In the setting of ‘‘affine’ ’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

Abstract: This paper examines the joint time series of the S&P 500 index and near-the-money short-dated option prices with an arbitrage-free model, capturing both stochastic volatility and jumps. Jump-risk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jump-risk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of cross-sectional options data.

I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.

A regime switching model in continuous time is introduced where a variety of jumps are allowed in addition to the diffusive component. The characteristic function of the process is derived in closed form, and is subsequently employed to create the likelihood function. In addition, standard results of the option pricing literature can be employed in order to compute derivative prices. To this end, the relationship between the physical and the risk adjusted probability measure is explored. The generic relationship between Markov chains and [jump] diffusions is also investigated, and it is shown that virtually any stochastic volatility model model can be approximated arbitrarily well by a carefully chosen continuous time Markov chain. Therefore, the approach presented here can be utilized in order to estimate, filter and carry out option pricing for such continuous state-space models, without the need for simulation based approximations. An empirical example illustrates these contributions of the paper, estimating a stochastic volatility jump diffusion model.