Like most academics, I have given many talks over the years, the majority of which are based on some finished paper. But sometimes the topic of the talk is more exploratory, and for whatever reason there is no related paper. Nevertheless, in a few cases the topic is of sufficient interest to post it here.
- A Bayesian solution to the equity premium puzzle, with Arnaud Jobert and Alessandro Platania. The equity premium puzzle dates back to a celebrated 1985 paper of Mehra & Prescott, who pointed out that annualized returns on US stocks from the late 19th century were too high relative to riskless returns to be explained by conventional von Neumann-Morgenstern agents with reasonable values of relative risk aversion. The economics profession has proposed many possible resolutions of this apparent paradox, and continues to do so. There will never be an accepted answer to the question, because too many economists have invested too much of their time in their own pet explanations, and because the data set is too small to permit any strong conclusion. This was our attempt to explain the paradox, probably no better nor worse than many others; but the pain of trying to argue this past some bigoted referees dissuaded us from tidying it up and submitting it.
- Perpetual defaultable callable convertible bonds, with Jon Heritage and Gunther Leobacher. This study was begun in Bath with PhD student Jon Heritage, as a simple topic for a final chapter of his thesis, which I envisaged would take him no more than a couple of months to complete. Though he made good progress on the question, it resisted our efforts, so when Gunther came to Cambridge to work with me for a year, we attacked it from every direction we could think of, yet a satisfactory conclusion eluded us. Numerics were too unstable, theory spawned too many cases to be sure that we had covered everything. In the end, after trying off and on for at least seven years, I abandoned the project – it had cost too much time. But maybe you will have better success …
- Reflections on modelling, arbitrage and equilibrium. This study of the foundations of mathematical finance was presented at the Festkolloquim for Hans Follmer in Berlin, June 2007. You can see how far it got – quite far – but how far there remains to go – moving from a representative agent to a multi-agent equilibrium.
- Diverse beliefs, with Angus Brown. This actually did get published, but the result presented in this talk is hidden away in the text of the paper among other results, and it deserves to be better known. In studies of equilibrium, it is often assumed that all agents have common knowledge about the randomness in the model, and all see the same things. Now this is quite unrealistic, and there are two main approaches to relaxing this restrictive assumption. One proposes private information (PI models), where different agents see common market variables, but also receive private information about those variables. The other proposes that agents have diverse beliefs (DB models), where all agents see the same random processes, but have different views about the probability laws governing their evolution. PI models are very hard to work with, and outside of a linear multivariate gaussian setting it is essentially impossible to do anything – and even within such a stylized setting it is hard to get hold of results that can be sensibly interpreted. DB models are much easier to work with, because everyone has the same filtration, just different probability measures. The conclusion of this study is that in finite-horizon discrete-time Lucas tree economies (quite a general class of examples), any equilibrium which came from a PI model could have been generated by a DB model. This is important because it says that (difficult) PI models are observationally indistinguishable from (easy) DB models, so we may as well make our lives easier and only work with DB models. But again, too many economists have invested too much intellectual capital in PI models for them to disappear rapidly – as Samuelson said, `Science advances funeral by funeral’.