Optimal hedging of Derivatives with transaction costs

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black-Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages, submitted to International Journal of Theoretical and Applied Finance (IJTAF

The Markowitz Category

We give an algebraic definition of a Markowitz market and classify markets up to isomorphism. Given this classification, the theory of portfolio optimization in Markowitz markets without short selling constraints becomes trivial. Conversely, this classification shows that, up to isomorphism, there is little that can be said about a Markowitz market that is not already detected by the theory of portfolio optimization. In particular, if one seeks to develop a simplified low-dimensional model of a large financial market using mean--variance analysis alone, the resulting model can be at most two-dimensional.Comment: 1 figur

Bond Market Completeness and Attainable Contingent Claims

A general class, introduced in [Ekeland et al. 2003], of continuous time bond markets driven by a standard cylindrical Brownian motion \wienerq{}{} in $\ell^{2},$ is considered. We prove that there always exist non-hedgeable random variables in the space \derprod{}{0}=\cap_{p \geq 1}L^{p} and that \derprod{}{0} has a dense subset of attainable elements, if the volatility operator is non-degenerated a.e. Such results were proved in [Bj\"ork et al. 1997] in the case of a bond market driven by finite dimensional B.m. and marked point processes. We define certain smaller spaces \derprod{}{s}, $s>0$ of European contingent claims, by requiring that the integrand in the martingale representation, with respect to \wienerq{}{}, takes values in weighted $\ell^{2}$ spaces $\ell^{s,2},$ with a power weight of degree $s.$ For all $s > 0,$ the space \derprod{}{s} is dense in \derprod{}{0} and is independent of the particular bond price and volatility operator processes. A simple condition in terms of $\ell^{s,2}$ norms is given on the volatility operator processes, which implies if satisfied, that every element in \derprod{}{s} is attainable. In this context a related problem of optimal portfolios of zero coupon bonds is solved for general utility functions and volatility operator processes, provided that the $\ell^{2}$-valued market price of risk process has certain Malliavin differentiability properties.Comment: 27 pages, Revised version to be published in Finance and Stochastic