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Explain the deterministic volatility in an option-pricing

Explain the deterministic volatility in an option-pricing.

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The deterministic volatility surfaces is the concept that volatility is not constant, or though, only a function of time, but an identified function of stock price and time, σ(S, t). Now there the word ‘known’ is a bit misleading. What we really understand are the market prices of vanilla options, a snapshot at one immediate in time. Now we should figure out the correct function σ(S, t) that the theoretical value of our options matches the market prices. It is mathematically an inverse problem; essentially get the parameter, knowing several solutions, volatility and market prices. That model may capture the volatility surface exactly at an instant in time, but this does an extremely poor job of capturing the dynamics, which is, how the data change along with time.

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