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ABSTRACT This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state‐dependent arrival intensity. We discuss how to perform likelihood‐based inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.