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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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 2 / 25
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”
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 3 / 25
This Talk’s Aim?
H&K’s paper mathematically intensive and formal
Will try to cut through and distill the intuition
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 4 / 25
Let’s Talk Financial Modeling
Currently, two most popular approaches to investing:
Average-case: The “classical” modelWorst-case: Universal Portfolio Selection
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 5 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 6 / 25
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)
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 7 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 8 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 9 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 10 / 25
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)
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 11 / 25
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)
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 12 / 25
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]
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 13 / 25
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)
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 14 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 15 / 25
What are the Expectations?
If an investor trades more frequently and stock returnsequence exhibit low variation, then algorithm shouldperform better
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 16 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 17 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 18 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 19 / 25
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
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 20 / 25
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]?
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 21 / 25
Questions & Answers?
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 22 / 25
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.
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 23 / 25
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.
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 24 / 25
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.
Tam Le (Columbia University) Bridging Investment Models November 22, 2011 25 / 25
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