Sense, nonsense and the S&P 500

The theory of financial markets is well developed, but before any of it can be applied there are statistical questions to be answered: Are the hypotheses of proposed models reasonably consistent with what data shows? If so, how should we infer parameter values from data? How do we quantify the error in our conclusions? This paper examines these questions in the context of the two main areas of quantitative finance, portfolio selection and derivative pricing. By looking at these two contexts, we get a very clear understanding of the viability of the two main statistical paradigms, classical (frequentist) statistics, and Bayesian statistics

Optimal investment: bounds and heuristics

This is the author accepted manuscript. The final version is available from Incisive Financial Publishing via http://www.risk.net/journal-of-computational-finance/technical-paper/2432431/optimal-investment-bounds-and-heuristicsHigh-dimensional optimal investment/consumption problems are hard to deal with, not least because of the difficulty in characterizing the value function. This paper tries to offer ways to determine an approximately optimal policy, and to estimate its performance using duality methods. Though the value function is required as a concept in developing the theory, it plays no part in the computation, nor is it necessary to have global knowledge of the policy; it is enough to determine the policy along the realized sample path

INVESTING AND STOPPING

In this paper we solve the hedge fund manager's optimization problem in a model that allows for investors to enter and leave the fund over time depending on its performance. The manager's payoff at the end of the year will then depend not just on the terminal value of the fund level, but also on the lowest and the highest value reached over that time. We establish equivalence to an optimal stopping problem for Brownian motion; by approximating this problem with the corresponding optimal stopping problem for a random walk we are led to a simple and efficient numerical scheme to find the solution, which we then illustrate with some examples.This is the author accepted manuscript. The final version is available from the Applied Probability Trust via http://projecteuclid.org/euclid.jap/142176331

Trading to Stops

The use of trading stops is a common practice in financial markets for a variety of reasons: it reduces the frequency of trading and thereby transaction costs; it provides a simple way to control losses on a given trade, while also ensuring that profit-taking is not deferred indefinitely; and it allows opportunities to consider reallocating resources to other investments. In this paper, we try to explain why the use of stops may be desirable, by proposing a simple objective to be optimized. We investigate a number of commonly used rules for the placing and use of stops, either fixed or moving, with fixed costs, showing how to identify optimal levels at which to set stops, and compare the performance of different rules and strategies.This is the final published version of the paper. First Published in Siam Journal on Financial Mathematics in 2014, published by the Society of Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited

Estimate nothing

In the econometrics of financial time series, it is customary to take some parametric model for the data, and then estimate the parameters from historical data. This approach suffers from several problems. Firstly, how is estimation error to be quantified, and then taken into account when making statements about the future behaviour of the observed time series? Secondly, decisions may be taken today committing to future actions over some quite long horizon, as in the trading of derivatives; if the model is re-estimated at some intermediate time, our earlier decisions would need to be revised - but the derivative has already been traded at the earlier price. Thirdly, the exact form of the parametric model to be used is generally taken as given at the outset; other competitor models might possibly work better in some circumstances, but the methodology does not allow them to be factored into the inference. What we propose here is a very simple (Bayesian) alternative approach to inference and action in financial econometrics which deals decisively with all these issues. The key feature is that nothing is being estimated.This is the author accepted manuscript. The final version is available from Taylor & Francis via http://dx.doi.org/10.1080/14697688.2014.95167

The least favorable noise

Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the “the least favorable perturbation” ˆ Y ε , which maximizes the prediction error E ( X − E ( X | X + Y ) ) 2 in the class of Y with v a r ( Y ) ≤ ε . We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse

Combining different models

Portfolio selection is one of the most important areas of modern finance, both theoretically and practically. Reliance on a single model is fraught with difficulties, so attempting to combine the strengths of different models is attractive; see, for example, Geweke and Amisano (J Econom 164(1):130–141, 2011) and the many references therein. This paper contributes to the model combination literature, but with a difference: the models we consider here are making statements about different sets of assets. There appear to be no studies making this structural assumption, which completely changes the nature of the problem. This paper offers suggestions for principles of model combination in this situation, characterizes the solution in the case of multivariate Gaussian distributions, and provides a small illustrative example

Conditional Sampling for Max-Stable Processes with a Mixed Moving Maxima Representation

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive.Comment: 35 pages; version accepted for publication in Extremes. The final publication is available at http://link.springer.co

The strong weak convergence of the quasi-EA

In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance

The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time T . The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see a units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresigh