Assume that the time dependent volatility function (t) is deterministic. Suppose we write imp (t, T)

Assume that the time dependent volatility function σ(t) is deterministic. Suppose we write σ_imp (t, T) as the implied volatility obtained from the time-t price of a European option with maturity T , for T > t. Show that

In real situations, we may have the implied volatility available only at discrete times T_i,i = 1, 2, ··· ,N. Assuming the volatility σ(T) to be piecewise constant over each time interval [T_i−1,T_i],i = 1, 2, ··· ,N, show that

The implied volatility σ_imp(t, T ) and the time dependent volatility function σ(t) are related by

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o (T) = 2 ə omp(t, T) +2(T − 1)¤¡mp(t, T); Timp (t, T).