Below is a paragraph from an academic paper. Polish the writing to meet the academic style, improve the spelling, grammar, clarity, concision and overall readability. When necessary, rewrite the whole sentence. Note in particular that the polished version of the paragraph should first be written in full, then list all modification and explain the reasons to do so in markdown table.
For options on the maximum or the minimum of two prices with default risk, \citet{NAME_1valuing} consider vulnerable options at the option's expiry, \citet{NAME_2valuation} derived the closed-form pricing formulae under GARCH models. Do not say the word "writing" in the response
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