate the iv for 1m and 3m and try to interpolate between these values (linear or otherwise). The interpolation curve should be such that the implied vol surface is arbitrage-free. As for actual volatility, we don't know what actually volatility is, we can estimate it for example by standard deviation of return, as for the difference between the two, IV is dependent upon the valuation model i.e. Black Scholes while the other is just a statistical estimate. IV is also widely used to quote options, instead of prices.
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