Merton’s equation and the quantum oscillator II: option pricing

Merton has proposed a model for contingent claims on a firm as an option on the firms value, and is based on a generalization of the Black-Scholes stochastic equation (Merton, 1974). A special case of Merton’s model is proposed – based on the quantum oscillator – for pricing options. Two cases of the option price are obtained: both these cases yield possible candidates for the generalization of the Black-Scholes option pricing formula. However, one of the proposed option prices does not obey the martingale condition and the other does not yield the correct discounting of future cash flows. For these reasons, the option prices do not obey put-call parity. The options can, however, be used to approximately price market traded options. The oscillator model for the option price has an extra parameter that is absent for the Black-Scholes case. Similar to the model studied by Baaquie et al. (2014), which that does not obey put-call parity, the option’s price can be studied empirically and the extra parameter in the model could, in principal, generate implied volatility.