We analyze continuous approximations of discrete choice models with a large number of options. We start with a discrete choice model where agents choose between different options, and where each option is defined by a characteristic vector and a utility level. For each option, the characteristic vector and the utility level are random and jointly dependent. We analyze the optimal choice, which we define as the characteristic vector of the option with the highest utility level. This optimal choice is a random variable. The continuous approximation of the discrete choice model is the distributional limit of this random variable as the number of offers tends to infinity. We use point process theory and extreme value theory to derive an analytic expression for the continuous approximation, and show that this can be done for a range of distributional assumptions. We illustrate the theory by applying it to commuting data. We also extend the initial results by showing how the theory works when characteristics belong to an infinite-dimensional space, and by proposing a setup which allows us to further relax our distributional assumptions.