Three essays on stopping

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic m/s2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatovi´c and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017)

Leveraged funds: robust replication and performance evaluation

Leveraged and inverse exchange-traded funds seek daily returns equal to fixed multiples of indexes’ returns, but the ensuing rebalancing costs create a tension between a high correlation with the index and a low average deviation from the leveraged index’ performance. With proportional trading costs, we find that the optimal replication policy is robust to the index’ dynamics and obtain a sufficient statistic for index replication performance, the implied spread, which is insensitive to risk-premia and enables comparisons of funds tracking different factors of an index. Overall, the impact of trading costs on replication performance is comparable to or higher than the effect of management fees </p

The limits of leverage

When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long‐term returns given average volatility. As leverage increases, rising rebalancing costs imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding the Sharpe ratio constant, higher asset volatility leads to superior returns through lower costs

Rogue traders

Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons and higher volatility all lead to honesty on a wider range of capital allocations between the traders. </p

Informational efficiency and welfare

In a continuous-time market with a safe rate and a risky asset that pays a dividend stream depending on a latent state of the economy, several agents make consumption and investment decisions based on public information–prices and dividends–and private signals. If eachinvestor has constant absolute risk aversion, equilibrium prices do not reveal all the privatesignals, but lead to the same estimate of the state of the economy that one would hypothetically obtain from the knowledge of all private signals. Accurate information leads to low volatility, ostensibly improving market efficiency, but also reduces each agent’s consumption through a decrease in the price of risk. Thus, informational efficiency is reached at the expense of agents’ welfare.</p