Studies of asset returns time-series provide strong evidence that at least two stochastic factors drive volatility. The first essay investigates whether two volatility risks are priced in the stock option market and estimates volatility risk prices in a cross-section of stock option returns. The essay finds that the risk of changes in short-term volatility is significantly negatively priced, which agrees with previous studies of the pricing of a single volatility risk. The essay finds also that a second volatility risk, embedded in longer-term volatility is significantly positively priced. The difference in the pricing of short- and long-term volatility risks is economically significant - option combinations allowing investors to sell short-term volatility and buy long-term volatility offer average profits up to 20% per month.

Value-at-Risk measures only the risk of loss at the end of an investment horizon. An alternative measure (MaxVaR) has been proposed recently, which quantifies the risk of loss at or before the end of an investment horizon. The second essay studies such a risk measure for several jump processes (diffusions with one- and two-sided jumps and two-sided pure-jump processes with different structures of jump arrivals). The main tool of analysis is the first passage probability. MaxVaR for jump processes is compared to standard VaR using returns to five major stock indexes over investment horizons up to one month. Typically MaxVaR is 1.5 - 2 times higher than standard VaR, whereby the excess tends to be higher for longer investment horizons and for lower quantiles of the returns distributions. The results of the essay provide one possible justification for the multipliers applied by the Basle Committee to standard VaR for regulatory purposes.

Several continuous-time versions of the GARCH model have been proposed in the literature, which typically involve two distinct driving stochastic processes. An interesting alternative is the COGARCH model of Kluppelberg, Lindner and Maller (2004), which is driven by a single Levy process. The third essay derives a backward PIDE for the COGARCH model, in the case when the driving process is Variance-Gamma. The PIDE is applied for the calculation of option prices under the COGARCH model.