Analytic approximations for multi-asset option pricing

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi- or single-asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi-asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single-asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi-asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi-asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi-asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.

How often does direct real estate increase the risk-adjusted performance of the US mixed-asset portfolio?

The “case for real estate” in the mixed-asset portfolio is a topic of continuing interest to practitioners and academics. The argument is typically made by comparing efficient frontiers of portfolio with real estate to those that exclude real estate. However, most investors will have held inefficient portfolios. Thus, when analysing the real estate’s place in the mixed-asset portfolio it seems illogical to do so by comparing the difference in risk-adjusted performance between efficient portfolios, which few if any investor would have held. The approach adopted here, therefore, is to compare the risk-adjusted performance of a number of mixed-asset portfolios without real estate (which may or not be efficient) with a very large number of mixed-asset portfolios that include real estate (which again may or may not be efficient), to see the proportion of the time when there is an increase in risk-adjusted performance, significant or otherwise using appraisal-based and de-smoothed annual data from 1952-2003. So to the question how often does the addition of private real estate lead to increases the risk-adjusted performance compared with mixed-asset portfolios without real estate the answer is almost all the time. However, significant increases are harder to find. Additionally, a significant increase in risk-adjusted performance can come from either reductions in portfolio risk or increases in return depending on the investors’ initial portfolio structure. In other words, simply adding real estate to a mixed-asset portfolio is not enough to ensure significant increases in performance as the results are dependent on the percentage added and the proper reallocation of the initial portfolio mix in the expanded portfolio.