On option pricing when volatility is proportional to stock price

We study option pricing in local volatility model with volatility function proportional to stock price, which can be regarded as a special case of the CEV model. We use the properties of modified Bessel functions of the first kind with half-integer order and derive simplified forms of the transition probability density function and European call option price. We demonstrate that the considered pricing model produces fat-tailed distribution with tail index 3 and build implied volatility surface with skew effect.