2017年7月1日 星期六

20170701 Kelly Criterion 在股票的應用

一直以來有留意Kelly Criterion 在股票的應用。在看過池兄的blog文後http://poolshunter.blogspot.hk/2017/06/kelly-criterion.html ,便在網上找尋"simultaneous kelly for stock portfolio",找到一些更豐富的內容,現在綜合一下,請各位指教。

1. How to apply Kelly criterion to a portfolio made by a stock plus a option?
https://quant.stackexchange.com/questions/26324/how-to-apply-kelly-criterion-to-a-portfolio-made-by-a-stock-plus-a-option

I know that by a portfolio made by only by one stock (and a risk free bond) I can use the formula:

f* = (R-Rf)/d^2

f* - Wealth fraction that maximize the log return
R - Asset Return
Rf - Risk-free return d - Standard Deviation

Answer:
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Kelly is mostly based upon assets with zero correlation made independent of each other.
The way I approximate Kelly for multiple bets with correlation is:

Assume after your first bet the capital is gone.
Place a second bet based upon the Kelly of the remaining capital.
Factor in correlation..
Part 3 is the challenging part. I assume that with multiple bets at zero correlation placed simultaneously that I would bet the full Kelly per bet made. I assume that with multiple bets at a correlation of 1, I would divide the Kelly by the number of bets. So if for example I were to make 5 bets with a Kelly of 20%...
a correlation of 1 would be 20% divided by 5 or 4% per bet. A correlation of zero would be 1-(0.80^5)
to determine total capital at risk and then divide by 5 which is ~13.45% per bet.
A correlation of 50% is the average of the two or ~8.7% Anything else is a weighted average
but you have to be careful not to get the weightings backwards.
For example a correlation of 20% you take 80% of the Kelly amount 13.45 and 20% of 4% and sum them together.

小記:看不懂Part 3

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2.

Quantitative Trading: Kelly vs. Markowitz Portfolio Optimization

In my book, I described a very simple and elegant formula for determining the optimal asset allocation among N assets:

F=C-1*M   (1)

where F is a Nx1 vector indicating the fraction of the equity to be allocated to each asset, C is the covariance matrix, and M is the mean vector for the excess returns of these assets. Note that these "assets" can in fact be "trading strategies" or "portfolios" themselves. If these are in fact real assets that incur a carry (financing) cost, then excess returns are returns minus the risk-free rate.
小記:這就是終極版的「發達公式」,非我等凡人可用+-*/或excel計算出來的!
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3. Simple closed form solution for unconstrained Simultaneous bet Kelly staking
So given, for example, events A, B, C, D, and E, with corresponding single-bet Kelly stakes of κA, κB, κC, κD, and κE,
then the Kelly stake for the 1-team parlay consisting of only bet A would be:
κA * (1-κB) * (1-κC) * (1-κD) * (1-κE)
While the Kelly stake for the 3-team parlay consisting of bets A, B, and C would be:
κA * κB * κC * (1-κD) * (1-κE)
Much simpler, no?
小記:這個可用!已改了我的組合程式上試試!

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4. Algorithms for optimal allocation of bets on many simultaneous events
Chris Whitrow 

Conclusions
When the number of bets is small, the optimal sizes of bet seem to be almost exactly proportional to the Kelly stakes on individual bets. 

小記:這個最簡單!已改了我的組合程式上試試! 

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(註:我的組合程式上的個別kelly是用以下的公式的:

Sharpe ratio S = (R-Rf)/d,
f = (R-Rf)/d^2 = S/d

據說the maximum compounded growth rate g is given by g=r+S^2/2. 
We usually drop the risk-free rate, so we have g=S^2/2.

現時盈富(2800.hk)的sharpe ratio大約是0.57


4 則留言:

  1. 嘩。內容好豐富,要學習下先。
    池某用慣第3種於波馬,但不認為適用於股票,因為single-bet Kelly stakes於某隻股票這個前提無法解決。

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    1. 池兄的所謂single-bet Kelly stakes於某隻股票這個前提無法解決,是指沒法算出p和odd吧!我的理解和處理是用William Sharpe 和 Edward Thorp的公式來估算某隻股票的single-bet kelly:
      Sharpe ratio S = (R-Rf)/d,
      f = (R-Rf)/d^2 = S/d

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    2. 後記:因為kelly的前提是要有edge,所以我只買盈富,reit和公用股,(或加上恆生和中銀),不買A股有關(越房405除外)。

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