Department of Economics, UCSD

Suppose that X and Y are random variables. We define a replicating function to be a function f such that f(X) and Y have the same distribution. In general, the set of replicating functions for a given pair of random variables may be infinite. Suppose we have some objective function, or cost function, defined over the set of replicating functions, and we seek to estimate the replicating function with the lowest cost. We develop an approach to estimating the cheapest replicating function that involves minimizing the cost function over an estimate of the set of replicating functions. Our estimated set of replicating functions is obtained by considering the functions f in some sieve space for which the empirical distributions of f(X) and Y are close. Under suitable conditions, we show that our estimated function comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, X is the market return, Y is the return from a hedge fund or other asset, and our estimation procedure amounts to choosing the cheapest portfolio of options on X such that the returns from our portfolio have the same distribution as the hedge fund returns.