Jumps, or significant discontinuities, exhibit in many financial variables such as stock prices, currencies, interest rates and volatilities. To date, research on jumps, while using advanced and complicated models, is atheoretical, being based on purely mathematical and statistical techniques. This thesis aims to provide a theoretical framework to tie jumps into a fundamental economic model of valuation—the Arrow-Debreu state preference approach.
Three essays of the thesis apply the state preference approach to jump risk at both market and individual security levels. The first essay focuses on the development of a theoretical framework to value market-wide downside jump risk using the state preference approach. We introduce a concept of catastrophe bonds that offer a $1 payoff in states when jumps occur and zero otherwise. In this essay, state prices and prices of catastrophe bonds are estimated using the Black-Scholes (1973) risk neutral framework. Based on the difference between prices of the catastrophe bonds under the conditional model (taking advantage of the CBOE volatility index, VIX) and the unconditional model (using historical volatility), we construct out-of-sample predictors of the S&P 500 downside jump risk. Our predictors provide good explanatory power in predicting the future realised downside jump risk. This shows the appropriateness of using the state preference approach to measure downside jump risk. The result implies that VIX is a better volatility measure than historical volatility in predicting future downside jump risk. Furthermore, based on our model’s predictions, we develop trading strategies using one-month futures contracts. These strategies are shown to generate positive profit over the whole sample period and they perform even better during the global financial crisis (GFC) when investors need to be protected the most.
The second essay also applies the state preference approach to measure the market-wide downside jump risk. However, this essay moves one step forward from the log normal price assumption in Black and Scholes (1973) employed in the first essay and estimates state prices and the prices of catastrophe bonds using the Merton (1976) risk neutral jump-diffusion framework. We investigate two conditional sub-models including (i) the model conditioning only the instantaneous volatility on VIX, and (ii) the model conditioning both the instantaneous volatility and jumps volatility on VIX. Similar to the findings in the first essay, we find that downside jump predictors possess high predictability and VIX does a better job than historical volatility in forecasting future downside jump risk. The results from the two conditional sub-models are consistent. However, the higher McFadden R-squared and adjusted R-squared obtained in the second sub-model lead to the conclusion that the second sub-model outperforms the first one in predicting future downside jumps. Profitable trading strategies are also developed based on our model’s predictions.
The third essay extends essays one and two to predict downside jump risk at the individual security level. This essay applies the state preference approach in a three-fold manner. First we measure the individual expected volatility, which is a counterpart of VIX at the individual security level. It is demonstrated to provide an unbiased estimate for the realised volatility. A horse race is also conducted between our state price volatility measure, the 30-day historical volatility and GARCH(1,1) volatility. We conclude that our state price volatility measure provides a better forecast than the other two alternative volatility measures. Second, we use our state price volatility measure to predict future downside jump risk of the sample banks. In line with the findings in the previous essays, our downside jump risk predictors, on average, have high out-of-sample predictability. Finally, we develop a monthly Value at Risk (VaR) estimate at a 95% confidence level using our state price volatility. The test of model accuracy indicates the high quality of our VaR estimate. Moreover, we also find that our method yields a better 1 month/95% VaR than the historical simulation, which is attributable to the good performance of our state price volatility in predicting the future monthly volatility.