Abstract
A function which transforms a continuous random variable such that it has a specified distribution is called a replicating function. We suppose that functions may be assigned a price, and study an optimization problem in which the cheapest approximation to a replicating function is sought. Under suitable regularity conditions, including a bound on the entropy of the set of candidate approximations, we show that the optimal approximation 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, the optimal approximation corresponds to the cheapest portfolio of market index options which delivers the hedge fund return distribution.
Highlights
A key difference between the approach to distributional replication proposed in this paper, and the approach taken by Amin and Kat [6], is that we do not assume that the cheapest replicating function is nondecreasing
We construct a pseudometric on the set of Borel measurable functions mapping the support of one random variable to the support of another, and we define a criterion function that identifies the set of replicating functions
The iid condition comes into play in the proof of Proposition 6, in which results in empirical process theory are used to establish a uniform bound on the error in the approximation of M by Mn over our sieve space Θn
Summary
A key difference between the approach to distributional replication proposed in this paper, and the approach taken by Amin and Kat [6], is that we do not assume that the cheapest replicating function is nondecreasing. Amin and Kat propose to implement the desired payoff function θ by engaging in a continuous time hedging strategy, trading market shares and cash. They introduce a “reserve asset” with payoff Z, and seek to find a bivariate function θ such that the joint distribution of θ ( X, Z ) and X is the same as the joint distribution of Y and X This replicating payoff function is implemented in practice using a continuously rebalanced portfolio formed by trading market shares, cash, and the reserve asset. Proofs of all numbered propositions may be found in Appendix A
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