How Default Jump Assumptions Affect Convertible Bond Greeks

Summary

The document asks how changing the assumed stock-price loss at default affects the Greeks of a convertible bond under a risk-neutral jump-diffusion model. It presents a stock-price process with a diffusion component, a Poisson default event, and a drift term that depends on the default intensity and assumed jump size. The author compares two loss assumptions and asks whether the larger loss also produces larger sensitivities such as delta.

No pricing method, numerical results, or answer to the sensitivity question is included. The stated expectation that bond value rises with the drift is an assumption in the question, not evidence establishing how the price or Greeks respond. The effects depend on the full convertible bond valuation setup, including conversion terms, recovery and default treatment, and model conventions. In particular, the jump-size sign convention and risk-neutral drift specification need careful definition before drawing conclusions from parameter comparisons.

Key ideas

• The model combines continuous stock-price diffusion with a Poisson default jump.
• The question varies the assumed stock-price decline at default and asks how Greeks change.
• The document does not provide calculations or establish the direction of delta sensitivity.
• Convertible valuation results depend on contract terms and the model’s default conventions.

Tags

# Greeks for Pricing Convertible Bond Using Jump Diffusion Model


# Greeks for Pricing Convertible Bond Using Jump Diffusion Model












I'm learning the jump diffusion model used to price a convertible bond, and got the following stochastic differential equation under risk neutral measure:

$$dS = (r+\lambda*p)Sdt + \sigma*SdW+Sdq$$

$\lambda$ is intensity of default

$dq$ is a Poisson process used to model default

$p$ is stock prices drop amount upon default

Assume we have two cases:

- stock price drops 30% upon default, $p$ is 30%

- stock price drops 50% upon default, $p$ is 50%

I know the convertible bond price under the second assumption should be higher since its drift is higher, what about the Greeks between these two assumptions, the second assumption creates higher Greeks (Ex. delta)?

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