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.

The consistency of private and public real estate within mixed-asset portfolios

This study considers the consistency of the role of both the private and public real estate markets within a mixed-asset context. While a vast literature has developed that has examined the potential role of both the private and public real estate markets, most studies have largely relied on both single time horizons and single sample periods. This paper builds upon the analysis of Lee and Stevenson (2005) who examined the consistency of REITs in a US capital market portfolio. The current paper extends that by also analyzing the role of the private market. To address the question, the allocation of both the private and traded markets is evaluated over different holding periods varying from 5- to 20-years. In general the results show that optimum mixed-asset portfolios already containing private real estate have little place for public real estate securities, especially in low risk portfolios and for longer investment horizons. Additionally, mixed-asset portfolios with public real estate either see the allocations to REITs diminished or eliminated if private real estate is also considered. The results demonstrate that there is a still a strong case for private real estate in the mixed-asset portfolio on the basis of an increase in risk-adjusted performance, even if the investor is already holding REITs, but that the reverse is not always the case.

The impact of real estate on the terminal wealth of the UK mixed-asset portfolio

The argument for the inclusion of real estate in the mixed-asset portfolio has concentrated on examining its effect in reducing the portfolio risk - the time series standard deviation (TSSD), mainly using ex-post time series data. However, the past as such is not really relevant to the long-term institutional investors, such as the insurance companies and pension funds, who are more concerned the terminal wealth (TW) of their investments and the variability of this wealth, the terminal wealth standard deviation (TWSD), since it is from the TW of their investment portfolio that policyholders and pensioners will derive their benefits. These kinds of investors with particular holding period requirements will be less concerned about the within period volatility of their portfolios and more by the possibility that their portfolio returns will fail to finance their liabilities. This variability in TW will be closely linked to the risk of shortfall in the quantity of assets needed to match the institution’s liabilities. The question remains therefore can real estate enhance the TW of the mixed-asset portfolio and/or reduce the variability of the TW. This paper uses annual data from the United Kingdom (UK) for the period 1972-2001 to test whether real estate is an asset class that not only reduces ex-post portfolio risk but also enhances portfolio TW and/or reduces the variability of TW.

The impact of real estate on the mixed-asset portfolio in periods of financial stress

The case for holding real estate in the mixed-asset portfolio is typically made on its stabilising effect as a result of its diversification benefits. However, portfolio diversification often fails when it is most needed, i.e. during periods of financial stress. In these periods, the variability of returns for most asset classes increases thus reducing the stabilising effect of a diversified portfolio. This paper applies the approach of Chow et al (1999) to the US domestic mixed-asset portfolio to establish whether real estate, represented by REITs, is especially useful in times of financial stress. To this end monthly returns data on five assets classes: large cap stocks, small cap stocks, long dated government bonds, cash (T-Bills) and real estate (REITs) are evaluated over the period January 1972 to December 2001. The results indicate that the inclusion of REITs in the mixed-asset portfolio can lead to increases or decreases in returns depending on the asset class replaced and whether the period is one of calm or stress. However, the inclusion of REITs invariably leads to reductions in portfolio risk that are greater than any loss in return, especially in periods of financial stress. In other words, REITs acts as a stabilising force on the mixed-asset portfolio when it is most needed, i.e. in periods of financial stress.