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ssrn-id285117

# ssrn-id285117 - Statistical Mechanics of Portfolios of...

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Statistical Mechanics of Portfolios of Options Lester Ingber Lester Ingber Research POB 06440 Sears Tower, Chicago, IL 60606 and DRW Inv estments LLC 311 S Wacker Dr, Ste 900, Chicago, IL 60606 ABSTRACT The essential math-physics and associated numerical algorithms underlying a reasonable approach to trading a portfolio of options (PO) is outlined. A description is given of risk- slides, asset disbursement, dynamic balancing, and value indicators. Ke ywords: Statistical Mechanics; Trading Financial Markets; Simulated Annealing 1. INTRODUCTION The essential math-physics and associated numerical algorithms underlying a reasonable approach to trading a portfolio of options (PO) is given. To perform these analyses, we use methods of nonlinear nonequilibrium multivariate statistical mechanics and associated numerical algorithms which have been tested in several systems, e.g., that typically arise in such diverse fields as finance [1-4], neuroscience [5-11], and combat simulations [12,13]. All aspects presented here have been tested with working code on real market data. However, we do not give numerical results as that specificity is currently considered to be proprietary. This paper stresses hedging a -hedged option on an index with baskets of -hedged options on its components. However, it is clear that the same methodology can be applied to any option portfolio management, simply by altering specific constraints imposed by the definition of an index. 2. BASIC FORMALISM 2.1. Index-Component Volatility Relationship Consider an index B with N components S i . B = N i Σ ω i S i (1) This underlying constraint is true in differential form, dB = N i Σ ω i dS i (2) Consider each of these N + 1 markets as defined as a stochastic process, consistent with the stochastic processes traders use to define their option processes. dB = f B dt + ˆ g B dw B dS i = f i dt + ˆ g i dw i < dw i > = 0 < dw i ( t ) dw j ( t ) > = ρ ij δ ( t - t ) dt ρ ii = 1 (3)

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Statistical Mechanics of Portfolios of Options - 2 - Lester Ingber The variance of dB is calculated straightforwardly in terms of its correlated components, < dB 2 > - < dB > 2 = ( ˆ g B ) 2 < dB 2 > - < dB > 2 = < ( i Σ w i dS i ) 2 > - < i Σ w i dS i > 2 ( ˆ g B ) 2 = i Σ j Σ ω i ω j ρ ij ˆ g i ˆ g j (4) 2.2. Specific Option Models For the Black-Scholes (BS) model, f B = b B B ˆ g B = σ B B f i = b i S i ˆ g i = σ i S i ( σ B B ) 2 = i Σ j Σ ω i ω j ρ ij σ i σ j S i S j (5) As another example, the Ornstein-Uhlenbeck model is defined with a constant diffusion term. In general, we can treat quite generally nonlinear models of underlyings and their derivatives [4,14]. 2.3. Probability Distributions To dev elop probability distributions, the correlated dw processes are decomposed into independent dz processes, ˆ g i dw i = m Σ ˆ g i m dz m < dz m > = 0 < dz m ( t ) dz n ( t ) > = δ ( t - t ) δ mn dt (6) The covariance matrix is defined by g ij = m Σ ˆ g i m ˆ g j m (7) The short-time conditional probability is developed as P [ S ; t | S ; t ] = 1 (2 π dt ) N /2 g 1/2 exp( - Ldt ) L = 1 2 i Σ j Σ ( dS i dt - f i ) g ij ( dS j dt - f j ) g = det( g ij ) ( g ij ) = ( g ij ) - 1 (8) where L is the proper Lagrangian of the system.
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ssrn-id285117 - Statistical Mechanics of Portfolios of...

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