Download KDS Quantum Option Model

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KDS Quantum Option Model
Chris Pei, Hanson Zhang, Leo Liu, John Wang
Knowledge Decision Securities, LLC, San Francisco, California, USA
Abstract: KDS option model is based on the quantum electrodynamics (QED). The classical random
walks are extended to quantum walks. The quantized stock price states are modeled along onedimensional tight-binding quantum chain. The wave function of such a quantum chain is described by
the Schrodinger equation and the probability density function is governed by the generalized master
equation of quantum diffusion. Numerical solutions can be obtained via Monte–Carlo simulations. The
resulting density distribution has well defined front shape. One key feature of our model is the
introduction of asymmetric quantum diffusion along positive and negative states to account for investor
asymmetric risk preference. Our QED-based model unifies many classical option models (BS, VG, DE)
as special cases but most importantly extends to other endless possibilities from quantum ballistic limit
to quantum super-diffusive regime.
Keywords: QED, Option pricing, Trading, Real time, Calibration, Monte-Carlo
1 Introduction
The Pricing of Options has long been a valuable topic [Black and Scholes 1973; Merton 1973].
Recently, Quantum theory was brought into this field with the purpose of better pricing [Baaquie2004;
Tsay 2005].
In this paper, we present a new pricing model of options based on the quantum electrodynamics (QED).
Stock prices are modeled as discrete states of quantum wave propagation along a quantum chain. The
wave function is described by the Schrodinger equation and the resulting probability density function
is governed by the generalized master equation of quantum diffusion. One noble feature of our model
is the introduction of asymmetric quantum diffusion to account for investor psychology of asymmetric
biases for positive and negative outcomes. Our QED based model unifies many classical option models
as special cases but most importantly extends to other endless possibilities from quantum ballistic limit
to quantum super-diffusive regime in the description of stock price movement.
2 QED Trading Model
The QED trading model utilizes the quantum field theory path integral formulation. The calibration
schema leverages the massively parallel Monte Carlo framework that utilizes real-time high frequency
market data. Our real-time trading execution architecture is shown in Figure 1.
The asymmetric nature of price jumps up and jumps down, occurring both intra-day and at market open,
was modeled using strong and weak forces to characterize and quantify the jump behaviors exhibited
in the quantum option chains and underlying indexes. This allows us to model the quantumized
structures from which we can simultaneously determine mispricings, particularly in highly unstable
market conditions.
To leverage the QED pricing model for equity derivatives trading applications, we must fully
understand and leverage the use of manifolds and fractals. The concept of manifolds within the QED
based universe is indispensable to the understanding of advanced mathematical physics, from
elementary particle physics, quantum electrodynamics, cosmological universe, to the grand unification
of the fundamental laws of nature. Manifolds allow physical processes in both the infinitesimally small
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and infinitely large scales to be expressed in a unified differential geometry, with more clarity in
understanding and greater accuracy in modeling. With respect to the unification of the fundamental laws
of nature, manifolds serve as the smooth phase spaces in both infinitesimal and infinite structures of
the universe, in which our views of space, time, momentum, and energy in general relativity have been
transformed from the classical Euclidean space to the four-dimensional Lorentzian manifolds. Figure 2
is a visualization of the manifolds.
Figure 1 KDS real time trading platform
Figure 2 QED real time surface topology
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In the context of quantum options trading subject to the risk neutrality constrains function, a manifold
represents the differentiable short range structures in a stochastic diffusion processes.
The four key components of the QED pricing framework which are important for derivatives trading
applications are the following:
• Monte-Carlo Simulation Calibration for all derivatives securities, such that the pricing model is
continuously calibrated using terabytes of real-time high frequency market data as well as historical
paths.
• Option trading volume and bid-ask spread data are taken into account in the calibration process.
• Using the entire spectrum of at-the-market (ATM) to far out-of-the-money (OTM) strikes within the
calibration process to account for the entire full volatility skew exhibited in the market prices and bidask data, and thereby allow accurate modeling of stochastic volatility.
• Advanced nonlinear optimization techniques and maximum-likelihood estimation methods are used to
optimize calibrated parameters and ensure the Global Minimum Solution (GMS).
In addition to the above factors for trading, the framework allows in-depth dissection of the computed
data under various scenarios, which can be used for market analysis, risk management, and academic
research purposes. This ability to "slice and dice" the pricing results, include the premium decay timeseries of trades and portfolios, as well as additional Monte Carlo analytics and calibration data are all
streamlined into a proprietary database system, so that extremely fast and accurate real-time slicing of
data can be done for both trading and risk management purposes, including client/investor ad hoc
requests. Profit and loss (P&L) of equity derivatives trading strategies can be described as Leptokurtic
distributions. The Leptokurtic P&L Concentration functions and graphs are fully integrated within the
QED pricing, and provide traders the real-time distribution on an asset- and portfolio- level in order to
better gauge skew of how much or little a given trade is affecting overall portfolio probability. Figure 3
use TNA data to illustrate this integration. When the real time surface topology trade selection is
identified via the curvature implied mispricing, a further "drill-down" analysis of the trade shows the
P&L profile of that exact trade with respect to the real-time probability distribution.
The equity-based trading strategies and mispricing signals are amplified when combined with set of
leading signal based models that utilize cross-asset correlation, such as leveraging interest rate models,
real-time credit default swap (CDS) market quotes, and correlation analysis of CDS against other asset
classes. In our analysis, we have found that CDS provides extremely valuable signals integrated within
the equity derivatives and futures trading framework.
The QED framework utilizes multiple time dependent factors to allow for an extremely accurate
description of short-range jump dynamics, long-range jumps dynamics, volatility skew and ability to
precisely capture the pair-wise interactions between long vs. short index exchange-traded funds (ETFs),
and leveraged ETFs.
Among the trading strategies within which the QED model serves as a powerful trading tool are
American short-range jump techniques. With respect to the American short-range universe, the QED
achieves the following strategies:
• High Probability Mean Reversion (HPMR) strategies
• Time decay (Theta) strategies
• Spread based strategies (vertical/calendar spreads)
• Underlying long/short strategies
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Figure 3 TNA Sep12-100x 40P / -100x 65C / 100x 71C – iron condor
The above strategies utilize elements from the traditional fund categories including Volatility Arbitrage,
Statistical Arbitrage, Algorithmic Trading, and Long/Short Equity Growth.
With respect to the Theta (Time Decay) trades, we isolate the theta risk by trading in the 80-90%
confidence range (at maturity). This trade strategy is leveraging a much shorter time period for option
prices to converge. Thereby, we can construct premium-based trading strategies (i.e. Sell Put/Call trades)
to take advantage of not only the probability of non-exercise, but also the speed of decay. Timing of
entry is very critical as confidence levels can quickly shift when market experiences short range jumps.
Figure 4 shows implied price distribution for SPY, which can be used to guide trading.
The Spread strategy utilizes the HPMR strategy at lower probabilities, and in exchange for higher
premium received, the trader can then buy-back risk at even further OTM strikes. This selling and
buying back of different strikes results in a "spread" which can be done vertically (i.e. same maturity) or
calendar style (i.e. various maturity). Thereby allowing traders to utilize theta and vega combinations in
these trades, as well as creating various limited downside scenarios for risk management.
Lastly, we shall discuss the applications of QED with respect to Delta/Gamma Hedging and Mark-toMarket (MTM) Risk Management. Using QED Greeks sensitivity analysis, we are able to select a
"Delta Neutral" and "Gamma Neutral" hedge which is calculated and identified by the process as the
optimal pairing. This calculation of exact dollar amount of Delta and Gamma hedges per respective
short-range strategy gives traders definite MTM risk management solution, while also giving traders the
flexibility to tailor the hedge corresponding to the fund’s required hedge ratio. The Gamma risk is the
second-order effect (similar to bond convexity) which is quite noticeable when high confidence trades
move to lower confidence levels, and the offsetting hedges no longer "neutralize" the MTM risk. To
avoid this Gamma risk, we can use QED to employ the Gamma neutral strategy which is derived as an
exact dollar amount which the trader needs to adjust in order to keep the portfolio neutral.
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(a) 10-day histogram based on 3-month calibration
(b) 60-day histogram based on 3-month calibration
(c) Comparison between 1-month and 3-month calibration
Figure 4 Implied price distributions for SPY
By analyzing the manifolds and fractals in the context quantum electrodynamics field theory, we can
very quickly and clearly determine the structure of the financial markets function. Furthermore, this tells
us when trading opportunities as well as abnormal activities leading to "black swan events" can happen.
Not only are we monitoring the signals in context of quantum field theory in real-time in order to detect
the market abnormal signals (and hence black swan events), we are applying and triggering business
logics that mitigate risks while setting up strategies to further profit from a variety of market
abnormalities and potential risk signals. These applications are not only powerful in the equity
derivatives space, but also highly applicable to index futures markets, credit and bond markets, and
over-the-counter (OTC) markets which are across various sectors and asset classes.
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3 Results
The KDS quantum option model offers extremely accurate real-time tracking of options pricing
topology and associated Greeks. The pricing accuracy is comparable to the spread of market data, as
illustrated in Figure 5 and Figure 6. The real-time tracking is achieved well within one-minute resolution.
To the best of our knowledge, none of the existing options pricing models in the market can match our
proprietary model in terms of the accuracy and speed to track and trade the market data.
This QED based model, together with the host of products and services provided by KDS, is expecting
to bring significant value to institutional and retail investors.
(a) The OTM option price as a function of strike for
Apple stock. The solid line represents the model and
the circles are the market data on January 22, 2013
for options with 62 days of maturity
(a) The OTM option price as a function of strike for
the SPY exchange traded fund. The solid line
represents the model and the circles are the market
data on August 30, 2012 for options with 58 days of
maturity
(b) The OTM option price as a function of strike for
Apple stock. The solid line represents the model and
the circles are the market data on January 22, 2013
for options with 37 days of maturity
(b) The out-of-money (OTM) option price as a function of
strike for the SPY exchange traded fund. The solid line
represents the model and the circles are the market data
on August 30, 2012 for options with 26 days of maturity
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(c) The OTM option price as a function of strike for
Apple stock. The solid line represents the model
and the circles are the market data on January 22,
2013 for options with 18 days of maturity
(c) The OTM option price as a function of strike for the
SPY exchange traded fund. The solid line represents the
model and the circles are the market data on August 30,
2012 for options with 16 days of maturity
Figure 5 Results for Apple stock
Figure 6 Results for SPY exchange traded fund
References
[1]. BLACK, F., and SCHOLES, M. 1973. The Pricing of Options and Corporate Liabilities. The
Journal of Political Economy 81, 3, 637–654.
[2]. MERTON, R. C. 1973. Theory of Rational Option Pricing. Bell Journal of Economics 4, 1, 141–
183.
[3]. BAAQUIE, B. E. 2004. Quantum Finance. Path Integrals and Hamiltonians for Options and
Interest Rates. Cambridge: Cambridge University Press. xv, 316 p.
[4]. TSAY, R. S. 2005. Analysis of Financial Time Series (Wiley Series in Probability and Statistics),
2nd ed. Wiley-Interscience.
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