Bridging Investment Models - Columbia Universityaho/cs6998/lectures/11-11-22_Le_Investment.pdf ·...

Bridging Investment Models

Tam Le

Columbia University

November 22, 2011

Tam Le (Columbia University) Bridging Investment Models November 22, 2011 1 / 25

Reference Paper

On Stochastic and Worst-case Models forInvesting

Elad Hazan & Satyen Kale

Presented December 22, 2009 at Conference on NeuralInformation Processing Systems (NIPS)

http://books.nips.cc/papers/files/nips22/NIPS2009 0470.pdf


Paper’s Aim?

In practice, much of mathematical finance uses twoapproaches to model stock prices and deviseinvestment strategies

Hazan & Kale tie these two approaches to get“best of both worlds”


This Talk’s Aim?

H&K’s paper mathematically intensive and formal

Will try to cut through and distill the intuition


Let’s Talk Financial Modeling

Currently, two most popular approaches to investing:

Average-case: The “classical” modelWorst-case: Universal Portfolio Selection


Average-case Investing

Long-standing model for stock prices[Bac1900] [Osb1959]

Probabilistic, using Geometric Brownian Motion

Has enjoyed great predictive success

Trillions of dollars traded every year

Black–Scholes [BS1973] won Nobel Prize for work onstock pricing options using this model


The Basic Math...

For stock price S at time t

dSt = St(µdt+ σdWt)

µ – drift, long-term trend of stock prices

σ – volatility, deviations from long-term trend

Wt – Wiener process, Brownian motion

Can be shown that

St = S0 exp((µ− σ2/2)t+ σWt)


Drawbacks?

Very susceptible to major deviations from the model

1987 stock market crash (Black Monday)

1997 Asian Financial Crisis → 1998 RussianFinancial Crisis

2007-Present global financial crisis


Another Approach: Worst-case Investing

T. Cover’s Universal Portfolio Selection [Cov1991]

Motivation? Fragility of average-case againstinfrequent but dramatic deviations

No statistical assumptions like GBM

An“Online” learning approach


The Worst-case Investing Approach

Investor iteratively distributes wealth over n assetsbefore observing change in price

At each period t = 1, 2, ... investor commits ton-dimensional distribution of wealthpt ∈ ∆n ≡ {

∑i pi = 1 and pi ≥ 0 ∀i}

Investor observes price relative vector rt, where

rt(i) =closing price of the ith asset at trading period t

opening price


The Worst-case Investing Approach

t 1 2 3 4

rt 1.5 1 1 0.5

pt 0.5 0 0.5 0

⇒ New wealth = Old wealth ×(1.5 · 0.5 + 1 · 0 + 1 · 0.5 + 0.5 · 0) = 1.25

Overall change in wealth is∏

t(rt · pt)

Objective is to maximize wealth:

log∏t

(rt · pt) =∑t

log(rt · pt)


Universal Portfolio Selection

Cannot hope to maximize wealth, so compare to abenchmark: Constant Rebalanced Portfolio (CRP)

Investor’s regret is difference between actual selectionsand benchmark CRP in hindsight:

regret = Optimal wealth− Algorithm wealth

≡ maxp∗∈∆n

∑t

log(rt · p∗)−∑t

log(rt · pt)


Drawbacks?

Dependence of regret on number of trading periods

Why does online algorithm have regret growth ratetied to number of periods?

Increase trading frequency ⇒ increases T ⇒ expecthigher regret?

Not the case at all [AHKS2006]


Comparison of Selection Algorithms

[Cov1991]Regret: O(n log T )Runtime: O(T n)

[HSSW1996]Regret: O(

√T log(n))

Runtime: O(n)

[KV2003] Cover implementationRuntime: O(n7T 8)

[HKKA2006] Online Newton algorithmRegret: O(n log T )Runtime: O(n3)


Tying Together Approaches: What’s the Objective?

Show regret of a good online algorithm depends ontotal variations in sequence of stock returns ratherthan on number of iterations


What are the Expectations?

If an investor trades more frequently and stock returnsequence exhibit low variation, then algorithm shouldperform better


How was Objective Achieved?

Used online convex optimization problem [Zin2003]which generalizes the universal portfolio model

In particular, for portfolio selection problem, use − logas the convex cost function


Some of the Math...

Regret bounds expressed as function of quadraticvariability Q of returns rt

Q(r1, ..., rT ) = minµ

T∑t=1

||rt − µ||2

Minimized at µ =1

T

T∑t=1

rt

How was regret bounded by observed variation instock returns?

Technical meat of paper, won’t go into detail hereEssentially used a generic Follow-The-Leader (FTL) onlineoptimization algorithm


The Results?

Regret ≤ O

(log∑t

||rt − µ||2)

= O(logQ)

= Q ≤ T (after normalization)

Improves on log T

Runs in time O(n3), i.e. efficientImplications for the GBM

Variation “essentially” independent of trading frequencyIncreasing trading frequency lowers risk of the best CRP


Conclusions...

Bridged two well-known financial modelingapproaches, retaining benefits while minimizingdrawbacks

Presented efficient algorithm minimizing regret,improving on state of the art

For portfolio selection problem, regret now bounded interms of observed stock return variations, NOTtrading frequency


Future Work

[DKM2006] presented game-theoretic framework foroptions pricing — perhaps analysis here can beapplied to their framework?

Assumed no transaction costs for trades — extend thismodel to take cost into account similar to [BK1997]?


Questions & Answers?


References

[Bac1900] Bachelier, L. (1900) Theorie de la speculation.

Annales Scientifiques de l’Ecole Normale Superieure, 3(17):21-86.

[Osb1959] Osborne, M.F.M. (1959) Brownian motion in thestock market. Operations Research, 2:145-173.

[Osb1959] Black, F. and Scholes, M. (1973) The pricing ofoptions and corporate liabilities. Journal of Political Economy,81(3):637654.

[Cov1991] Cover, T. (1991) Universal Portfolios. MathematicalFinance, 1:1-19.


References

[AHKS2006] Agarwal, A., Hazan, E., Kale, S., & Schapire, R.E.(2006) Algorithms for Portfolio Management Based on theNewton Method. In ICML, 9-16, 2006.

[HSSW1996] Helmbold, D. P., Schapire, R. E., Singer, Y., andWarmuth, M. K. 1996) Online portfolio selection usingmultiplicative updates. In ICML, pages 243251.

[KV2003] Kalai, A. and Vempala, S. (2003) Efficient algorithmsfor universal portfolios. J. Mach. Learn. Res., 3:423440.

[HKKA2006] Hazan, E., Kalai, A., Kale, S., & Agarwal, A.(2006) Logarithmic Regret Algorithms for Online ConvexOptimization. In COLT, 499-513.


References

[Zin2003] Zinkevich, M. (2003) Online Convex Programming andGeneralized Infinitesimal Gradient Ascent. In ICML, 928-936.

[DKM2006] DeMarzo, P., Kremer, I., and Mansour, Y. (2006)Online trading algorithms and robust option pricing. In STOC06: Proceedings of the thirty-eighth annual ACM symposium onTheory of computing, pages 477-486.

[BK1997] Blum, A. and Kalai, A. (1997) Universal portfolios withand without transaction costs. In COLT, pages 309-313.


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