The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black-Scholes Model

Takaoka (Asia-Pacific Financial Markets 11:431-444, 2004) proposed a generalization of the Black-Scholes stock price model by taking a weighted average of geometric Brownian motions of different variance parameters. The model can be classified as a local volatility model, though its local volatility function is not explicitly given. In the present paper, we prove some properties concerning the instantaneous volatility process, the implied volatility curve, and the local volatility function of the generalized model. Some numerical computations are also carried out to confirm our results.