Jimmy E. Hilliard

Robust binomial lattices for univariate and multivariate applications: choosing probabilities to match local densities

A wide variety of diffusions used in financial economics are mean-reverting and many have state- and time-dependent volatilities. For processes with the latter property, a transformation along the lines suggested by Nelson and Ramaswamey can be used to give a diffusion with constant volatility and thus a computationally simple binomial lattice. Drift terms in mean-reverting and transformed processes frequently result in either ill-defined probabilities or complex grids. We develop closed-form, legitimate probabilities on a simple grid for univariate and multivariate lattices for well-posed smooth diffusions. The probabilities are based on conditional normal density functions with parameters determined by the diffusion. We demonstrate convergence in distribution under mild restrictions and provide numerical comparisons with other univariate and multivariate approaches.

Estimating Early Exercise Premiums on Gold and Copper Options Using a Multifactor Model and Density Matched Lattices

We use the standard geometric Brownian motion augmented by jumps to describe the spot underlying and mean regressive models of interest rates and convenience yields as state variables for gold and copper prices. Estimates of parameters of the diffusion processes are obtained by the Kalman filter. Using these estimates, jump parameters are estimated in the second stage by least squares. Early exercise premia on puts and calls are computed using a lattice with probabilities assigned by the density matching technique. We find that while deep in the money options have greater absolute early exercise premiums, the early exercise premium is roughly constant as a percent of option price. Our findings also confirm that gold behaves like an investment asset and copper behaves like a commodity.

A Comparison of Rebalanced and Buy and Hold Portfolios: Does Monetary Policy Matter?

We examine buy and hold and a number of rebalancing strategies on a portfolio of indices that are tracked by ETFs. The indices include Barkleys treasuries and MSCI indices on emerging markets, Pacific and European markets, value funds, and growth funds. Portfolios are rebalanced using threshold schemes and compared with portfolios rebalanced using weights from different points on the Markowitz efficient frontier. We also examine portfolios that are rebalanced across monetary policy regimes. We find that portfolios rebalanced using Markowitz weights that allow for shifts in monetary policy regimes easily outperform both threshold rebalancing and buy and hold strategies.

Pricing American options when there is short–lived arbitrage

Models in financial economics derived from no-arbitrage assumptions are standard fare among theoreticians and practitioners. However, several authors have investigated the impact of short lived arbitrage on European options using models borrowed from disequilibria in physics. In this paper, we extend that research by assessing the impact of arbitrage on American options. We use an Ornstein-Uhlenbeck Bridge as a model for the arbitrage process and calculate American option prices using a bivariate lattice with density matched probabilities. We find that arbitrage correlated with the underlying has an economically meaningful sizeable impact on option prices. Short lived arbitrage has essentially virtually no economic impact on option prices when the arbitrage is uncorrelated with underlying asset returns.

Timing versus Buy and Hold: A Model for Determining Predictive Accuracy Required for Superior Performance

In this application, we develop a model to simulate the decisions of a trader whose subjective distribution of returns may be correlated with realized stock returns. Using empirically estimated parameters from stocks in the CRSP database, we obtain performance data on a number of measures, including mean returns, volatility, the Sharpe measure, and the probability of a correct trading decision. The model suggests that daily trading of a portfolio of 20 volatile stocks gives a Sharpe measure better than that of buying and holding the S&P 500 when timing accuracy is 53% or better.

Rebalancing versus Buy and Hold: Theory, Simulation and Empirical Analysis

We consider returns from rebalanced and buy and hold portfolios consisting of the same stocks. Theoretical properties are derived using Jensen’s inequality and the Holder’s Defect Formula. Simulations are used to confirm theory and to investigate ambiguous cases where theory is silent. Rebalancing decreases total return volatility, while buy and hold produces greater expected return. Results are more opaque with respect to Sharpe Ratios and expected geometric means. Our empirical tests are based on portfolios composed of the risk-free asset, CRSP market value returns and returns from five Fama–French industries. While rebalancing reduces volatility and momentum effect, our tests largely favor the buy and hold strategy due to the high relative returns enjoyed by stocks vis-a-vis the risk-free asset. Transactions cost for rebalancing the portfolio are economically negligible.

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favor among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein-Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that liquidity factors significantly effect arbitrage model parameters.

The VIX, VXO and realised volatility: a test of lagged and contemporaneous relationships

The VIX has traditionally been considered a forward indicator of realised volatility. This follows from its original formulation as the implied volatility of an option on the S&P 100 index and its later incarnation based on the fair price of a realised volatility swap. We focus on the related issue of Granger causality. Our results suggest that realised volatility Granger causes the VIX. In fact, it appears that the VIX lags realised volatility by about one month. Overall, our results are consistent with the notion that participants rely more on objective probabilities derived from past observations and less on future subjective probabilities. We also use threshold analysis to investigate asymmetric relationships between realised and implied volatility.

Matching non-synchronous observations in derivative markets: choosing windows and efficient estimators

Equilibrium and arbitrage-based option pricing models are based on the assumption that the derivative and its underlying asset are simultaneously observable. However, empirical testing with transactions data must deal with less than perfect synchronicity and windows defining a ‘match’ between the derivative and its underlying must be specified. A narrow window minimizes measurement error at the expense of a smaller sample size. The analysis in this paper assumes Poisson transaction arrivals and smooth diffusion price processes. Optimal windows and efficient estimators are derived and further evaluated by simulation. Benchmarks options are chosen using data from pit-traded S&P 500 futures options and Globex traded Euro options.