Yahoo’s stock dropped roughly 19%-20% this morning at the open, shaving $8.7 billion off its value, its first installment on the price of its independence from Microsoft.

Yahoo, whose position improved a couple of percentage points in the first half-hour of trading, is being held up by investor confidence that Microsoft will be back after the stock sinks, a widely held view, or alternatively that Yahoo will align with Google. At around 10 o’clock it was around $23.70.

Yahoo’s co-founder Jerry Yang and its board succeeded in running Microsoft off over the weekend by demanding more money than Microsoft was willing to pay, at least $37 a share, or $53 billion, Microsoft CEO Steve Ballmer said, although Yang really wanted $38, according to reports, down from $40.

And a Wall Street Journal blog claimed “Yahoo requested other unspecified costs that Microsoft was unwilling to accept.” Elsewhere the Journal as well as BusinessWeek said the sides never reached accord on issues like a regulatory approach or price guarantees should Microsoft stock fall again.,

Microsoft was only willing to up the ante to $$33 a share, up $5 billion for a total of $47.5 billion, representing a 70% premium over Yahoo’s stock price of $19.18, when the chase began three months ago. It is unclear if they were talking cash.

Microsoft even refused to pay Yahoo the complement of making good on its threat of a proxy fight for control of Yahoo.

Ballmer said in the letter to Yang Saturday giving notice that he was quitting the field that the week’s negotiations had opened his eyes “for the first time with real clarity on what is and is not possible” and that he understood that Yahoo would rub lye in its face if he went directly to Yahoo’s shareholders and do a deal with Google, a step, he said, that “would make Yahoo undesirable as an acquisition for Microsoft.”

So he formally withdrew Microsoft’s proposal to acquire Yahoo like it was tainted goods.

And it appears that Yahoo, which was unable to find a knight in shining armor anywhere during the last three months, has run out of alternatives to a deal with Google, its worst enemy whether it admits it or not.

Yang has promised shareholders a turnaround without Microsoft – painting a picture of Yahoo revenues up 25% in 2009 and 2010 based on improved ad revenue – and that looks impossible without a potentially illegal axis with Google.

It is believed the pair will try sprinkling the word “non-exclusive” over a deal that would have Yahoo outsource key paid search terms to Google to sell ads around, expecting the incantation to get them past Justice Department scrutiny.

It has been reported that later this week they will announce a deal to create a real-time auction system “that would choose the most lucrative ads for any given consumer query from among those sold by Yahoo, Google and any of their competitors” – even Microsoft’s ads could theoretically be sold alongside Yahoo’s search results.

Citigroup has estimated that outsourcing search advertising completely to Google would increase Yahoo’s cash flow by a billion dollars a year and that’s what Yang needs to save himself and his administration.

The problem is Google and Yahoo together represent 83% of US search advertising, according to one estimate.

The pair already conducted a two-week experiment, apparently to the DOJ’s consternation – at least it raised concerns and questions – but otherwise seems to have produced happy results for Yahoo.

As Forbes observed, “That kind of cooperating doubtlessly required more information sharing than most rivals do” and suggested that Google “probably now knows more about Yahoo’s advertising results than a competitor should.”

And Ballmer in his letter reeled off a list of problems with such a gambit that are – whatever side of the political divide you’re on – fairly hard to argue with.

It would, he said, “fundamentally undermine Yahoo’s own strategy and long-term viability by encouraging advertisers to use Google as opposed to [Yahoo’s own] Panama paid search system” and “fragment [Yahoo’s] search advertising and display advertising strategies and the ecosystem surrounding them.”

It would also “undermine the reliance on [Yahoo’s] display advertising business to fuel future growth.”

As a result Yahoo would have trouble retaining the engineers working on advertising systems that are important to Microsoft and let Google “set the prices for key search terms on both their and your search platforms and, in the process, raise prices charged to advertisers on Yahoo.”

Ballmer said that above and beyond the legal issues such a policy “seems unwise from a business perspective unless in fact one simply wishes to use this as a vehicle to exit the paid search business in favor of Google.”

Microsoft’s stock is rising, up around 2.5%, or 75 cents, like the smart money said it would. Ballmer said Yahoo would have accelerated Microsoft’s strategy but now it’s going to build scale organically and “potentially through strategic transactions with other business partners.”

Observers have been throwing names around like Facebook, AOL and News Corp’s MySpace, all names Ballmer has recently conjured with.

If Microsoft goes after AOL it would displace a potential deal between Yahoo and Time Warner

Besides a Google alliance, Yahoo has appeared enamored lately with the idea of acquiring AOL and giving Time Warner 20% of the combined company. Google owns 5% of AOL.

Analysts have upgraded Google.