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
[email protected], [email protected]
ABSTRACT
The essential mathphysics 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 mathphysics 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 [14], neuroscience [511], 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. IndexComponent 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 BlackScholes (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 OrnsteinUhlenbeck 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 shorttime 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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 Winter '11
 BARNARD
 Physics, mechanics, pH, Statistical Mechanics, The Land, Probability theory, Stochastic process, Mathematical finance, Stochastic volatility, Lester Ingber, L. Ingber

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