application of information theory in finance

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



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




Introduction to Horse Race ProblemMaximizing ReturnGambling and Side Information

Outline








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.





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.





Outline








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




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.





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 .





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!





Outline








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).





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.




Terminology: Stock Market and PortfolioLog-Optimal PortfolioOptimality of Log-Optimal PortfoliosSide Information and Doubling Rate

Outline








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 .





Outline








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∗).





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 ∗.





Outline








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





Outline








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)





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 ).




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.




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.


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