# Black-Scholes formula with non-constant volatility (function of time)

Let's have the following stochastic process:

$$dS_t = r S_t dt + σ(t) St dW_t$$

where $W_t$ is the Brownian motion, r the drift and $σ(t)$ the volatility, a deterministic function of the time.

Applying Ito's lemma, I have reached that :

$$S_t = S_0 e^{rt − \frac{1}{2} \int_0^t \sigma (s)^2 ds +\int_0^t \sigma (s) dW_s}$$

Now, I have to obtain the price of a call option, that is :

$$e^{-r(T-t)}\mathbb{E}^*(S_T-k)_+|S_t=x)$$

I have tried doing it directly, but I think it can be done using Black-Scholes formula. Any hint? Thanks! :)

Define $\bar \sigma =\sqrt{\frac{1}{T}\int_0^T\sigma(t)^2dt}$ and plug it into the B-S formula. The idea is that, as long as you have the same distribution of S at T, you should get the same (European) option price.
• Thanks @Jay.H. However I do not get to understand that beeing $S_t$ log-normal, I can use the same way that with constant volatility to get the answer. – Edin_91 Jan 16 '16 at 17:26