On Arbitrage Possibilities via Linear Feedback in an Idealized Market B. Ross Barmish James A. Primbs University of Wisconsin Stanford University barmish@engr.wisc.edu japrimbs@stanford.edu Workshop on Uncertain Dynamical Systems Udine, Italy, 23-26 August, 2011
Outline for This Talk • Feedback Control for Stock Trading • Classical Linear Feedback in C1 Markets • Geometric Brownian Motion Markets • Practical Considerations and Back-Testing • Concluding Remarks and Future Research 2
Feedback Control View Reactive Versus Predictive Modelling 3
Feedback Control in Idealized Markets • Idealized Market Model • Proving Ground Concept • Continuously Differentiable • Geometric Brownian Motion • Price Differentiation Issue • LTI Feedback Controllers • Simultaneous Long-Short • Any Arbitrage Possibilities? 4
Arbitrage-Like Results • Get g(t) > 0 or E(g(t)) > 0 • Cannot exploit IM Model • Still not quite an arbitrage • Role of Risk-free return • Stock Correlation to Market • The CAPM Theory to Consider 5
Sampling Background Literature Finance: Some highlights include Fama and Blume (1966), Gencay (1988), Brown and Jennings (1989), Brock et al (1992), Frankel and Froot (1996), Neely (1997), Lo et al (2000) Control: Meindl and Primbs (2004, 2008), Primbs (2007), Primbs and Sung (2008), Mudchanatogskul et al. (2008), Barmish (2008, 2010), Califiore (2008, 2009), Iwarere and Barmish (2010), Barmish and Primbs (2011) plus literature on stochastic differential equation approaches. Some Key Issues Addressed: market efficiency, chart patterns, moving average rules, benefits of timing, portfolio optimization, volatility, hedging, pairs trading (closest in flavor to today’s talk) 6
Minimal Notation p(t) = PRICE OF THE STOCK I(t) = AMOUNT INVESTED IN STOCK g(t) = GAIN or LOSS ON STOCK V(t) = ACCOUNT VALUE I(t) < 0 means short sale 7
Idealized Market • Continuous trading a la Black-Scholes • Costless Trading (no commissions, fees) • Perfect Liquidity (bid, ask, fills) • Trader is assumed to be price taker • Adequate Resources (no margin calls) • Class of Market Price Variations • C1 and Geometric Brownian Motion 8
Outline for This Talk • Feedback Control for Stock Trading • Classical Linear Feedback in C1 Markets • Geometric Brownian Motion Markets • Practical Considerations and Back-Testing • Concluding Remarks and Future Research 9
Idealized Market Dynamics To illustrate for simplest case with continuously differentiable prices Let Other Idealized Markets • Geometric Brownian Motion • Classes of Bull Markets • Classes of Bear Market • Many Others Possible with initial conditions
Special Case: Linear Feedback With linear control and adequate resources the state equation solution becomes • Signs of I_0,K • Long or Short • Can lose money with resulting investment • Round trip g(T) • How to win? • Arbitrage Thm 11
Simultaneous Long-Short (SLS) Linear Feedback Superposition of two linear feedbacks: with trades individually satisfying
Simultaneous Long-Short (SLS) Concept The long trade: The short trade: The aggregate trade: This can be implemented as single trade by “netting out” as above. 13
Simultaneous Long-Short (SLS) Gains and Losses Summing the two solutions And w.l.o.g. taking p(0) = 1, I(0) = 1, obtain Arbitrage-Like Result 14
Why Arbitrage-Like? Consider the function More formally, F(p) is minimized at p = 1 15
Outline for This Talk • Feedback Control for Stock Trading • Classical Linear Feedback in C1 Markets • Geometric Brownian Motion Markets • Practical Considerations and Back-Testing • Concluding Remarks and Future Research 16
Idealized GBM Market The incremental return is given by dZ is a standard Brownian motion which can be viewed as N(0,dt) with drift mu and volatility sigma. With investment I, the incremental trading gain or loss is IMPORTANT The stock trader should not assume knowledge of the drift Note that I can be a feedback; that is, mu and the volatility sigma. This amounts to cheating. and we obtain In the linear feedback case,
Theorem 1: Trading Gains Along a GBM sample path p(t), the SLS feedback controller strategy leads to trading gain Note that the trading gain is independent of the Brownian motion drift parameter mu. We now look at the trading gains as a function of sigma and K. 18
Trading Gains and Loss Well ISSUES TO CONSIDER • Probability of Win • Mean Trading Gain • Variance of Trading Gain • More Generally, the PDF 19
Theorem 2: Free Lunch? The expectation and variance resulting from the SLS feedback controller are given by and Moreover, Robustly Non-negative! TRUE ARBITRAGE? • Depends on Risk-Free Return 20 • Invariance to Market: CAPM • My Current Work with Primbs
PDF for Trading Gain Preliminaries: Define the quantities Can you remember all this? 21
Theorem 3: PDF for Trading Gain For ,the probability density function f(x,t) for the SLS trading gain g is given as follows: For f(x,t) = 0 and for Now, what does this all mean in terms of the attractiveness of an SLS trade? 22
PDF For Trading Gain We plot the PDF for and obtain REMARKS • For SLS: wlog, mu > 0 • When mu large: Look @ P(win) • When mu small: Look @ P(win) • Limited loss due to g* • E[g] as a function of mu 23
Attractiveness of SLS Attractiveness governed by For example, when This implies 45% plus return When mu is small P(win) can be small but recall P(g < g*) is zero. For example, in the driftless case with mu = 0, we have g* = -.07. This means worst case loss is 7%. 24
Attractiveness of SLS (2) With parameters given by We use the pdf of Theorem 3 to generate
Trading GBM Sample Pathes • Massive Monte Carlo Simulation • More than 2M Sample Pathes • Pick uniform distribution • Pick 40K sigma values in • Use I_0 = 1, K = 4, N = 252 • Run GBM + SLS Simulation • Keep Track of Statistics 26
Evaluation of Performance For each value of mu, let the i-th trading gain and let be the associated average investment. Now we define raw return We define weighting factor Convergence: Partial Sums of R and weighted raw return 27
Evaluation of Performance (2)
Evaluation of Performance (3)
Outline for This Talk • Feedback Control for Stock Trading • Classical Linear Feedback in C1 Markets • Geometric Brownian Motion Markets • Practical Considerations and Back-Testing • Concluding Remarks and Future Research 30
Some Practical Considerations • Use of Leverage and Margin • Controller Saturation • Controller Reset • Inclusion of Risk-free Asset 31
Improvement: Control Reset Motivation provided by simple intuitive example When t = 50, long position is thriving and short position is diminished nearly to zero. If we leave the controller alone, we will arrive at break-even by t = 100. So we reset the controller back to initial investment values when either of the conditions below is satisfied: 32
Sinusoidal Prices With Reset Gain K = 4, initial investment I(0) = 10,000 and reset triggered when long or short position goes below 2,000 33
A Little Back-Testing Discrete-Time Implementation The Basic Equations: Add Sign Requirements: Add Margin Requirements: 34
Back-Testing: Trading the Nasdaq • Exchange traded fund (ETF) called QQQQ tracks • So we trade the “quadruple Q’s” instead of Nasdaq • Two year period beginning September 1, 2006 • Precedes beginning of the market crash in 2008 and includes some nice round trips for testing. SLS Controller Parameters We take control gain K = 8, initial investment and the account value to be $10,000, leverage = 2V, so gamma = 2 and reset the trade when either the long or the short investment falls below 20% of its initial value; that is below $2000. 35
Simulation Results Green = LONG Red = SHORT Black = TOTAL 36
IBM Two Year Bull Market Begins 8/13/05 Parameters K = 4, V(0) = 10,000, Delta = 1.333, r = 0.05 37
IBM Bear: Reverse Prices Parameters K = 4, V(0) = 10,000, Delta = 1.333, r = 0.05 38
Outline for This Talk • Feedback Control for Stock Trading • Classical Linear Feedback in C1 Markets • Geometric Brownian Motion Markets • Practical Considerations and Back-Testing • Concluding Remarks and Future Research 39
Concluding Remarks • Positive Gain g(t) Versus Arbitrage • Discounting Using Risk-free Return • CAP Consideration: dp/p = beta*dM/M + phi • Adaptive Control; e.g., K(history) • Consideration of Portfolio; e.g., S&P 40
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