幾個星期前,博文「數學『諾貝爾獎』揭曉:又沒中國人啥事」引起不小的討論。一些人在問那些數學頂尖的人才都去幹什麼了?其實,數理好的孩子是有很多職業道路可選的,有些選擇了學術道路,讀博當教授;有些轉工程,計算機,進入高科技;有些學經濟,MBA,走管理的道路;一些為CIA,FBI,NSA,DOD工作,搖身為間諜或網路安全專家;也有些進入華爾街街,成為了寬客(quants)。Quants是做定量分析(Quantitative Analysis)的人的昵稱。
美國是自由國家,走哪條道路完全是個人的選擇,本身並沒有高低貴賤之分。認為去了華爾街就是出賣靈魂的看法也是一種偏見。可喜的是,美國孩子的選擇還是相當多樣的,這種多樣性是社會均衡發展的重要保證。美國還有一個不算秘密的武器,那就是全世界的頂尖人才都來美國求學,學成后很多選擇留在美國發展。這樣也在一定程度上緩解了本土人才不足的問題。
那些在華爾街的寬客們又在做什麼呢?我想起了小寶在高中時為一家英文報紙所寫的專欄 「It』s not QUANTum physics anymore」,翻譯出來,以餐讀者。小寶的專欄涉及很廣,但其中不少是寫數學和科學的。
小寶對數學的興趣在很早的時候就展示出來。在初中是曾經是北加州MATHCOUNTS的地區冠軍(三個縣),數十年來第一位女生獲此榮譽。她在高中時又花過兩個暑假在夏令營學習數論,開始的時候小寶並不想去,寶爸只好說:「你不是喜歡密碼學嗎(cryptology)?加密要用到素數(prime number),而數論則是研究素數的。」當然數論不只是研究素數,她聽了居然就去了。小寶對純數學的興趣不大,但數學夏令營的孩子們卻是她喜歡的。按照她的說法,數學男孩多半有些怪,數學女孩卻大多正常,只是很聰明。第二年她又去了,倒不是素數有這麼大的魅力,和她那幫狐朋狗友們在一起很開心。
夏令營的一個後果是小寶對建立數學模型(Mathematical models)解決複雜的問題產生了興趣,這種喜好也影響到她的專業選擇,以及後面的職業選擇。
是誰搞垮華爾街的?
小寶,寫於2009年
在過去十幾二十年裡,粒子物理學家開始在所有的頂級金融機構里鎮守攤位。他們不再研究大統一理論(General Unified Theory),夸克,或粒子加速器。他們使用自己在數學和邏輯推理方面的專業知識來預測經濟下一步會怎樣運行。他們的目的不僅僅是買或賣 - 還包括分析相關性(correlations)和風險因素,解讀鐘形曲線,還有操縱細微的優勢為他們的僱主和客戶賺錢。
令人驚訝,是不是?如果我們回顧一下這些工作的來源,其實很正常。半個世紀前美國和蘇聯的冷戰培育整整一代的數學和科學家...當冷戰結束時,大學只能容納這麼多教授。於是許多科學家決定放棄純科學的「美」,轉而挺進向華爾街。他們成了「寬客」- 定量分析和風險管理的專家。
舉個例子來說,伊曼紐爾·德曼(Emanuel Derman),一個有哥倫比亞學位和在牛津大學工作過的學者,在高盛(Goldman Sachs)度過了17個春秋 [注一]。事實上近年來麻省理工學院的數學和物理的畢業生有很大比例在華爾街工作,像浮士德一樣把靈魂換成錢了。
金融工程師利用複雜的,滿是希臘字母的數學公式來梳通,理解股市 - 不幸的是,有時會有不曾預想到的後果。例如,李祥林(David X. Li)的高斯Copula函數,這使得非常複雜的風險,可以輕鬆和具有相當精度地進行建模。當它首先發明時被金融界視為天賜之物。因為這個方程式「可以讓交易員賣出數量龐大的新證券,將金融市場擴大到無法想象的地步。」
李祥林幾年前是列在諾貝爾經濟獎的短名單上。然而今天,他據傳躲在中國。一些人指責他的公式的巨大裂痕導致了2008年美國經濟的次貸危機和隨後的經濟衰退的。
其實,這並不一定是李祥林和他的方程式的錯,而是那些過度信任數學公式的交易員和金融大師們的錯。就連李祥林本人,他已經不在華爾街工作,也說過,「最危險的部分是當人們相信從公式里出來的一切東西。」如今金融領域裡寬客的作用的整個問題已引發了巨大的爭議。一方面,部分人認為,如果被視為一種模型,作為一種指引,而不是刻在石頭上的,金融工程師做的工作是非常有用的。在另一方面,我們有納西姆·尼古拉斯·塔勒布(Nassim Nicholas Taleb)。
塔勒布是個有話就說的人,「我認為物理學家應該回到物理系去,別再纏著華爾街」。不同於許多寬客,他們對股市的假設是基於鐘形曲線公式 – 平均值是最有可能的,這位前股票交易員認為我們國家的金融歷史是更多地基於不可預測,無法建模的事件而非平均。富豪慈善家巴菲特顯然同意這種說法,他在2008年查理·羅斯節目上(Charlie Rose Show )就說過 –「我能說的是,留神那些穿著公式的極客們」(beware of geeks bearing formulas)
這句話基本上總結了進口的物理學家在金融界的地位層次。他們做了很多工作,但他們並沒有像交易員或經紀人受到公司的重視。而到了最後,經濟不景氣罪魁禍首的頭銜又被放在他們身上。有些人說,他們應該能預測下沉的房地產和下沉的經濟之間的相關性。「他們忘記了這些數字代表的具體的現實。」華爾街是在告訴寬客們:這些東西不是像高中微積分課程那麼容易。
然而,這也許是問題的一部分。太多鼻孔朝天的銀行家已經不記得了他們的高中微積分課程。他們不會讓寬客做任何實際的決定,他們也不問這些數學模型在那些情況下可以適用。信用研究公司CreditSights的吉爾克斯(Kai Gilkes)說,「我們應該責怪那些誤讀它的銀行家。」
整個情況就像坐在你身邊的孩子抄了你所有的作業,也不問你是怎麼得出結果的。當要考試的時候,他就死定了。
所以人們不應該把寬客們從華爾街趕出來,因為他們在做有價值的工作–大部分時間是對的。然而,寬客們對定量財務分析技術的信心應該有所動搖。一個很大的問題是當所有的人都同時使用李祥林的公式或者「寬客的黃金標準」,Black-Scholes模型 [注二],當船破的時候,所有人都淹死。
缺少遺傳多樣性的物種不能活的長久。也許華爾街需要更多的寬客和更多的數學模型,而不是更少。
白露為霜注:以下是李祥林毀掉華爾街的公式。他用高斯Copula函數來衡量事件的關係。用數學名詞,隨機的經濟事件之間的相關性,可表示為:
通俗地說,他提出兩件不同但有關的事件的關係,比如「樓A」違約和「樓B」違約可以用相關性來衡量。計算相關性的公式似乎在大部分時間是相當準確的,在某些特殊情況下則沒有用。最根本的問題:過去的歷史終究無法預測未來,數學模型的局限性通常只在最糟糕的時候才暴露出來。
注一:伊曼紐爾·德曼寫過一本書「我的寬客生活」(My life as quants).
注二:Black-Scholes模型是常用的計算期權(option)價格的公式。
火辣的數學女
怪發的數學女
華爾街的寬客
李祥林
股票市場是不可預測的
上了時代周刊的寬客
It』s not QUANTum physics anymore
For more than two dozen years, there have been particle physicists manning the booths at all of the top financial organizations. They are no longer studying General Unified Theory, quarks, and particle accelerators. They are using their expertise with mathematics and logical reasoning to forecast what the economy is going to do next. Their game isn』t just buying and selling – it』s correlations and risk factors, interpreting bell curves and manipulating the slightest percentage of an advantage to make money for their employers and clients.
Surprising, isn』t it? Not so much, if we go back to the root of the job. America and Russia』s little Cold War half a century ago bred an entire math and science generation…when the Cold War ended, there were only so many professors needed. Many of the scientists decided to forgo the 「beauty」 of pure science and made their way to Wall Street. They became 「quants」, masters of Quantitative Analysis and 「risk management」.
Take, for example, Emanuel Derman, an academic with a degree from Columbia and a job at Oxford, who spent 17 years at Goldman Sachs. Indeed, a large percentage of M.I.T.』s math and physics graduates went on to work in Wall Street in recent years, bartering away their souls for money.
Quants utilize complicated mathematical equations involving the greek alphabet to detangle the stock market – unfortunately sometimes with the bad consequences. For example, David X. Li』s Gaussian copula function, which allowed hugely complex risks to be modeled with ease and supposed accuracy, was seen as a godsend by the Wall Street community when it was first invented. After all, the equation 「made it possible for traders to sell vast quantities of new securities, expanding financial markets to unimaginable levels.」
Li was on the short list for a Nobel Prize just a few years ago. Today, however, he has gone into hiding in China; some are blaming the huge cracks in his function for the sub-prime mortgage meltdown and subsequent downturn of America』s economy.
Nevertheless, it was not necessarily the fault of Li and his equation, but rather the traders and financial gurus who placed an excess amount of trust in a mathematical formula. Even Li himself, who is no longer working in the field, said that 「the most dangerous part is when people believe everything coming out of it」. Now the entire issue of having quants in the financial sector at all has sparked a giant debate. On one side, some argue that the work that quants do is useful if seen as a model, a guideline, not something set in stone. On the other hand, we have Nassim Nicholas Taleb.
Taleb is a man of set opinions: 「I think physicists should go back to the physics department and leave Wall Street alone.」 In contrast to many quants, who base their stock-market assumptions on the bell curve formula where the average is most likely, the former trader believes that our nation』s financial history is based far more upon unpredictable, unmodeled events than on the mean. The charitable Warren Buffet apparently agreed last fall on The Charlie Rose Show, when he said 「All I can say is, beware of geeks bearing formulas」.
That sentence might best sum up the placing on the hierarchy of the imported physicists. They do a lot of work, but they do not get as much credit as the traders or the brokers who are the face of the company. And in the end, much of the blame for economic downturn is being placed on them. People say that they should have been able to perceive the correlations between the sinking housing market and the sinking economy. 「They forget about the concrete reality the figures are supposed to represent.」 Basically, Wall Street is telling the quants that stuff here isn』t as easy as it is in a high school calculus class.
However, this might be part of the problem. Too many bankers, who look down their noses on the quants, do not remember anything from their high school calculus classes. They don』t let any of the quants make the actual decisions, and they don』t ask questions about the applicability of the models in certain contexts. Kai Gilkes of CreditSights, a credit research film, says that 「we should blame the bankers who misinterpreted it.」
The entire situation is like the kid who sat next to you and copied all your homework without asking how you got to that point. When the test comes around, he was totally screwed.
The quants should not be run out of Wall Street, because ultimately they were doing valuable work that was correct a majority of the time. However, their faith in tenets of quantitative financial analysis should be shaken a little bit. The main problem with force-feed equation』s like Li』s or 「the quant』s gold standard」, the Black-Scholes Model, is that everyone started using it at once, so everyone sunk with the ship. A species can』t live on without genetic diversification.
Maybe what the Wall Street needs are more quants and more models, not less.