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Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Application of Information Theory in Finance
Kashyap AroraGuide: Dr. Andrew Thangaraj
Department of Electrical EngineeringIIT, Madras
11th September 2006
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Horse Race Problem
Simple horse race problem with m horses.
i th horse wins with a probability pi and payoff = oi .
bi=fraction of wealth invested in horse i .∑bi=1 and bi ≥ 0 for i = 1, 2, . . . , m
bioi = fraction of money received if the i th horse wins.
Wealth Relative (Sn) after n races
Sn =n∏
i=1
S(Xi)
where X1, X2, . . . , Xn are race outcomes.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Doubling Rate
Doubling Rate for a horse race is defined as
W (b, p) = E [log(S(X ))] =m∑
k=1
pk log bkok
It can be proved using the weak law of large numbers
Sn = 2nW (b,p)
Wealth increases exponentially with WAim is to maximize the doubling rate such that the wealthrelative at the end of n races is maximum.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Optimum Doubling Rate
We define the optimum doubling rate W ∗ as
W ∗(p) = maxb
W (b, p) = maxb
∑pi log bioi
b : bi ≥ 0,∑
bi = 1
Proof: Using the Langrange Multiplier
J =∑
pi log bioi + λ(∑
bi − 1)
∂J∂bj
= 0 ⇒ bj = −pj
λ
Using∑
i
bi = 0 we get λ = −1 ⇒ bj = pj
Makes intuitive sense!Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Optimum Doubling Rate
We define the optimum doubling rate W ∗ as
W ∗(p) = maxb
W (b, p) = maxb
∑pi log bioi
b : bi ≥ 0,∑
bi = 1
Proof: Using the Langrange Multiplier
J =∑
pi log bioi + λ(∑
bi − 1)
∂J∂bj
= 0 ⇒ bj = −pj
λ
Using∑
i
bi = 0 we get λ = −1 ⇒ bj = pj
Makes intuitive sense!Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
continued
Alternate Proof:
W (b, p) =∑
pi logbi
pipioi
=∑
pi log pioi − D(p ‖ b)
≤∑
pi log pioi
It can again be seen that maximum value of W (b, p) existswhen b = p.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Fair Odds
ri = 1oi
represents the bookies estimates of win probability.
W (b, p) =∑
pi log bioi =∑
pi logbi
ri
=∑
pi logpi
ri
bi
pi
= D(p ‖ r)− D(p ‖ b)
Intuitively: To outperform the bookies b has to be closer to pthan r .
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Even Odds
Consider a case with even odds i.e. each horse has oddsm for 1 (⇒ ri = 1
m )
W ∗(p) = D(p ‖ 1m
) =∑
pi log pim
= log m − H(p)
⇒ W ∗(p) + H(p) = log m = constant
Sum of the optimum doubling rate and entropy is constant.Low entropy races are more profitable!
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Side Information
Suppose the gambler has some information relevant to theoutcome of the gamble.e.g. Performance of the horse in the previous race.
What is the increase in wealth that can result from suchinformation?What is the increase in the doubling rate due to thatinformation?
Let horse X = {1, 2, . . . , m} win the race and pay odds ofo(x) for 1. Let (X , Y ) have joint probability mass functionp(x , y).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Side Information
Let b(x |y),∑
x b(x |y) = 1 be an arbitrary conditionalbetting strategy, based on the side information Y .Let the unconditional and conditional doubling rates be
W ∗(X ) = maxb(x)
∑x
p(x) log b(x)o(x),
W ∗(X |Y ) = maxb(x |y)
∑x ,y
p(x , y) log b(x |y)o(x)
and let
∆W = W ∗(X |Y )−W ∗(X ).
Increase in doubling rate (∆W ) due to the presence ofside information is equal to the mutual information betweenthe side information and the horse race.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
Proof
With side information, the maximum value of W ∗(X |Y ) isachieved by conditionally proportional gambling i.e.b∗(x |y) = p(x |y).
W ∗(X |Y ) = maxb(x |y∑
p(x , y) log o(x)b(x |y)
=∑
p(x , y) log o(x)p(x |y)
=∑
p(x) log o(x)− H(X |Y ).
also
W ∗ =∑
p(x , y) log o(x)− H(X ).
Therefore
∆W = W ∗(X |Y )−W ∗(X ) = H(X )− H(X |Y ) = I(X ; Y ).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
What is the stock market?Stock Market is a vector of stocks represented as
X = (X1, X2, . . . , Xm) Xi ≥ 0 , i = 1, 2, · · · , m
m: Number of stocks Xi : price relativeX ∼ F (x) Joint distribution of the vector of price relatives
What is portfolio?Allocation of wealth across the various stocks.Mathematically we can define a portfolio as
b = (b1, b2, . . . , bm) bi ≥ 0,∑
bi = 1
where bi is the fraction of wealth invested in stock i .
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
What is the stock market?Stock Market is a vector of stocks represented as
X = (X1, X2, . . . , Xm) Xi ≥ 0 , i = 1, 2, · · · , m
m: Number of stocks Xi : price relativeX ∼ F (x) Joint distribution of the vector of price relatives
What is portfolio?Allocation of wealth across the various stocks.Mathematically we can define a portfolio as
b = (b1, b2, . . . , bm) bi ≥ 0,∑
bi = 1
where bi is the fraction of wealth invested in stock i .
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
What is the stock market?Stock Market is a vector of stocks represented as
X = (X1, X2, . . . , Xm) Xi ≥ 0 , i = 1, 2, · · · , m
m: Number of stocks Xi : price relativeX ∼ F (x) Joint distribution of the vector of price relatives
What is portfolio?Allocation of wealth across the various stocks.Mathematically we can define a portfolio as
b = (b1, b2, . . . , bm) bi ≥ 0,∑
bi = 1
where bi is the fraction of wealth invested in stock i .
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
What is the stock market?Stock Market is a vector of stocks represented as
X = (X1, X2, . . . , Xm) Xi ≥ 0 , i = 1, 2, · · · , m
m: Number of stocks Xi : price relativeX ∼ F (x) Joint distribution of the vector of price relatives
What is portfolio?Allocation of wealth across the various stocks.Mathematically we can define a portfolio as
b = (b1, b2, . . . , bm) bi ≥ 0,∑
bi = 1
where bi is the fraction of wealth invested in stock i .
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Log-Optimal Portfolio
S= Ratio of wealth at the end of the day to the beginning ofthe day. S = bT XAim is to maximize S in some sense.
Doubling Rate for stock markets is defined as
W (b, F ) =
∫F (x) log bT X = E(log bT X).
Optimal Doubling Rate is defined as
W ∗(F ) = maxb
W (b, F )
b which achieves W ∗ is called log-optimal portfolio (b∗).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Log-Optimal Portfolio
S= Ratio of wealth at the end of the day to the beginning ofthe day. S = bT XAim is to maximize S in some sense.
Doubling Rate for stock markets is defined as
W (b, F ) =
∫F (x) log bT X = E(log bT X).
Optimal Doubling Rate is defined as
W ∗(F ) = maxb
W (b, F )
b which achieves W ∗ is called log-optimal portfolio (b∗).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Log-Optimal Portfolio
S= Ratio of wealth at the end of the day to the beginning ofthe day. S = bT XAim is to maximize S in some sense.
Doubling Rate for stock markets is defined as
W (b, F ) =
∫F (x) log bT X = E(log bT X).
Optimal Doubling Rate is defined as
W ∗(F ) = maxb
W (b, F )
b which achieves W ∗ is called log-optimal portfolio (b∗).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Doubling Rate and Wealth
Let X1, X2, . . . , Xn be i.i.d. random variables according toF (x).Wealth Relative at the end of n days Sn using a constantportfolio b∗ can be represented as
S∗n =
n∏i=1
(b∗)T Xi
It can be proved using strong law of large numbers that1n log S∗
n −→ W ∗ with probability 1.S∗
n increases exponentially with W ∗.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Kuhn-Tucker Characterization of Log OptimalPortfolios
The necessary and sufficient conditions for log optimumportfolio are
E
(Xi
(b∗)T X
)= 1 if b∗
i > 0< 1 if b∗
i = 0
From above it follows that
E( S
S∗
)≤ 1 and also E
(log
SS∗
)≤ 0
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Outline
1 Information Theory in GamblingIntroduction to Horse Race ProblemMaximizing ReturnGambling and Side Information
2 Information Theory in Stock MarketTerminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
3 Direction of Future Work
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Use of Alternate Probability Distribution
b∗f be the log-optimal portfolio according to f (x).
b∗g be the log-optimal portfolio according to g(x) (some
other density).
The increase in doubling rate by using b∗f instead of b∗
g canbe expressed as
∆W = W (b∗f )−W (b∗
g)
It can be proved that ∆W is bounded on the top byD(f ‖ g)
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate
Extending to Side Information
Say, the investor knows the outcome of some event Y = y .
He now uses the probability distribution f (x |y).
Use the result in the preceding slide, it can be shown thatincrease in doubling rate due to the side information isbounded by I(X ; Y ).
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Immediate Goals
Complete the problem sets given in ’Elements ofInformation Theory’ and the additional problems availableat http://www-isl.stanford.edu/ jat/eit2/download.htm .T. Cover. Universal Portfolios. Mathematical Finance, 1(1):1-29, January 1991.T. Cover and E. Ordentlich. Universal Portfolios with SideInformation. IEEE Transactions on Information Theory,42(2):348-363, March 1996.A comparative study of mean variance model(Markowitz),log-optimal portfolio’s, and Cover’s Universal Portfolio(CUP).Implementing these 3 models within the frame work ofIndian Market(or maybe the US market) and compare theirperformance. (Implementation in MATLAB or C++)Expected Time : Around 6 -7 weeks.
Kashyap Arora Dual Degree Seminar
Information Theory in GamblingInformation Theory in Stock Market
Direction of Future Work
Long-Term Goal
A comprehensive understanding of various work done inthis area. It is a very niche area with not many peopleworking on it. I would like to have a really good idea of thelimitations as well as the advantages of using concepts ofInformation Theory in the Stock Market.
Kashyap Arora Dual Degree Seminar