Option pricing with a quadratic diffusion term

Several authors have derived closed-form option prices in models where the underlying financial variable follows a diffusion process with the following two characteristics: (i) the process has natural upper and lower boundaries; (ii) its diffusion coefficient is quadratic in the current value of the variable. The present paper uses a probabilistic change-of-numeraire technique to compute the corresponding option price formula. In particular, it shows how to interpret the formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios.