I am new to Black Scholes, and trying to use it to model a clawback in private equity. Essentially, a clawback gives the "limited" partners in the deal the option to pull some funds away from the "general" partner. I am essentially trying to value a put option, to assess the likelihood of the put option being in the money (i.e. $\mathcal{N} \left( -d_2 \right)$).
I believe the option in this case would rely on: Underlying value (i.e. "S"): net asset value of the investments + the distributions earned by the "limited" partners Exercise ("K"): the capital contributions (with a preferred growth rate) of the "limited" partners
My concern is primarily with the portion of "S", and not as much with "K". "S" has 2 separate components: the net asset value which has a volatility, and the distributions which does not have a volatility. I'm essentially trying to figure out how to model using Black Scholes when my "S" has a component with and without volatility. Has anyone had to deal with this before?
Thank you!
Update to question: thanks LocalVolatility! (I am now logged in using my own account).
Two clarifications:
1) I do not understand what the "+" after the brackets mean in your answer below? Thought maybe it would help that I understood it moving forward. Example: $\left( \hat{K} - X_T \right)^+$
2) Also, there will be times when the distribution $Y$ is larger than $K e^{\gamma T}$; so $\hat{K}$ will be less than 0. I think this is OK. $X$ (in this case the net present value, which follows a Brownian motion) is never less than 0. So, when $\hat{K}$ is less than 0 because $Y$ is larger than $K e^{\gamma T}$, the payoff $\left( \hat{K} - X_T \right)^+$ will end up being negative. We can just interpret this payoff as being 0. Further, the actual calculation of $\mathcal{N} \left( -d_2 \right)$ would just lead to an error. So, we can also interpret $\mathcal{N} \left( -d_2 \right)$ as being 0. This is because the likelihood of being in the money in this case is just 0. Does this align with your thinking?