human
Below is a paragraph from an academic paper. Please paraphrase the entire paragraph(rewrite it if necessary), and ask not to have too many repetitions of the original sentence. Then, 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.
In this paper,we investigate the price for options on the maximum or the minimum of several assets with counterparty default risk before maturity. To give the price, we give the joint distribution of the minimum value of one Brownian motion and the terminal values of all Brownian motions for correlated multidimensional Brownian motion. This result can also address other pricing problems of multi-asset options with default risk before maturity. In virtue of the measure-change technique, we obtain the analytical valuation formula for the options. Finally, we present numerical results.

gpt
--- Reasoning ---
1. Is the question multiple-choice question?
No
2. Is it implied multiple-choice question?
No
3. If two experts were independently asked the question, will the final answer be exactly the same?
No

--- Answer ---
open