Reconstructing distribution from option prices. (2013)
AuthorsSung, Jackyshow all
Option pricing has been a popular topic in the financial industry. If there were an effective way to price options correctly, it could help to identify potential profits and risks in the options market. The risk-neutral density (RND), if it exists, leads to calculation of fair prices of options by taking the expected value of the payoff function under the RND, which can be reconstructed nonparametrically from the market data alone. Jaynes (1963) argues that, out of the infinitely many density functions, there is a unique and most preferred way to choose the density: which is via the maximum entropy principle, and hence, the density obtained is called maximum entropy density (MED). A classical approach of finding the MED is by maximising the Lagrangian function with Lagrange multipliers; however, due to potential numerical difficulties, this is reformulated under the duality result by Borwein et al. (2003). This thesis carries out a simulation study to explore the properties of the MED estimators proposed in the literature. With the framework given, some data were simulated with a log-normal distribution and found that the MED constructed converges to the original distribution when the data is convex and noiseless. However, it is inevitable for the market data to be noisy, simulated noise is added and explores effective methods that would not only filter out the noise but also guarantee the existence of MED were explored. Many possible strategies that deal with noisy and non-convex data including the Tikhonov regularisation, polyhedral set projection, convex hull methods, and the cubic spline smoothing methods have been attempted. As a result of 1000 replications of simulated experiments, the cubic spline smoothing method outperforms the other methods yielding the lowest mean integrated squared error. Some of the reconstructed densities give relatively accurate results. This method was then applied to real VIX indices data, the results obtained, however, depended on the choice of the mixing parameter p, which could be subjective at times.