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Market Frictions in Mathematical Finance

Final Report Summary - MFMF (Market Frictions in Mathematical Finance)

This project investigates the role of frictions in mathematical models of financial markets. It aims at i) developing new theoretical tools for optimization and equilibrium problems involving frictions, ii) solving such problems as explicitly as possible, seeking approximations when necessary, and iii) exploring the implications of these solutions for the problems considered. The project encompasses frictions that result either from the costs of trading, as in models of bid-ask spreads and price-impact, or from incentives contracts and, by extension, from distortions of preferences.
One stream of research investigates the role of transaction costs. We prove the Fundamental Theorem of Asset Pricing, characterizing the absence of arbitrage in terms of the existence of strictly consistent price systems, developing in the process an extension of the integral that is central to modeling transaction costs with discontinuous prices. More concretely, in a market with constant investment opportunities, we derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume for an investor with a long horizon and constant relative risk aversion. We fine a universal relation, whereby the liquidity premium equals the spread, times share turnover, times a universal constant. We also find that leverage can scale an asset's return only up to a maximum multiple, which is sensitive to the asset's volatility and liquidity. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines.
A second stream involves price impact and liquidity. At the theoretical level, we characterize superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. In concrete models, we derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a representative investor, with a long horizon and constant relative risk aversion, trades safe and risky assets. Trading volume approximately follows a mean-reverting diffusion, and increases with depth, volatility, and risk aversion.
The third stream encompasses incentive and preferences. We study the consequences of the high-water mark contracts that are prevalent in the hedge-fund industry, solving the portfolio choice problem from the viewpoints of managers and investors. We find that managers with constant relative risk aversion and constant investment opportunities, maximizing utility of fees at long horizons, choose constant Merton portfolios, with an effective risk aversion shrunk towards one in proportion to performance fees. Investigating the potential effect of managers’ private investments, we find that the fund's portfolio depends only on the fund's investment opportunities, and the private portfolio only on private opportunities.