yin-yang volatility surface evolution - a dynamic structure for

Yin-Yang Volatility Surface Evolution- A dynamic structure for modeling

multi-level trends, swings and breakouts

Prof Dr PAN Heping, DirectorPrediction Research CenterUniversity of Electronic Science & Technology of China (UESTC)Chinese Institute of Intelligent FinanceEmail: [email protected]

Workshop on Quantitative Behavioral Finance (QBF)8.-10. Dec, 2010, Nice Sophia Antipolis University

http://math.unice.fr/CIF/

Pan Heping [email protected] 2010-12-8 2

About Me – PAN Heping 潘和平

I have three parallel life experiences

1. A Scholar- Philosopher, Scientist, Professor, ……- Chair of the first 3 International Workshops on Intelligent Finance IWIF-I 2004, Melbourne, sponsored by ANZ BankIWIF-II 2007, Chengdu, sponsored by SWUFE IFIF-III 2009, Beijing, sponsored by China Finance Online (NASDAQ)

2. An Engineer- Designer & Developer of computational systemsfor financial market prediction & trading

3. A Trader- Multi-Frequency Trading in global financial markets, mainlyForex – Foreign ExchangeStock Index FuturesStocks – Australia, China, US, ……Financial Options, ……

（Pan Swingtum Master Trading Framework)


Economic Finance

Behavioral Finance

Strategic Finance

My Understanding to Finance in General


My Understanding to Behavioral Finance

We are actually immersed in the

- Water

- Sea

- Ocean

of Behavioral Biases

We include- Governments - Media- Banks - Financial Scholars- Firms - Financial Engineers- Funds - Financial Information Providers- Investors - Financial Software Developers- Traders -- Speculators


My Understanding to Behavioral Finance

We are actually immersed in the

- Water We Need A Reference System

- Sea for modeling market dynamicsas a product of all market drivers

- Ocean Behavioral Biases &Patterns included

of Behavioral Biases

We include- Governments - Media- Banks - Financial Scholars- Firms - Financial Engineers- Funds - Financial Information Providers- Investors - Financial Software Developers- Traders -- Speculators


Yin-Yang Volatility Surface Evolution - Contents

1. Duality of Trend versus Volatility in Scale Space– A Reference System for Stochastic-Dynamics of Price

2. Existing Models in Quantitative Finance and Econophysics3. Yin-Yang Volatility (YYV) on a Single Level of Time Scale4. Yin-Yang Volatility Surface (YYVS) in the Scale Space

5. Modeling YYVS Evolution (YYVSE) with YYV GARCH Models6. Predicting YYVS Evolution with Implied YYV from Option Market

7. Forex Trading Strategies based on Yin-Yang Volatility8. Technical Trading Strategies based on Two-Level YYV9. Yin-Yang Fractals for Predictive Trading Strategies10. Causal Process Models of Financial Market Prices11. Implicit World Currency - What behind the Dollar Index 12. Conclusions


1. Duality of Trend versus Volatility in Scale Space

We only study global liquid financial marketsincluding currencies, equity indexes, commodity futures, etcWe recognize that market prices are stochastic and dynamicWe are looking for more sensible models beyond random walk

Volatility is a central notion in academic finance – option pricing, portfolio theory, risk metrics, etcTrend is a central notion in professional finance – technical analysis, trend following, momentum trading, strength investing, etc

Volatility is defined by the variance of the price around the trend axis.Trend is defined by the bias in the volatility of the price

Neither can be well-defined without the otherTrend and Volatility form a unity of signal-noise duality


Trend and Volatility cannot be defined without each other

Definition of Trend in Technical Trading:

Up-Trend: Price exhibits Higher Highs and Higher LowsDown-Trend: Price exhibits Lower Lows and Lower Highs


Volatility is defined by variance of price returns around average (MA) of price returns


A Dynamic Structure for Modeling Price Process

Definition of Trend and Volatility depends on the Time ScaleSo what could be an objective Reference System for expressing the stochastic-dynamic characteristics of the market prices ?We reckon there is no absolute Reference System, and it must be relativistic to the recent history of the market prices and economic information flows. If this Reference System is relativistic to the recent price history, it should correspond to a Dynamic Structure of price time series.Such a Dynamic Structure should be expressive enough for modeling typical tendencies (market regimes) of price process – Trend, Swing and Breakout.

We found that Yin-Yang Volatility Surface Evolution (YYVSE)provides such a Dynamic Structure.


2. Existing Models in Quantitative Finance and Econophysics

Two Typical Schools of ThoughtEfficient Market Theory Complex Dynamic Systems

Two Typical Models of Stochastic-Dynamic Process of PriceGeometric Brownian Motion (GBM)Log-Periodic Power Laws (LPPL)


Geometric Brownian Motion(Bachelier, Samuelson, Black, Scholes, … since 1900, 1973, …)

Brownian Motion – Wiener Process

W

W tεΔ = Δ ~ (0,1)ε N

~ (0, )W tΔ ΔN

( ) (0) ~ (0, )W T W T− NGeometric Brownian Motion (- a power law with exponential noise ?)

dS S dt SdWμ σ= +2

0 exp2t tS S t Wσμ σ

⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦=

( )22 20 0~ , 1t t t

tS S e S e eμ μ σ⎡ ⎤ − ⎣ ⎦

N


Volatility in the sense of Geometric Brownian Motion

0rT

TS S e=

0

1 ln TSrT S

=

2 2

~ ,2

rT

σ σμ⎡ ⎤⎛ ⎞

− ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

N

std( )T rσ =

td( )s rσ = when T = 1 (year)


Log-Periodic Power Laws in Econophysics(Sornette, Johansen, Bauchaud, Andersen, Zhou, … since 1996)

The “Linear” Log-Periodic Power Law （LPPL）（Fractal of Complex Dimension – extending GBM)

]})log(cos[1{)()](log[ φωβ +−+−+= ttCttBAtp cc

The “Nonlinear” Log-Periodic Power Law

A Renormalization Group Model with the Weierstrasse Function

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

+Δ

+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

+

−+=

β

ωβ

β

βω

2

21log

2)log(cos1

1

)()](log[t

cc

t

c

c ttttCtt

ttBAtp

⎟⎠

⎞⎜⎝

⎛ℜ+−+= ∑

=

−N

n

sin

mc

nn xeCttBAtp1

)()](log[ ϕ


LPPL- Fractal of Complex Dimension


Log-Periodic Power Laws as a kind of Trend and Cycle for Intelligent Dynamic Portfolio Management

Discontinuation of LPPL


3. Yin-Yang Volatility on a single level of time scale

Moving Averages: SMA, EMA or Wavelet Scaling

( ), σ σ σ± + −=

( ) ( )1

22

0

1( )m

n i n n i ni

n x x x xm

σ δ−

+ − −=

= − −∑

( ) ( )1

22

0

1( )m

n i n n n ii

n x x x xm

σ δ−

− − −=

= − −∑

Yin-Yang Volatility of Scalar Price Time Series

1

0

1 m

n n ii

x xm

−

−=

= ∑

1 1(1 )n n nx x xλ λ− −= + −


Bias of yin-yang volatility expresses trend direction

( -1 ≤ d ≤ +1 )0 00

d trend upd d sideway

d trend down

σ σσ

+ −

>⎧− ⎪= ⇒ =⎨

⎪ <⎩

2 2 2 2σ σ σ σ± + −= = +


Relative Yin-Yang Volatility

( , ) ,n nx x

σ σρ ρ ρ + −± + −

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

1

ln nn

n

xyx −

⎛ ⎞= ⎜ ⎟

⎝ ⎠

( , ) ϕ ϕ ϕ± + −=

( ) ( )22

1

1( )m

n i n n i ni

n y y y ym

ϕ δ+ − −=

= − −∑

( ) ( )22

1

1( )m

n i n n n ii

n y y y ym

ϕ δ− − −=

= − −∑


Pan Bands of Yin-Yang Volatility

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

+

σσ

kxkx

ndPanLowerBandPanUpperBa

PanBandsn

n

n

nn

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

σσ

kxkx

andBollLowerBandBollUpperB

andsBollingerBn

n

n

nn


Pan Bands vs Bollinger Bands


Yin-Yang Volatility and Pan Bands vs Bollinger Bands


Philosophical and Psychological Underpinnings of YYV

Chinese Yin-Yang Philosophy (Book of Change, 2800BC-2737BC)

8 Trigrams (Source: http://baike.baidu.com)


(Source: http://image.baidu.com)


64 Hexagrams (Source: http://image.baidu.com)



Chinese Yin-Yang Philosophy (Book of Change, 2800BC-2737BCcan be summarized in 4 principles:

1. Everything has both yin and yang aspects, although yin or yang elements may manifest more strongly in different objects or at different times or situations.

2. Yin and yang arise together from an initial quiescence or emptiness, and continue moving in tandem until quiescence is reached again

3. Yin and yang transform each other. Whenever one quality – yin or yang – reaches its peak it will naturally begin to transform into the opposite quality.

4. Yin and yang constantly interact, never existing in absolute stasis. The interaction of the two gives birth to new things.



Western Dialectics Philosophy (Socrates (469 BC – ) and Plato, Hegel and Marx)can be summarized in 4 principles:

1. Everything is transient and finite, existing in the medium of time.2. Everything is made out of opposing forces or opposing sides

(contradictions)3. Gradual changes lead to turning points, where one force overcomes

the other (quantitative change leads to qualitative change)4. Change moves in spirals or helices, not circles (referred to as

“negation of the negation”).


4. Yin-Yang Volatility Surface in Scale Space

( ) ( )22, ,

1

1( , )m

t i t m t i t mi

t m x x x xm

σ δ+ − −=

= − −∑

( ) ( )22, ,

1

1( , )m

t i t m t m t ii

t m x x x xm

σ δ− − −=

= − −∑

,1

1 m

t m t ii

x xm −

=

= ∑

(Virtually) Continuous Scale Space (m=2, 3, …, 250, …)



Discrete Scale Space (Time Compression Sampling)(3 Yin-Yang Volatility Surfaces of high, low, close)

1( . , . , . , . ),k k k k k k kx x o x h x l x c t t t−= < ≤

monthlyTS94-hourlyTS615-minuteTS3

weeklyTS8hourlyTS55-minuteTS2

dailyTS730-minuteTS41-minuteTS1

each transaction

TS0

DataLow-FreqTime Scale

DataMid-FreqTime Scale

DataHigh-FreqTime Scale



Discrete Scale Space (Time Compression Sampling)(3 Yin-Yang Volatility Surfaces of high, low, close)

1( . , . , . , . ),k k k k k k kx x o x h x l x c t t t−= < ≤

( ) ( )1 22

, , , ,0

1( )m

z z n i z n z n i z ni

n x x x xm

σ δ−

+ − −=

= − −∑

( ) ( )1 22

, , , , ,0

1( )m

z z n i z n z n z n ii

n x x x xm

σ δ−

− − −=

= − −∑

( ) ( , ), , , (high,low,close)z z zn z h l cσ σ σ+ −= =



1. Discrete Scale Space=> Multi-Level Yin-Yang Volatilities

1a: Regular Discrete Scalesm = 2, 2+a, 2+2a, …, 2+ka

1b: Fibonacci Discrete Scalesm = 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

2. Continuous Scale Space

m = 2, 3, 4, …, 250


5. Modeling Yin-Yang Volatility Surface Evolution (YYVSE)with Yin-Yang Volatility GARCH Model - YGARCH

The General Form of GARCH(p, q) Model

2 2 20

1 1

p q

n i n i j n ji j

uσ ω α σ β− −= =

= + +∑ ∑ n n nu y y= −

Yin-Yang Volatility GARCH Model – YGARCH on a given time scale

( ) ( )2 2 2 2 2, 1 , 2 , 1 2

1 1( ) ( )

p q

n i n i i n i j n j j n ji j

u u u uσ ω α σ α σ β δ β δ+ + + − − − − −= =

= + + + + −∑ ∑

( ) ( )2 2 2 2 2, 3 , 4 , 3 4

1 1( ) ( )

p q

n i n i i n i j n j j n ji j

u u u uσ ω α σ α σ β δ β δ− − + − − − − −= =

= + + + + −∑ ∑


6. Predicting Yin-Yang Volatility Surface Evolution with Implied Yin-Yang Volatility from Option Market

Black-Scholes Option Pricing Formula

0 1 2( ) ( )f dr T r Tc S e N d Ke N d− −= −

2 0 1( ) ( )fd r Tr Tp Ke N d S e N d−−= − − −

( ) 20

1

ln / ( / 2)d fS K r r Td

Tσ

σ

+ − +=

( ) 20

2 1

ln / ( / 2)d fS K r r Td d T

Tσ

σσ

+ − −= = −

0fd r Tr Tc Ke p S e−−+ = +


Implied Volatility from Option Market Prices

1ˆ ( , , , , , | )o d fc S K T t r r cσ − ′=

1ˆ ( , , , , , | )o d fp S K T t r r pσ − ′=

Implied Yin-Yang Volatility from Option Market Prices

0a bK S K< < ( , ) ( , )a bput K call KΔ = Δ

1ˆ ( , , , , , | )a o a d f ap S K T t r r pσ − ′=

1ˆ ( , , , , , | )b o b d f bc S K T t r r cσ − ′=

( ) ( )ˆ ˆ ˆ ˆ ˆ, ,b a σ σ σ σ σ± + −= =


7. Forex Trading Strategies based on Yin-Yang Volatility

Performance Measures for a trading strategyARI – Annual Return of Investment for a continuous period of 12 months

in percentageMRDD – Maximum Relative Draw Down of the capital MSR – Maximum Survival Risk in percentage CRRR – Capital Risk Reward Ratio during the test periodSRRR – Survival Risk Reward Ratio during the test period

1/ (1 )MSR MRDD= −

/CRRR ARI MRDD=

/SRRR ARI MSR=


Pan Yin-Yang Line (YYL) and Pan River Indicator

2 2100 100YYL d σ σ

σ σ+ −

+ −

−= × = ×

+

⎩⎨⎧

<+≥+

=−+

−+

σσσσ

ifndPanLowerBaMAifndPanUpperBaMA

PanRiver2/)(2/)(


Pan Bands, Pan River, Pan Yin-Yang Line Signals


Markov Chain of Forex Market Regime Shift

The down trend is confirmeddowntrending8

Market has left the top, is trending downtrendingdown7

Market is starting to turn down from a toptopdown6

Market is overbought, reached a toptopped5

The up trend is confirmeduptrending4

Market has left the bottom, is trending uptrendingup3

Market is starting to turn up from a bottombottomup2

Market is oversold, reached a bottombottomed1

MeaningSignalsRegimeIndex

95YYL ≤ −

95YYL ↑ −

95 0 YYL YYL YYL− < < ∩ >

0 95 YYL YYL YYL≤ < ∩ >

95YYL ≥

95YYL ↓

0 95 YYL YYL YYL< < ∩ <

95 0 YYL YYL YYL− < < ∩ <


Entry Conditions

If the market has been ‘bottomed’ and then is ‘bottomup’ with a high probability Prob(‘bottomup’| ‘bottomed’, prices, MA, PanRiver, PanLowerBand)

Then go long with a market order with a stoploss level and a takeprofit level

If the market has been ‘topped’ and then is ‘topdown’ with a high probability Prob(‘topdown’| ‘topped’, prices, MA, PanRiver, PanHigherBand)

Then go short with a market order with a stoploss level and a takeprofit level


Exit Conditions and Position Management

Exit Conditions:If the position is long and

( the price crosses down its stoploss level or the price crosses up its takeprofit level or the market is ‘topdown’)

Then exit the long position.

If the position is short and( the price its stoploss level or the price its takeprofit level

or the market is ‘bottomup’)Then exit the short position.

Position Management:Bet a constant ratio of the current capital for position sizing,

and use trailing stop loss orders to manage the floating profits.

↓↓


Back Test of Swingtum YinYangVola for EURUSD on H1

5.7312.81123.4655.25707.7616.5312.8997.3649.33635.730.96.5411.8581.2344.82531.130.86.3810.8470.1041.21446.920.75.228.4161.1637.95318.970.65.818.3243.2530.19251.060.55.447.3134.4125.60187.120.45.226.5024.5819.73128.230.34.635.3515.6513.5372.460.24.004.307.486.9629.90.1

SRRRCRRRMSRMRDDARIRPS


Performance Measures of Swingtum YinYangVola


Equity Curves with Position Size = 0.1, 0.3 lots/$10,000

Equity curve with RPS = 0.1 lots / $10,000, ARI=29.90%, MRDD=6.96%, SRRR=4.0

Equity curve with PS = 0.3 lots / $10,000, ARI=128.23%, MRDD=19.73, SRRR=4.63


Equity Curves with Position Size = 0.8, 1.0 lots/$10,000

Equity curve with RPS = 0.8 lots / S10,000, ARI=531.13%, MRDD=44.82, but with best SRRR=6.54

Equity curve with RPS = 1.0 lots / $10,000, ARI=707.76%, MRDD=55.25%, SRRR=5.73


8. Technical Trading Strategies based on Two-Level YYV


9. Yin-Yang Fractal and Predictive Trading Strategies


10. Causal Process Model of Financial Market Prices


11. The Implicit World Currency – What behind Dollar Index


The Standard US Dollar Index

Standard Definition: Started in 1973 with a base of 100, the US Dollar Index (USDX) is a real-time measure of the value of the United States dollar relatives to a basket of foreign currencies – a majority of its most significant trading partners. Mathematically, it is calculated as a weighted geometric mean of the dollar’s value compared only with

EUR (57.6%), JPY (13.6%), GBP(11.9%), CAD (9.1%), SEK(4.2%), CHF(3.6%).

0.576 0.136 0.119

0.091 0.042 0.036

50.14348112 [ / ] [ / ] [ / ] [ / ] [ / ] [ / ]USDX EUR USD JPY USD GBP USD

CAD USD SEK USD CHF USD

− − −

− − −

= × × ×

× × ×


The Trade Weighted US Dollar Index

The Broad Index: the Trade Weighted US Dollar IndexCreated by Federal Reserve in 1998, the broad index was introduced for the introduction of the euro and for keeping pace with new developments in US trade.The basket is much broader, including Nt=26 currencies (CRCj, j=1,2,…,Nt).

Based on Nominal Exchange Rates

Based on Real Exchange Rates

( ),

1

[ / ]1 [ / ]

1

j ttj t

j t

wNCRC USD

t t CRC USDj

USDX USDX−−

=

= ×∏

,

1,

1 , 1

[ / ]1 [ / ]

1

j tCPIt tj t CPIt

CPI j tj t CPI j t

wNCRC USD

t t CRC USDj

USDX USDX −

− −

×

− ×=

⎛ ⎞= × ⎜ ⎟

⎝ ⎠∏


Any Currency Index CRCX

For any currency CRC, its Currency Index CRCX can be defined

by its exchange rate with USD and the US dollar Index USDX:

[ / ]t t tCRCX CRC USD USDX= ×


Wollar: the Implicit World Base Currency

1 1 1 1 1 1 1 1

t

t

t

t

t

t

t

t

USD USDX WEUR EURX WJPY JPYX WGBP GBPX WCHF CHFX WAUD AUDX WNZD NZDX WCAD CADX W

=

=

=

=

=

==

=

It is naturally valid to assume that there is an implicit unit underlying the US dollar Index, then, every currency index shares the same unit – Wollar – the World Dollar, such as

W


Currency Indexes – USDX, EURX, GBPX, CHFX, JPYX, AUDX, NZDX, CADX


A Dynamic Currency Allocation Model for Foreign Reserve Risk Management


12. Conclusions1. Trend and Volatility form a unity of Signal-Noise duality.2. Yin-Yang Volatility form a unity of Yin-Yang duality,

which expresses the duality of trend and volatility in computable form on any given scale of time.

3. Yin-Yang Volatility Surface (YYVS) expresses the YYV throughout the scale space of price-time.

4. The YYVS Evolution (YYVSE) provides a Dynamic Structure for Modeling the Behavioral Patterns of the price dynamics.

5. Even simple explorations of YYVSE give rise to useful trading strategies in forex market.

6. For the time being, the best technical trading strategies must be based on at least two-level YYV.


12. Conclusions (continued)

7. Future Research Possibilities on YYV include

- Implied YYV from Option Market and New Option Pricing Models

- Modeling YYVSE in YGARCH – A New Family of GARCH Models

- New Market Prediction Models based on YYVSE

- New Dynamic Portfolio Theory Minimizing Yin-Volatility of Returns

- Yin-Yang Fractals as extension of Yin-Yang Volatility

- Yin-Yang Fractals of Complex Dimension as extension of Yin-Yang Fractals

- Financial Information Fusion on the Dynamic Structure of YYVSE


China-Europe Institute of Finance (CEIF) – a proposal

Vision: 4 Roles of CEIF• China-Europe Academy of Finance• China-Europe School of Finance• China-Europe Research Institute of Finance• China-Europe Financial Institution

Mission: New Science, Engineering, Standards and Leadership• Promoting New Science of Finance

- An evolutionary framework of finance via an interdisciplinary fusion of all-source information and knowledge

• Promoting New Engineering of Finance- Network-centric causal process modeling for market intelligence, risk management and financial operations

• Developing New Standards for a better financial industry• Leading the Financial World with real-world examples in intelligent investing,

trading, risk management and financial operations


Thanks for Your Attention!

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