According to the doc here: http://faculty.baruch.cuny.edu/jgatheral/JumpDiffusionModels.pdf.
Formula 7 specifies that the option value under jump diffusion model becomes:
So when the default intensity lambda is high, the equity option seems to become ITM and you will soon have option value equal to the stock price. This seems to be contradicted to my understand that if default intensity goes high, you expect a default event which would likely drag the company price to go lower.
So my question is when you price an option with these kind of model, how do you do credit adjustment? It seems that the stock process, if follow non-compensated jump diffusion process, which means there is no lambda drift term, makes more sense.