  We study convexity and monotonicity properties of option prices in a model
with jumps using the fact that these prices satisfy certain parabolic
integro-differential equations. Conditions are provided under which
preservation of convexity holds, i.e. under which the value, calculated under a
chosen martingale measure, of an option with a convex contract function is
convex as a function of the underlying stock price. The preservation of
convexity is then used to derive monotonicity properties of the option value
with respect to the different parameters of the model, such as the volatility,
the jump size and the jump intensity.
