In an extension of the Kyle (1985) model of continuous insider trading, it is shown that asymmetric information can make it impossible to price options by arbitrage. Even when an option would appear to be redundant, its introduction into the market can cause the volatility of the underlying asset to become stochastic. This eliminates the potential for dynamically replicating the option. The change in the price process of the asset reflects a change in the information transmitted by volume and prices when the option is traded. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.