This paper investigates the pricing of options written on non-traded assets and trading strategies for the stock and option in an exponential utility maximization framework.Under the assumption that the option can be continuously traded without friction just as the stock,a dynamic relationship between their optimal positions is derived by using the stochastic dynamic programming techniques.The dynamic option pricing equations are also established.In particular,the properties of the associated solutions are discussed and their explicit representations are demonstrated via the Feynman-Kac formula.This paper further compares the dynamic option price to the existing price notions,such as the marginal price and indifference price. This paper investigates the pricing of options written on non-traded assets and trading strategies for the stock and option in an exponential utility maximization framework. Under the assumption that the option can be continuously traded without friction just as the stock, a dynamic relationship between optimal positions is derived by using the stochastic dynamic programming techniques. The dynamic option pricing equations are also established. In particular, the properties of the associated solutions are discussed and their explicit representations are made via the Feynman-Kac formula. This paper further compares the dynamic option price to the existing price notions, such as the marginal price and indifference price.