Hedging equity portfolios against the risk of large drawdowns is notoriously difficult and expensive. Holding, and continuously rolling, at-the-money put options on the S&P 500 is a very costly, if reliable, strategy to protect against market sell-offs. Holding ‘safe-haven' US Treasury bonds,...

We study the econometric properties of dynamic risk parity, which volatility scales to equalise risk through time using the precision process, the inverse of the time-varying volatility. A particular focus is on the impact of the Sharpe ratio. We give necessary and suffcient conditions that...

In the late stages of long bull markets, a popular question arises: What steps can an investor take to mitigate the impact of the inevitable large equity correction? However, hedging equity portfolios is notoriously difficult and expensive. We analyze the performance of different tools that...

Recent studies show that volatility-managed equity portfolios realize higher Sharpe ratios than portfolios with a constant notional exposure. We show that this result only holds for “risk assets”, such as equity and credit, and link this to the so-called leverage effect for those assets. In...

We model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the asset is driven by Brownian motion, an associated "master...

We consider a full equilibrium model in continuous time comprising a finite number of agents and tradable securities.We show that, if the agents' endowments are spanned by the securities and if the agents have entropic utilities, an equilibrium exists and the agents' optimal trading strategies...

A heat kernel approach is proposed for the development of a flexible and mathematically tractable asset pricing framework in finite time. The pricing kernel, giving rise to the price system in an incomplete market, is modelled by weighted heat kernels which are driven by multivariate Markov...