Why Implied Volatility Curves Help Interpolate Option Quotes

Summary

The document explains why option traders fit a curve to implied volatilities when pricing or comparing vanilla options. Market quotes are sparse across strike, maturity, and forward level, so a fitted curve lets a trader interpolate values at points where no quote is available. Working in implied volatility space can make this multidimensional interpolation more manageable than fitting prices directly, while Black–Scholes provides a convenient mapping back to prices.

A useful curve also needs to respect no-arbitrage features such as monotonicity and convexity in strike. The answers note that relatively simple parameterizations can work in some markets and that their parameters can have interpretable effects on skew and smile. Fitting is also useful for comparing standardized points over time or across markets, such as fixed moneyness or delta levels that may not have direct quotes. The discussion is conceptual: it gives no calibration procedure or evidence that one curve form is best, and the choice of model and extrapolation remains market-dependent.

Key ideas

• Fitted implied volatility curves interpolate option values across strikes and maturities where quotes are unavailable.
• Implied volatility space can simplify fitting across several option dimensions compared with price space.
• A useful fit should preserve no-arbitrage properties such as strike monotonicity and convexity.
• Curve parameters can summarize changes in the smile or skew.
• Interpolation enables comparisons at standardized moneyness or delta points even without direct quotes.

Tags

# What is the point of volatility curve fitting?


# What is the point of volatility curve fitting?












What is the point of fitting curves to the implied smile in the market? (Other than pricing exotics where the hedging instruments are vanillas). How does fitting a vol curve help you trade/market make vanillas? Isn't it rather tautological to use the market's implied vol as your theoretical value when trading?

## Answer by alexprice (score 4)

https://quant.stackexchange.com/a/55368

Pricing of vanillas is basically interpolation of existing (or past) quotes. It is easier to interpolate in implied volatility space , than in price space. Reasons are we need to interpolate in multidimensional space (maturity, strike,forward, etc) and satisfy non-arbitrage conditions.

Using Black-scholes formula is convenient mapping which would also simplify satisfying the non-arbitrage conditions (positive density (convexity) , monotonicity vs strike).

In price space parametrisations would get more complex than in implied volatility terms (you can get good match in say some FX markets with just simple quadratic implied vol interpolation in log(K/F)/sqrt(T) strike (with some simple wings extrapolation).

Also implied vol parametrisations have advantage that parameters usually have easily understood effect to implied vol smile/skew.

## Answer by nodesr (score 3)

https://quant.stackexchange.com/a/55362

If you want to compare quotes across markets or over time it can be useful to use fixed points: eg the 110%/90% points to compute skew or the +/-25 delta points for risk-reversal. You can't rely on quotes existing at exactly those points so you would want to interpolate.

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