In mathematical finance, models for pricing derivative instruments such as options are most frequently formulated in terms of continuous-time dynamics of the underlying stock price, in order to allow use of machinery such as Ito calculus and martingale theory or PDEs. Since closed-form solutions are frequently unavailable, the study of the convergence of objects (such as contingent claims, their prices, the corresponding hedging strategies or gains processes) defined in approximating discrete-time models is a matter of some interest.

The simplest example is the convergence of a suitably chosen sequence of Cox-Ross-Rubinstein (CRR) pricing formulae to the famous Black-Scholes formula based on geometric Brownian mtion. A new and simpler proof of this result will be given.

For the convergence of more complex objects, a mode of convergence stronger than weak convergence which remains stable under the operations of the stochastic calculus is described. This leads to applications in other contexts, such as convergence of the critical prices for the American put option and the minimal martingale measures defined in incomplete pricing models. While the formulation of the results is couched in 'standard' terminology, the only currently available proofs of these results use the ideas of nonstandard analysis, i.e the theory of infinitesimals.