Improving the Design of Financial Products in a Multidimensional Black-Scholes Market

Abstract

Using various techniques, Cox and Leland (1982,2000), Dybvig (1988a, 1988b), Vanduffel et al. (2009) and Bernard and Boyle (2010) have shown that in onedimensional markets, complex (path-dependent) contracts are generally not optimal for rational consumers.

In this paper, we generalise these results to a multidimensional Black-Scholes market. In such a market, we discuss optimal contracts for investors who prefer more to less and have a fixed investment horizon T>0. First, given a desired probability distribution, we give an explicit form of the optimal contract that provides this distribution to the consumer. Second, in the case of risk averse investors, we are able to propose two ways of improving the design of financial products. In all cases, the optimal payoff can be seen as a path-independent European option that is written on the so-called "market portfolio".

We illustrate the theory with a few well-known securities and strategies. For example we show that a buy-and-hold investment strategy can be dominated by a series of power options written on the underlying market portfolio. We also analyse the inefficiency of a widely used portfolio insurance strategy called Constant Proportion Portfolio Insurance.

Keywords: cost-efficiency, optimal design, Black-Scholes