财新传媒
位置:博客 > 王烁 > 怎么最大限度下注又避免自我毁灭?

怎么最大限度下注又避免自我毁灭?

赌博是最古老的人类活动,几千年来的发展主要由那些想发财的科学家特别是数学家推动。Fortune's Formula《财富公式》,讲几名当代科学家的努力:发明信息论的香农(Shannon)、香农在贝尔实验室的同事凯利(John Kelly)、在MIT的同事Ed Thorp,成果是凯利判据。

    凯利判据解决投注问题:胜算不同,赔率各异,应该怎样下注?
    假设一种赌局:押对翻倍,押错赔光,连续博弈,最佳下注比例如图中所示公式(截自维基百科):
    f是下注比例,b是收益,本例中为1,p是胜率,q是败率。假设胜率为60%,公式告诉你理想的下注比例是20%。如果胜率是100%,可以全部押上去。但生活中没有100%的事,不要把9成9把握当成必然。把握9成9,只应投下98%的筹码。
    凯利判据下注有两大好处:在长期中能获得最高的复利增长率;永远不会输掉全部本金。有两大坏处:尽管数学保证长期最高复利增长率,但这个长期可以长到地老天荒,如果赌徒足够倒霉,可能穷其一生都等不到。另外,按凯利判据下注,净值波动总是很大。
    凯利判据最适用于同一赌局,重复进行,每局结果相互独立,所以只能近似地应用于投资。
    面临多个投资机会的时候,应如何选择?按照各个投资机会出现各种可能结果,取其几何平均值,从中选择几何平均值最大的那个投资,这种判断方法与凯利判据在数学上等价。
    主流的资产定价模型CAPM由均值/方差模型发展而来,以算术平均值为基础。算术平均值总是大于几何平均值,但无法对抗黑天鹅。黑天鹅出现,可能赔光。以算术平均值为基础的下注法,有时是疯狂的。凯利式下注永远贪婪,但从不疯狂,数学保证其不会赔光。
    哪些投资比较适合凯利判据?指数应该是一类。还有哪些?还得把连续下注变成同时下注,人生苦短。
    凯利本人早逝。
    Thorp在三人中最热中发财。他带人用凯利判据扫荡了21点赌场,著书Beat the Dealer《战胜庄家》。21点不许赌客算牌,应该跟他有关。他创建了最早的量化对冲基金,追逐涡轮、可转债的错价,十余年间稳定地获取绝对收益,又著书Beat the Market《战胜市场》。Black-Scholes期权定价公式在学术杂志上发表之前,Thorp就在实战中自己发明并运用了同样的定价方法。他不在乎credit,在乎赢钱。萨缪尔逊及其得意弟子罗伯特·默顿与凯利判据派打了十几年的笔仗,互斥对方为愚不可及。在学界,Thorp等人无法与萨缪尔逊的威望对抗,但赌博这种事最终由实绩说话。默顿与Scholes加入长期资本管理公司,两人因期权定价公式上的贡献获得1997年诺贝尔奖,但长期资本次年爆仓。Thorp说:“默顿这帮人没按凯利判据下注,赌得太大。”
    香农始终没出校园。投资之道,他不是凯利型赌徒,而是买入持有型赌徒,20余年间复利年收益率是28%,最好一笔投资获利2000倍。不过,能找到的香农的投资对账单显示总额不过50来万美元。赌博小道,数学最高。
    下面是用iPad+ibooks+evernote做的零星摘录。有兴趣者应读全书,William Poundstone是个很好的作者。
 

扫瞄关注BetterRead公号,有理有趣有用的英文作品推介

 
 
William Poundstone
 
Bernoulli’s thesis was that risky ventures should be evaluated by the geometric mean of outcomes.
 
There is a deep connection between Bernoulli’s dictum and John Kelly’s 1956 publication. It turns out that Kelly’s prescription can be restated as this simple rule: When faced with a choice of wagers or investments, choose the one with the highest geometric mean of outcomes. This rule, of broader application than the edge/odds Kelly formula for bet size, is the Kelly criterion.
 
When the possible outcomes are not all equally likely, you need to weight them according to their probability. One way to do that is to maximize the expected logarithm of wealth. Anyone who follows this rule is acting as if he had logarithmic utility.
 
In short, the Kelly criterion may risk money you need for gains you may find superfluous; it may sacrifice welcome gains for a degree of security you find unnecessary. It is not a good fit with people’s feelings about the extremes of gain and loss.
 
The promises of the Kelly criterion recall those tales of mischievous genies granting wishes that never turn out as planned. Before you wish for maximum long-term return and zero risk of ruin, Samuelson is saying, you had better make sure that is exactly what you want—because you may get it.
 
There is a catch. Life is short, and the stock market is a slow game. In blackjack, it’s double or nothing every forty seconds. In the stock market, it generally takes years to double your money—or to lose practically everything. No buy-and-hold stock investor lives long enough to have a high degree of confidence that the Kelly system will pull ahead of all others. That is why the Kelly system has more relevance to an in-and-out trader than a typical small investor.
 
The bankroll fluctuations in Kelly betting obey a simple rule. In an infinite series of serial Kelly bets, the chance of your bankroll ever dipping down to half its original size is…½.
 
This is exactly correct for an idealized game in which the betting is continuous. It is close to correct for the more usual case of discrete bets (blackjack, horse racing, etc.). A similar rule holds for any fraction 1/n. The chance of ever dipping to 1/3 your original bankroll is 1/3. The chance of being reduced to 1 percent of your bankroll is 1 percent.
 
The good news is that the chance of ever being reduced to zero is zero. Because you never go broke, you can always recover from losses.
 
The bad news is that no matter how rich you get, you run the risk of serious dips. The 1/n rule applies at any stage in the betting.
 
One is to stake a fixed fraction of the Kelly bet or position size. As before, you determine which opportunity or portfolio of opportunities maximizes the geometric mean. You then stake less than the full Kelly bet(s). A popular approach with gamblers is “half Kelly.” You consistently wager half of the Kelly bet.
 
This is an appealing trade-off because it cuts volatility drastically while decreasing the return by only a quarter. In a gamble or investment where wealth compounds 10 percent per time unit with full-Kelly betting, it compounds 7.5 percent with half-Kelly.
 
The core of John Kelly’s philosophy of risk can be stated without math. It is that even unlikely events must come to pass eventually. Therefore, anyone who accepts small risks of losing everything will lose everything, sooner or later. The ultimate compound return rate is acutely sensitive to fat tails.
 
Shannon was a buy-and-hold fundamental investor. From the late 1950s through 1986, Shannon’s return on his stock portfolio was about 28 percent.
 
推荐 50