State price density analysis of equity market volatility: Tools for fund managers

Abstract

This thesis relies on empirical application of state pricing theory (Arrow, 1964; Debreu, 1959) to develop tools for equity investors. Existing measures for forecasting volatility are ad hoc and lack theoretical basis. On investigation, they show bias and inefficiency and are inappropriate for pricing risk. By employing state pricing methodology, efficient and unbiased measures are developed which allow fund managers to develop trading and hedging strategies. In the first study, the Fund Volatility Index (the “FVX”) is proposed as a forward-looking measure of fund volatility, with applications in hedging and risk management. The method applies equity market state prices to individual fund payoffs. The FVX is validated as a predictor of short-term realised volatility for 30 Exchange Traded Funds (ETFs). The performance of the method is compared with existing methods using a dataset of 14,925 non-traded funds. The FVX exhibits lower bias and higher forecast accuracy than existing methods. It is a more general, theory-based measure, allowing for incorporation of terms to capture skewness of individual fund returns. It also allows for projection of higher moments of fund returns. The second study assesses deficiencies in the Chicago Board Options Exchange Volatility Index, (the “VIX”). The VIX is the most widely used measure of market volatility expectations. In practice, it is also applied as a forecast of realised market volatility. Here it is demonstrated that the VIX is mispriced by the market, being biased when applied to predict realised volatility. It is therefore unsuitable as a hedging instrument for fund managers. The over-pricing of the VIX can be attributed to buyers of ETFs who have become increasingly reliant on out-of-the-money index put options to hedge their equity exposure or portfolio insure. This study demonstrates the associated rise in volatility exposure for market makers, and the increasing costs of hedging this exposure using S&P 500 mini futures. An excess demand for index put options leads to increasing bias in the VIX, so opening potential trading strategies for market makers to sell volatility. However, it is also shown here that it is difficult for a market maker to exploit the bias in the VIX, after taking into account the costs of volatility hedging. The bias in the VIX has also noticeably increased since January 2009 with the introduction of VIX Exchange Traded Products (ETPs). In light of these findings, the VIX is considered to be an inappropriate volatility measure for funds. This finding justifies the development of alternative measures based on state pricing. The third study proposes a method for generating unbiased predictors of downside and tail volatility for individual mutual funds. It applies theoretical market state prices to fund payoffs. The method is validated here as a predictor of market downside and tail volatility. The Fund Volatility Index-Lower Partial Moment (the “FVX¯”) is proposed as a forward-looking hedge of downside volatility for managed funds. It is calibrated and assessed on a database of 13,202 individual funds. The index proves to be unbiased with high forecast accuracy, also capturing individual fund skewness. In a final study, state pricing methodology is used to derive a State Price Volatility (SPV) index for China's stock market which has similar properties to the VIX (the “China-SPV”). The index serves as a benchmark for China volatility expectations from 1996 to 2011. Construction relies on the implied volatility of the HangSeng China Enterprise Index (HSCEI). Historic open- high-low-close volatility on the Shanghai Composite Index (SHCI) is also used to extend the benchmark prior to the availability of HSCEI options data. The China-SPV successfully forecasts realised volatility for the Shanghai Stock Exchange. It serves as a “fear gauge” in that it responds to daily movements in the SHCI in the same way that the VIX responds to movements in the S&P 500 index (Whaley, 2009). The index can assist in risk management decisions made by investors. With more and more investors operating in multiple markets, and increasing foreign investment in China, the need for such an index is increasing. This study illustrates how a market maker can hedge the China-SPV using traded options on the HSCEI with daily balancing. It also illustrates how the China-SPV can be used to hedge fund volatility and downside volatility, using the FVX and FVX¯ methods developed earlier in the thesis. The development of a number of indices in this thesis evidences the power of state pricing as an empirical tool in financial economics. In each study described here, the application of state pricing methodology has led to relatively unbiased measures with strong predictive power. Bibliography Arrow, K. J. 1964. The Role of Securities in the Optimal Allocation of Risk-Bearing. Review of Economic Studies, 31, 91-96. Debreu, G. 1959. Theory of Value, Wiley, New York, NY. Whaley, R. E. 2009. Understanding the VIX. Journal of Portfolio Management, 35, 98-106.