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