Below is a paragraph from an academic paper. There are two tasks. First, paraphrase the entire paragraph(rewrite it if necessary), and ask not to have too many repetitions of the original sentence. Second, Polish the writing to meet the academic style, improve the spelling, grammar, clarity, concision and overall readability. When necessary, rewrite the whole sentence. Furthermore, list all modification and explain the reasons to do so in markdown table.“Once the total value of issuer's assets goes down and touches the certain barrier $L^*$ at time $\tau$, the option issuer will declare restructuring or bankruptcy. Mathematically,
$$
\tau:=\inf \left\{t>0 \mid V(t) \leq L^*\right\}
$$
stands for the time when the default event happens. If a credit default event occurs, the option issuer's assets are immediately liquidated and the scrap value is $(1-\alpha) V_\tau$, where $\alpha V_\tau(0<\alpha<1)$ is assumed to be the deadweight costs because of the bankruptcy or reorganization. The remaining value of $(1-\alpha) V_\tau$ is paid to the holders of the option and other liabilities, and hence the recovery rate for option holders equals $1-\alpha$.”