29 research outputs found
Optimal Investment with Transaction Costs and Stochastic Volatility
Two major financial market complexities are transaction costs and uncertain
volatility, and we analyze their joint impact on the problem of portfolio
optimization. When volatility is constant, the transaction costs optimal
investment problem has a long history, especially in the use of asymptotic
approximations when the cost is small. Under stochastic volatility, but with no
transaction costs, the Merton problem under general utility functions can also
be analyzed with asymptotic methods. Here, we look at the long-run growth rate
problem when both complexities are present, using separation of time scales
approximations. This leads to perturbation analysis of an eigenvalue problem.
We find the first term in the asymptotic expansion in the time scale parameter,
of the optimal long-term growth rate, and of the optimal strategy, for fixed
small transaction costs.Comment: 27 pages, 4 figure
Asymptotic Analysis for Optimal Investment in Finite Time with Transaction Costs
We consider an agent who invests in a stock and a money market account with
the goal of maximizing the utility of his investment at the final time T in the
presence of a proportional transaction cost. The utility function considered is
power utility. We provide a heuristic and a rigorous derivation of the
asymptotic expansion of the value function in powers of transaction cost
parameter. We also obtain a "nearly optimal" strategy, whose utility
asymptotically matches the leading terms in the value function.Comment: 22 page
Axioms for Automated Market Makers: A Mathematical Framework in FinTech and Decentralized Finance
Within this work we consider an axiomatic framework for Automated Market
Makers (AMMs). By imposing reasonable axioms on the underlying utility
function, we are able to characterize the properties of the swap size of the
assets and of the resulting pricing oracle. We have analyzed many existing AMMs
and shown that the vast majority of them satisfy our axioms. We have also
considered the question of fees and divergence loss. In doing so, we have
proposed a new fee structure so as to make the AMM indifferent to transaction
splitting. Finally, we have proposed a novel AMM that has nice analytical
properties and provides a large range over which there is no divergence loss
Optimal Investment with Correlated Stochastic Volatility Factors
The problem of portfolio allocation in the context of stocks evolving in
random environments, that is with volatility and returns depending on random
factors, has attracted a lot of attention. The problem of maximizing a power
utility at a terminal time with only one random factor can be linearized thanks
to a classical distortion transformation. In the present paper, we address the
problem with several factors using a perturbation technique around the case
where these factors are perfectly correlated reducing the problem to the case
with a single factor. We illustrate our result with a particular model for
which we have explicit formulas. A rigorous accuracy result is also derived
using a verification result for the HJB equation involved. In order to keep the
notations as explicit as possible, we treat the case with one stock and two
factors and we describe an extension to the case with two stocks and two
factors