2
$\begingroup$

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! :)

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Edin_91 Jan 16 '16 at 17:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.