# How option value default adjusted in jump diffusion model

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.

• If the stock price wouldn’t drift up by the hazard rate then the forward wouldn’t be correct anymore. In the absence of fixed cash dividends like in this example the forward is unaffected by default risk; you can easily verify this yourself. As a result, the non-default able asset’s drift (what goes into the B/S formula here) needs to increase by the hazard rate. – LocalVolatility Nov 6 '20 at 19:27
• @LocalVolatility I see the point and that is the nature of the jump diffusion process. However, my question is whether this is reasonable or not. If not, I am curious what people adjust the model so it reflects the market behaviour. As default goes up, naturally, we expect the stock to go down and option value to go down and that is also what market data shows. E.g. look at the Carnival stock during Covid and their CDS spread. – HoldBreath Nov 6 '20 at 21:49
• The current stock price and hazard rate are exogenous to the model. The stock price and default event are independent which is clearly unrealistic but keeps the model tractable. Note that hazard rates going up still causes the call price to decline in this model. The survival probability enters twice - as a prefactor and to scale the default-free asset price passed into the B/S formula. Since the call delta is less than one, the effect of the decrease of in the prefactor will dominate that of the increase in the default-free asset price. – LocalVolatility Nov 6 '20 at 23:00
• @LocalVolatility I don't think hazard rates going up still causes the call price to decline. It is easy to verify using the formula. an extreme case, for deep OTM option, if hazard rate is 0, your option price would be 0, if hazard rate is rocket high, e.g. 100%. Your stock price would go up and become ITM forward. Then you discount back. Eventually, it will give you option price = current stock price > 0. Actually, if you take derivatives, you will get sensitivity of lambda to be T(SN(d1) - V). V will always be smaller than S and d1 could go to 1 quickly if you have large lambda. – HoldBreath Nov 9 '20 at 16:30