The Black-Scholes model is an elegant model

It is well known that stock prices jump on occasions and do not always move in the smooth manner predicted by the GBM motion model.

The volatility surface is a function of strike, K, and time-to-maturity, T, and is defined implicitly

C(S, 0) := BS (S, T, r, q, K, σ(K, T))

where C(S, K, T) denotes the current market price of a call option with time-to-maturity T and strike K, and BS(·) is the Black-Scholes formula for pricing a call option.

In other words, σ(K, T) is the volatility that, when substituted into the Black-Scholes formula, gives the market price, C(S, K, T). Because the Black-Scholes formula is continuous and increasing in σ, there will always4 be a unique solution, σ(K, T). If the Black-Scholes model were correct then the volatility surface would be flat with σ(K, T) = σ for all K and T. In practice, however, not only is the volatility surface not flat but it actually varies, often significantly, with time.

In general, we often observe an inverted volatility surface with short-term options having much higher volatilities than longer-term options. This is particularly true in times of market stress.