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What is the difference between implied volatility and local volatility?
Implied volatility is a parameter used to back out option price by substitting the volatility parameter in the standard Black Scholes model. While local volatility actually governs the dynamics of how the stock price will evolve. Hence if an exotic option is to be priced using Monte carlo simulation we use volatility values from the local volatility surface and not from the implied volatility surface. The relationship between local and implied vol. is provided by Dupire formula.
