These unglamorous garments are a requirement to keep warm in an office kept deliberately on the chilly side - part of a strategy by Cohen to keep his staff focused. The billionaire investor, 57, once billed as the most talented trader of his generation, does not like his employees to get too comfortable.
Not that there is much danger of that any more. Last week SAC Capital admitted it was guilty of wire fraud and four counts of securities fraud, and agreed to pay a record US$1.8 billion ($2.2 billion) fine to settle an investigation into insider trading that has cast an inky cloud over the hedge fund for the past five years.
No charges were ever brought against Cohen, but he has been punished indirectly. Under last week's agreement, his hedge fund, which managed more than US$14 billion before investors began withdrawing their money this year, will be barred from managing any funds other than its own.
It is a spectacular downfall for a trader with Cohen's track record. SAC's returns over the past two decades have averaged 30 per cent, but they occasionally topped an astonishing 70 per cent - first riding the wave of the 1998 dotcom bubble and then a year later, when Cohen bet against those same companies and cashed in on the bubble bursting.
That sort of performance meant Cohen was able to collect a 3 per cent fee and a handsome 50 per cent of his clients' profits, rather than the Wall Street average of a 1.5 per cent management fee and 20 per cent share of any gains.
He also pumped valuable fees into brokerages. According to the Wall Street Journal, Cohen and SAC's 100 or so portfolio managers accounted for 2 per cent of all stock market activity on a typical day.
SUCH a record helped Cohen earn as much as US$1 billion a year, making him one of the richest men in America. He has spent some of that money on the largest mansion in Greenwich, Connecticut - a town which boasts more wealth per capita than any other in the US - where he lives with his second wife.
Jeff Koons' sculpture Balloon Dog stands in his driveway - part of an extraordinary art collection which also includes works by Andy Warhol and Matisse, and is estimated to be worth around US$1 billion.
Cohen's recent woes do not seem to have diminished his appetite for new pieces. Days after SAC agreed the first part of its US$1.8 billion fine - a US$616 million penalty payable to America's Securities and Exchange Commission - he splashed out US$155 million on Picasso's painting Le Reve.
Cohen's interest in trading began at an early age. He was born into a middle-class family in Long Island, with a piano teacher mother and a father who ran a garment factory, but as the third of eight children he had to learn to jostle for attention.
His mother had a saying: "Money makes the monkey jump." Cohen paid close attention.
At 14 he started to dabble in poker, discovered he had a natural gift, and was soon making between US$500 and US$1000 a night playing against his school friends.
A few years later, at the University of Pennsylvania, he started loitering outside the local Merrill Lynch branch, transfixed by the electronic ticker.
Between his classes he played a game of trying to guess the fluctuations in the prices of the different stocks he was watching. He found that he had a surprisingly good success rate, and set up a brokerage account with US$1000 of his tuition money.
Decades on, Cohen puts his success down to the same instinct for anticipating stock market movements.
US prosecutors, and many of his competitors, are less sure.
SAC was a "magnet for cheaters", according to Preet Bharara, the US attorney who led the inquiry.
Traders were encouraged to use "high-conviction tip-offs" to gain an "edge" over rivals, cultivating a culture of insider trading that was "substantial, pervasive and on a scale without precedent in the history of hedge funds", he claimed.
The firm fought the allegations for years, but SAC was finally forced to concede a grudging defeat after eight of its former staff were prosecuted. Six of them have pleaded guilty.
According to interviews at the time, Cohen's firm ran according to the credo "get the information faster than anyone else". SAC's traders expected to get the first calls from analysts who were downgrading or upgrading a given stock. They would bawl out those who did not oblige.
It was a brutal environment. Traders were paid according to the money they made on individual trades, rather than the performance of the fund. The company also operated on a "down and out" principle, meaning that traders who suffered one bad day too many were quickly removed from the business.
SAC employees felt their own stock within the company rise and fall according to the information they could get their hands on.
One of them, Richard Lee, admitted making US$1.5 million based on insider tip-offs from contacts in the technology industry. In one instance, a Microsoft executive told him the company was forming a search partnership with Yahoo! shortly before the deal was announced.
Another, Mathew Martoma, allegedly profited from secret information from doctors about trials of drugs produced by pharmaceutical companies such as Elan. He is fighting the charges.
However, if Bharara's portrait was of wilful, rampant wrongdoing, SAC's terse statement did not give quite the same impression. The company maintains that just a few rogue characters were involved in the insider trading. "We take responsibility for the handful of men who pleaded guilty and whose conduct gave rise to SAC's liability," the company said last week.
"The tiny fraction of wrongdoers does not represent the 3000 honest men and women who have worked at the firm during the past 21 years. SAC has never encouraged, promoted or tolerated insider trading."
Jack Schwager, who interviewed Cohen for his series of books about Market Wizards, says the hedge fund tycoon is one of the greatest traders of recent years, regardless of whether he was caught up in insider trading.
"This guy has made thousands of trades a year over more than 20 years. Maybe some percentage of those were insider trades, but do I have confidence that he could have a great trading record without insider trading? Yes - just from a statistical standpoint," he says.
"I am only addressing the skill question here, not the ethics. This is a guy who I don't think needed insider trading to have a great track record, so it wouldn't have made any logical sense to do it. Whether he did or didn't, I have no idea."
Schwager is one of only two journalists to have interviewed Cohen.
The billionaire is famously reclusive, going so far as to buy the copyright to most of the photographs that are taken of him in order that they do not make their way into the press.
He doesn't seem to have much regard for the media in any case. In the other interview he has given, to Vanity Fair in 2010, he said he felt like "Don Quixote fighting windmills".
"There's a perception, and I'm trying to fight that perception," he said. "I find it offensive that they lump SAC into these articles [about insider trading]. I really do. The press, I mean, they don't understand what the hell - they don't understand what they're writing about."
Video recordings of a 2011 civil lawsuit show Cohen describing the laws that prohibit insider tip-offs as "vague".
"I look at this as a manual, and it's a manual of guidelines," he said. "The vast majority of the time it is adhered to, but there are times - and I can think of times - when it doesn't have to be."