I am playing around with implied tree model using backward induction methodolgy to calibrate stock prices with option prices given. The implied risk-neutral probability density functions can be calculated from option prices. The theory is taken from Breeden and Litezenberger (1978): get the second derivative of the call function (call prices vs their strike prices, not vs the stock price)The problem i confront now is that there is no stochastic volatility incorporated in this model. And therefore, the stock prices are only calculated from observed option prices and strike prices. Is it reasonable? BTW: the stochastic volatility can be also computed only by option prices and strike price with appropirate risk-free interest rate discounted.

You're right; implied tree models traditionally correspond to deterministic local vol, not stochastic vol. You could try a two-dimensional implied tree of vol and spot moves, but that is probably more trouble than it's worth given other ways you could treat stochastic vol.I'd really like to know more about your technique for computing (assuming that doesn't mean "calibrating") stochastic vol from only option prices, strikes, and "the appropriate" risk-free rate; every SV model I know involves estimating the parameters by fitting their implied prices to observed vanilla prices, much like a non-linear regression with a fair amount of error.

Thanks for your reply. You are right the implied tree model usually corresponding to only local volatility. Derman and Kani (1998) extended the restrict assumption of deterministic volatility and develop a pricing model with stochastic volatility that fits current option prices. There is a good paper published by Britten-Jones and Neuberger(2000) on model-free volatility calibrated from given option prices. I am here trying to implement their option pricing model. But the model in attached file lacks the volatility. Let me figure out what kind of file can be attached here and how you the model