# Dual Delta - probability of exercise?

#### LVB

Hi all,

I am working on a project in the field of Convertible Bonds. Based on the paper by C.M. L.R.J. Rogalski and J.K. Seward (1996), I want to determine whether I can classify a convertible bond as debt or equity like. I currently replicate the embedded option of the convertible bond by creating an american option. To get the probability to exercise, I obtain the dual delta, rather than looking at the delta (due to the option/convertible bond's long maturity). The dual delta defined as: $\frac{\delta C}{\delta K}$
I obtained the dual delta for my whole data sample and notice all values range (roughly) between -0.2 and -0.3. If I understand the dual delta correctly, this would imply that the exercise probability of the option is between 20-30%. According to the paper mentioned above, this would make these convertible bonds debt-like (<40% = debt-like, 40-60% in between, >60% = equity-like). What bugs me is that I have seen the method of the paper above applied in numerous papers, and for all of them they find majority of their convertible bonds to be equity like (over 95% of their data sample is equity like) with a exercise probability averaging around 80%.

My question is as follows:
1. Am I applying the dual delta correctly? Does it indeed give me the exercise probability similar to the paper, or am I actually retrieving a completely different aspect of the option?
2. Am I reading the values correctly. Does -0.3 indeed tell me it is 30% probability of exercise, or should I actually read it as 1-30% = 70% probability exercise..?

Below the formula by C.M. L.R.J. Rogalski and J.K. Seward (1996) (see page 9), I am aware this is based on a European call option, but results shouldn't differ as I assume my convertible bonds (and thus the option aswell) to not be dividend paying.

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