A new take on mathematical finance
Mathematical finance, which takes observed market prices as input, represents an important field of applied mathematics that supports financial markets. For mathematical finance to succeed, however, it requires advanced probability theory where a stochastic or random process tracks the evolution of a system of random values. Against this backdrop, the EU-funded project 'Optimal portfolios with random environments, frictions and incentives' (PORTFOLIO) explored a specific class of challenges in choice of portfolios that arise in mathematical finance. Specifically, it looked at how a common relevance across these problems can be relevant for applications, bringing forth new mathematical questions in the area of stochastic processes. To achieve its aims, the project team investigated dynamic portfolio choice with random environments, outlining a way to derive optimal portfolios and risk premiums. It proved static fund separation theorems for investors with a long-term horizon and constant relative risk aversion, and with stochastic investment opportunities. The team also proved three kinds of portfolio turnpikes, in addition to examining consumption in incomplete markets. The second part of the project looked at different trading frictions, such as asset pricing under transaction costs, and the dynamics between transaction costs, trading volume and liquidity premium. Topics also included high risk aversion, endogenous spreads and dynamic trading volume. Lastly, the project studied incentive fees, particularly performance maximisation of actively managed funds, as well as incentives of hedge fund fees and high-water marks. It also explored topics such as robust portfolios and weak incentives in long-run investments, in addition to hedge and mutual funds' fees and the separation of private investments. Several publications and/or papers have been published on these separate topics, with more publications on the way. These analyses and investigations are expected to yield novel mathematical tools that will refine the study of financial markets and ultimately help improve how they operate.
Keywords
Mathematical finance, advanced probability theory, financial market, market price, applied mathematics, random values, portfolios, random environment, incentives, stochastic process, risk premium, risk aversion, investment
