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DETROIT — Detroit-based Rocket Cos. Inc. has begun trading today on the New York Stock Exchange. Rocket, which operates billionaire businessman Dan Gilbert’s Quicken Loans Inc. and Rocket Mortgage, is offering 100 million shares at $18 each for a valuation of $1.8 billion. The company has downsized its initial public offering (IPO) from an earlier filing that announced shares would be priced between $20 and $22. Investors viewed the business as more of a consumer finance company than a technology-driven organization, which contributed to the lower valuation, according to Bloomberg. The shares are trading under the ticker symbol RKT.