Local Versus Stochastic Volatility Models

Summary

The note contrasts two ways to represent volatility when modeling option prices. A local volatility model makes volatility a deterministic function of the underlying asset level and time. A stochastic volatility model instead allows volatility to evolve randomly, potentially with mean-reverting behavior. Both approaches are presented as ways to fit an observed implied-volatility surface.

The response gives a qualitative account of their reported strengths: local volatility is said to fit short-maturity options well but to have difficulty with longer maturities, while stochastic volatility is described as more effective for longer-term surfaces but less able to reproduce short-term skew or smile. These are broad claims from a brief explanation, not universal model properties or empirical results; the document offers no equations, calibration details, or supporting evidence to establish when the trade-offs hold.

Key ideas

• Local volatility is modeled as a deterministic function of the underlying price and time.
• Stochastic volatility allows volatility to vary randomly and may include mean reversion.
• Both model classes can be calibrated to an implied-volatility surface.
• The note claims different short- and long-maturity fitting strengths but provides no derivation or empirical support.

Tags

# Local volatility and Stochastic Volatility


# Local volatility and Stochastic Volatility












Please help me understand similarity and differences between local volatility and Stochastic Volatility both intuitively and mathematically.

## Answer by Crushh (score 1)

https://quant.stackexchange.com/a/49083

I'll try my best to explain them

Both of them aim to match the implied volatility surface as shown by the empirical data.

Local volatility process is a function of Stock and time without any stochastic term (not moving randomly). It changes with with different inputs of stock and time. It matches the implied volatility surface with short term maturity very well, but not well with option with long maturity.

Stochastic volatility is simply volatility that varies stochastically (it moves randomly). It can have mean reverting properties. It matches long term implied volatility surface well, but unable to replicate short term implied volatility skew or smile

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