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Explain deterministic model

Explain deterministic model.

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Deterministic models: The straightforward Black–Scholes formula assumes as volatility is constant or time dependent. Other than market data suggests as implied volatility varies along with strike price. That market behaviour cannot be consistent with a volatility which is a deterministic function of time. One way wherein the Black–Scholes world can be changed to accommodate strike-dependent implied volatility is to suppose that actual volatility is a function of time and the price both of the underlying. It is the deterministic volatility (surface) model. It is the simplest extension to the Black–Scholes world which can be made to be consistent along with market prices.
All it needs is that we have σ(S, t), and the Black–Scholes partial differential equation is even valid. The interpretation of an option’s value like the present value of the expected payoff within a risk-neutral random walk carries over also. Unfortunately the Black–Scholes closed-form formula is no longer correct. It is a simple and popular model, but this does not capture the dynamics of implied volatility thoroughly.

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