On Arbitrage Possibilities via Linear Feedback in an Idealized - - PowerPoint PPT Presentation
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,
Workshop on Uncertain Dynamical Systems Udine, Italy, 23-26 August, 2011
University of Wisconsin barmish@engr.wisc.edu James A. Primbs Stanford University japrimbs@stanford.edu
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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)
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To illustrate for simplest case with continuously differentiable prices with initial conditions
Other Idealized Markets
Let
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Superposition of two linear feedbacks: with trades individually satisfying
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The long trade: The short trade: The aggregate trade: This can be implemented as single trade by “netting out” as above.
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Arbitrage-Like Result
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Consider the function F(p) is minimized at p = 1 More formally,
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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. Note that I can be a feedback; that is, and we obtain In the linear feedback case,
IMPORTANT The stock trader should not assume knowledge of the drift mu and the volatility sigma. This amounts to cheating.
With investment I, the incremental trading gain or loss is
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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.
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ISSUES TO CONSIDER
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The expectation and variance resulting from the SLS feedback controller are given by Moreover, and
TRUE ARBITRAGE?
Robustly Non-negative!
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Preliminaries: Define the quantities
Can you remember all this?
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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
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We plot the PDF for and obtain
REMARKS
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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%.
We use the pdf of Theorem 3 to generate With parameters given by
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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 and weighted raw return
Convergence: Partial Sums of R
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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:
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Gain K = 4, initial investment I(0) = 10,000 and reset triggered when long or short position goes below 2,000
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Discrete-Time Implementation
The Basic Equations: Add Sign Requirements: Add Margin Requirements:
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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.
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Green = LONG Red = SHORT Black = TOTAL
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Parameters K = 4, V(0) = 10,000, Delta = 1.333, r = 0.05
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Parameters K = 4, V(0) = 10,000, Delta = 1.333, r = 0.05
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