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Unformatted text preview: Statistical Mechanics of Portfolios of Options
Lester Ingber
Lester Ingber Research
POB 06440 Sears Tower, Chicago, IL 60606
and
DRW Investments LLC
311 S Wacker Dr, Ste 900, Chicago, IL 60606
ingber@ingber.com, ingber@alumni.caltech.edu
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 riskslides, asset disbursement, dynamic balancing, and value indicators.
Keywords: 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 .
N B = Σ ω i Si (1) i This underlying constraint is true in differential form,
N dB = Σ ω i dS i (2) i 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 + g B dw B
ˆ
dS i = f i dt + gi dw i
ˆ
< dw i > = 0
< dw i (t ) dw j (t ′) > = ρ ij δ (t − t ′) dt
ρ ii = 1 (3) Statistical Mechanics of Portfolios of Options 2 Lester Ingber The variance of dB is calculated straightforwardly in terms of its correlated components,
< dB2 > − < dB >2 = ( g B )2
ˆ
< dB2 > − < dB >2 = < (Σ w i dS i )2 > − < Σ w i dS i >2
i i ( g B )2 = Σ Σ ω i ω j ρ ij gi g j
ˆ
ˆˆ
i (4) j 2.2. Speciﬁc Option Models
For the BlackScholes (BS) model,
f B = bB B
gB = σ B B
ˆ
f i = bi S i
gi = σ i S i
ˆ
(σ B B)2 = Σ Σ ω i ω j ρ ij σ i σ j S i S j
i (5) j 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 develop probability distributions, the correlated dw processes are decomposed into independent dz
processes,
gi dw i = Σ gim dz m
ˆ
ˆ
m < dz m > = 0
< dz m (t ) dz n (t ′) > = δ (t − t ′)δ mn dt (6) The covariance matrix is deﬁned by
gij = Σ gim g m
ˆ ˆj (7) m The shorttime conditional probability is developed as
1
exp(− Ldt )
P [ S ; t ′ S ; t ] =
(2π dt ) N /2 g1/2
L= 1 Σ Σ( 2i j dS i
dS j
− f i ) gij (
− f j)
dt
dt g = det( gij )
( gij ) = ( gij )−1 (8) where L is the proper Lagrangian of the system.
ˆ
The algebra above is developed in the prepoint Ito discretization, sufﬁcient for this exposition. Details of
the induced Riemannian algebra and deeper understanding of the metric gij are given in other Statistical Mechanics of Portfolios of Options 3 Lester Ingber references [1,1518].
2.4. Implied Volatility Metric
The only practical way of getting the covariance matrix is from historical data. Since the options are
taken on univariate underlyings, implied volatilities backed out from the option prices do not have these
correlations. However, we build a covariance matrix for implied volatilities by scaling historical
volatilities as they appear in bilinear form in gij ,
g IV i = si gi
ˆ
ˆ (9) This leads to new gij ’s and g’s. In this way we handle the covariances for different strikes regions.
2.5. Stochastic Volatility
Since we will want to examine scenarios that may change by standard deviations (StdDevs) of volatilities,
we formulate a 2 N + 2 system, including stochastic volatilities.
d σ i = ε dt + ξ dv i
{ dS i } → { dM i } = { dS i , d σ i } (10) Note that the structure of the extended 2 Nx 2 N symmetric covariance matrix is in terms of three NxN
matrices,
gij = A C sym C CoVar( dS dS ) CoVar( d σ dS ) =
B CoVar( dS d σ ) CoVar( d σ d σ ) (11) Experience suggests that the d σ distributions are much narrower than the dS distributions, so that we use
for the mean of the d σ the differenced usual historical or implied volatility of dS . We have examined this
in the context of full 2factor stochastic volatility models for some markets [19].
3. RISK SLIDE
3.1. Greeks
Consider a position Π containing an options and its underlying, perhaps in ∆hedged proportions. The
change in P/L associated with changes in time dt , underlying dS and volatility d σ is given by a sum over
the “Greeks,” as derived from a Taylor expansion of the differenced position,
dΠ = 2
2
3
∂Π
∂Π
∂Π
1∂Π
1 ∂Π
1 ∂Π
∂2 Π
dS 2 +
dσ +
dS d σ +
dσ 2 +
dS 2 d σ + . . .
dS +
dt +
2 ∂ dS 2
2 ∂σ 2
6 ∂ S 2 ∂σ
∂S
∂σ
∂t
∂ S ∂σ d Π = ∆ dS + 1
2 Γ dS 2 + Κ d σ + Θ dt + ∆′ dS d σ + 1
2 Κ′ d σ 2 + 1
6 Γ′ dS 2 d σ + . . . (12) This equation illustrates how different Greeks are properly scaled to appropriate “dollar” values [20]. The
Greeks and the covariance matrix are functions of strikes.
The Greeks are calculated using models selected by traders. They can draw from BS, OU or more general
unvariable or multivariable models [4,14,19].
3.2. Standard Deviation Moves
The question might be posed “if the index moves 1 or 2 StdDev in its underlying and/or in its volatility,
what will be the likely commensurate StdDev moves in its components?” At ﬁrst glance this question
appears to be illposed. The answer is not unique as there can be many combinations of moves in the
components that will give rise to a given move in the index. Thus, we approach this inverse problem.
Consider a trajectory through all components. On any one component node we can assume that a move in
StdDevs in units {2, 1, 0, +1, +2} is made in dS and/or d σ . (In practice, we have found it better to use
units of fractional StdDevs.) At each such node, we record the associated shorttime distribution, Statistical Mechanics of Portfolios of Options 4 Lester Ingber calculated by giving dM i the indicated movement. Associated with this trajectory is a move in the index
dB using Eq. (1), and in its volatility d σ B using Eq. (4). Consider taking a lot of such trajectories. We
coarsegrain into dB d σ B bins the associated values of the movements of the index.
Note that these coarsegrained bins are not used to represent any distribution of the index with respect to
dBd σ B . The appropriate distribution, Eq. (8), is already deﬁned by the speciﬁc option model used by
traders, which is used to develop Greeks, etc., e.g., the BS model.
3.3. Aggregation of Information
We gather and renormalize the nodeprobabilities in various ways. For example, the P/L, Eq. (12),
associated with a speciﬁc movement in the index can be calculated from its Greeks. This can be
compared to the likely movements in the components, by weighting by nodeprobabilities the component
P/L’s, Eq. (12), of only those paths that contribute to this speciﬁc move in the index. This information
also can be made Greekspeciﬁc, e.g., comparing Γ’s, etc.
3.4. MiniRisk Slide
If a basket of components smaller than the full set of components is chosen to represent the index, e.g.,
using algorithms described below under Dynamic Balancing, then a minirisk slide is developed using
only nodes of the riskslide paths corresponding to this basket.
After a basket is determined, a new coarsegrained set of index bins is easily developed using only this
subset of components. Their associated nodeprobabilities are used to develop a minirisk slide speciﬁc to
this basket.
3.5. Disbursement of Assets
First, the money invested in a basket of components should be related to the money invested in the index
B according to expected values,
B StdDev B = Basket StdDevBasket (13) This might be modiﬁed by the value indicator described below.
Second, the money invested in the basket of components should be dispersed according the expected
values of P/L’s of the components, with respect to the renormalized nodeprobability distributions of the
paths developed by the risk slide above.
3.6. ASA MULTI_MIN Sampling of Paths
The task of gathering trajectories for these risk slides is formidable. For example, consider 5 states each
for StdDev moves in dS i and d σ i , 25 states per component. For the DOW30, this would represent
2530 ≈ 1042 trajectories; for the SP500, this would represent 25500 ≈ 10700 trajectories!
We do not have to give up on this statistical mechanics approach. Rather, we can take advantage of this
formulation. We can use the nodeprobability distributions to deﬁne a “cost function” to use in a
maximumlikelihood importancesampling approach to ﬁnd a relatively few “good” paths, i.e., with not
too small probabilities, to effectively sample the huge combinatoric space. The cost function is the
effective action derived from Eq. (8),
Aeff = Ldt + 1
2 ln g + N
2 ln(2π dt ) (14) Adaptive simulated annealing (ASA) is a powerful sampling algorithm for statistically ﬁnding the global
minimum of a quite general nonlinear and/or stochastic system [21]. Among the over 100 OPTIONS
available for tuning systems is MULTI_MIN. When turned on with a coarse resolution speciﬁed for the
parameters to be optimized, a userdeﬁned number of close local minima are returned. We take the coarse
resolution to be simple integral values of the (fractional) StdDev moves. It seems even 100 trajectories
for the DOW30 gives a good representation useful for a risk slide. The actual number of trajectories
clearly is dependent on the market selected. Statistical Mechanics of Portfolios of Options 5 Lester Ingber The ASA OPTION ASA_SAMPLE illustrates how generatedparameter and acceptancecost probability
distributions can saved to use the power of ASA importancesampling for integrals. Here, we are
concerned with establishing a few good paths through the system, and we use the marketmodel
probability distributions for weights.
4. DYNAMIC BALANCING
In practice it is difﬁcult, if not desirable, to hedge the full set of components against the index. For
example, there are issue of liquidity, handling partial ﬁlls, not wanting to buy or sell particular
components at an unfavorable time, wanting to take some extra advantage of particular good buys at a
time, etc.
In practice, it may be most proﬁtable to dynamically hedge a basket of a subset of all the components of
an index, based on criteria of minimizing risk and maximizing proﬁt. The following algorithm does this
by “surﬁng” on a wobbly (stochastic) risk surface while presenting sorted and ranked alternatives of best
buys and sales of possible changes in baskets.
4.1. Minimizing Risk — StdDev Cost Function
The parameters ﬁt by ASA include the number of rounded lots of contracts bought or sold of the subset of
components. The cost function has two main parts, the constraints and the goodness of ﬁt function.
The goodness of ﬁt is the difference between the variance of the subset of components and the variance of
the full set of components the subset is trying to emulate.
There can be several kinds of constraints. To avoid multiple regions of similar volatility matches but with
different estimates of the index value, the sum of the weighted lots of components is constrained to be
within 10% of the index price. We can use a value indicator to ﬁx a speciﬁc component to be included or
not included in the basket optimization.
4.2. Maximizing Proﬁt — ASA MULTI_MIN Alternatives
MULTI_MIN is turned on during the basket optimization. Thus several minima are offered that have a
reasonable ﬁt to the volatility of the full set of components, i.e., all having some reasonable risk control.
All current bidask prices are used for each basket in the set that is returned. These are sorted and ranked
according to their current proﬁtability if the basket were to be changed. Another sorting and ranking is
performed of our value indicator, discussed in the next section, to offer a complementary guide as the best
current way to dynamically balance the portfolio.
5. GENERIC VALUE INDICATOR
6. Bubble Indicator
To take advantage of market inefﬁciencies, trades are initiated when it seem that an indicator, e.g., price,
volatility, volatility ratios, etc., are considered to be outliers and when it is expected that the indicator will
mean revert back to some statistical norm. We call the excursion out of and back into this statistical norm
a “bubble.”
The ﬁrst step is to use a stochastic model to determine the ﬁrst and second moments of the data. We can
use standard statistics which implicitly assumes a Gaussian normal model of the data, or we can use ASA
to ﬁt a relatively more general stochastic model to the data.
A window must be used to develop these moments. We can use traders’ judgments of these windows, or
use some autocorrelation analyses to try to ﬁnd some representative halflife of the system.
These kinds of ﬁts generally take in most of the data within a fraction of a StdDev of the data. Our
bubbles lie outside this set of data, so we perform secondary analyses/ﬁts on the outlier data to determine
its ﬁrst and second moments, histograms of heights, durations and areas of these bubbles to categorize
them to be used as trading indicators.
Since the market generally behaves asymmetrically when going up versus down, we typically develop two
sets of such bubble indicators. Statistical Mechanics of Portfolios of Options 6 Lester Ingber We can weight these indicators by the projected relatively longtime probability distributions of the
market variables, e.g., using our PATHTREE [14] or PATHINT codes [4] which have proven to be very
robust and precise in other systems as well [7].
6.1. CMI Correlates
The above bubble analyses are performed on insample training and outofsample testing data. However,
in realtime trading, is of course is to late to take advantage of a bubble if it already has reentered the
statistical norm.
Therefore, we look for correlations of another indicator with a shorter lifetime than the bubble indicator,
to use as a proxy for what is statistically expected of a given bubble as it leaves the statistical norm.
Good indicators to correlate with the duration or the area of bubbles are canonical momenta indicators
(CMI). These have proven useful in other market studies [3,22,23].
CMI are derived from our Lagrangian L . For example, the CMI of a given S i variable is
CMIi = dS i / dt − f i
∂L
=Σ
gij
∂S i
j (15) Even though the correlated cross terms are important, in practice it often is useful to collect the CMI
separately for each market, to compare with bubbles separately for each market.
In realtime, since we would like to sort and rank the bubbles as value indicators of expected payoffs, we
use the CMI as proxies for this purpose.
7. CONCLUSION
We have used methods of nonlinear nonequilibrium multivariate statistical mechanics to develop shorttime probability distributions, together with associated numerical algorithms, e.g., adaptive simulated
annealing (ASA), to give a detailed methodology for trading portfolios of options.
Riskslides are developed to assess distributions of P/L’s and of sets of speciﬁc Greeks across components
that are correlated with an index, as a function of fractional StdDev moves in underlyings and volatilities.
This approach also contributes to an algorithm for disbursement of component assets to be correlated with
a disbursement on the index.
We develop an algorithm for realtime dynamic balancing to determine sets of reasonably optimal subbaskets that satisfy risk constraints, while presenting alternative sorted and ranked buys and sells that
maximize proﬁts. This can be visualized as “surﬁng” a risk surface to beneﬁt from proﬁtable trades in the
riskmanagement process.
We develop a generic value indicators, based on Training and Testing sets of “bubbles” of exiting and
reentering statistical norms of indicators, e.g., prices, volatilities, ratios of other indicators, etc. Faster
indicators, canonical momenta indicators (CMI) are correlated to these bubbles to give sets of sorted and
ranked expected value indicators.
This methodology can be applied to any option portfolio management, simply by altering speciﬁc
constraints imposed by the deﬁnition of an index.
ACKNOWLEDGMENTS
I thank Donald Wilson for his ﬁnancial support. I thank Dave Hrencecin, Radu Mondescu, David
Muzzall, Mark Phelan, and Fred Schuster for useful discussions. Statistical Mechanics of Portfolios of Options 7 Lester Ingber REFERENCES
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