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Consider the stock price process satisfies the following SDE:

$dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $

and the mean return $\mu_t$ satisfies the following SDE:

$d\mu_t=(a-\mu_t)dt +dB_t, \mu_0=\mu$

where $W_t,B_t$ are two independent Brownian motions, and we consider the full-information case here.

Under the Black-Scholes model, we know that the stock return $\mu$ does not appear in the pricing formula. And my question is, if the stock return $\mu$ is not a constant, moreover if $\mu$ is a Ornstein–Uhlenbeck process.

How can we pricing the option? Is it similar to the Black-Scholes case?

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  • $\begingroup$ Yes, it is even exactly the same. But this is not really a research question. All what you have to do is to check that the integrability conditions in Girsanov's theorem are justified and this should be quite straight forward. $\endgroup$
    – Stephan Sturm
    Commented Nov 13, 2015 at 21:09
  • $\begingroup$ So, if the integrability conditions in Girsanov's theorem are justified, then the discounted stock price is a martingale, and using the Martingale pricing, we will get the same price with Black-Schole case? $\endgroup$
    – N.chan
    Commented Nov 14, 2015 at 2:56
  • $\begingroup$ Yes, exactly. That's what I meant. $\endgroup$
    – Stephan Sturm
    Commented Nov 14, 2015 at 3:18

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