This paper extends the Black-Scholes-Merton option pricing theory to markets with informed traders, addressing the discontinuity puzzle where option prices become discontinuous at extreme stock return means or probabilities. The authors propose that a trader with information on stock price direction (probability of upturn) or mean return can incorporate this information into option pricing. For a trader with directional information, the fair option price follows the Black-Scholes-Merton formula with an additional term interpreted as a continuous dividend yield, which depends on the trader's information intensity and the market price of risk. This resolves the discontinuity by smoothly connecting the noisy trader case to the fully informed trader case. For traders with information on the mean return, the option price differs from Black-Scholes-Merton only when continuous-time trading is not allowed. The paper also generalizes the model to continuous-diffusion processes. Using market option data, the authors estimate implied surfaces of the probability for a stock upturn, the implied mean stock return, and the implied trader information intensity. The methodology is based on dynamic asset pricing theory and the Kim-Stoyanov-Rachev-Fabozzi (KSRF) binomial option pricing model, which is extended to incorporate informed traders through optimal forward contracts. The results provide a theoretical foundation for option pricing under asymmetric information and offer empirical tools to infer market participants' information sets.