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A Volatility Skews- based Options Arbitrage Model

via Artificial Intelligence

Department & Graduate School of Business AdministrationCollege of ManagementNational Changhua University of Education

Shinn-Wen Wang

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outlineMotivationIntroductionEmpirical Study and EvidenceConclusions

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MotivationObservations

Black-Scholes formula real world considerations six unreasonable assumptions implied volatility skew

jump-grade (or the ranking system)

ObjectGa-Neural Modeling

jump-grade considerationsimplied volatility skewEasy to extend model

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Motivation (Cont.)

BSMC = ( ) )2(1 T-r dNekdNS ×−× (1)

d1 = TT

TrKS σσ 2

1)/ln(+

×+

d2 = Td σ−1

C :fair value of options; S :spot price of underlying; K: strike price; r : instantaneously risk free rate; T: maturity; σ: underlying return of instantaneously standard deviation; ln(.): natural-log; N (.): accumulated properties of standardize normal distribution

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Vo l. Smile (No . 05 43 :K =6 7.8 )

y = 0.0 00 7x 2 - 0.0 91 1x + 3 .64 05

0 .3

0.35

0 .4

0.45

0 .5

0.55

0 .6

0.65

0 .7

0.75

0 .8

0.81 0.86 0.91 0.96 1.01 1.06 1.11

U nd erlying Stock Pr ice (A d j .): No .13 03

Impl

ied

vol.

I m p . Vo l.

P o ly n o mi al

S/K

Fig.1 Case study of volatility smile (Taiwan Options Market) Chun- I 05: No.0543Basic data

underlying:Nan Ya(No.1303) strike price:67.8(to be issued at

20% outside of price) maturity:1999/11/18~2000/11/17 exercise ratio:1:1

No. 1303 Log-Return Described statistics

mean S.D. kurtosis skewness 0.00157 0.004 37.038 4.918

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IntroductionVolatility skew analysis

tree solutionsCRR

Cox, Ross & Rubinstein, 1979

local volatilityDerman & Kani, 1994; Dupire, 1994; Rubinstein, 1994

the implied trinomial tree Derman, Kani & Chriss, 1996

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Introduction (Cont.)Arch series theorem

Arch Model (autoregressive conditional heteroskedasticity) (Engle, 1982) Garch Model Generalized Garch Model (Bollershlew, 1986)

Igarch Model (Integrated Garch)(Nelson, 1990)Egarch Model (Exponential Garch)(Nelson, 1991) parameter estimating would influence the result a lot

Duan, 1995 estimating volatility

Heston, 1993 dynamic implied volatility function

Rosenberg, 2000 stochastic volatility model

Eisengberg & Jarrow, 1994

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Introduction (Cont.)the volatility estimating model constructed through analytic approach

Stein & Jeremy, 1991

Dufresne, Keirstead & Ross, 1999 complexity difficult to promote and understood

high frequency data analysisGavridis, 1998; Moody & Wu, 1998

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Introduction (Cont.)

Neural Networksneural network is better than non-traditional statistical model

multiple differential analysis Yoon & Swales, 1991

multiple regression analysisKimoto, Asakawa, Yoda & Takeoka, 1990

Logistic regression model and linear differential analysis

Tam & Kiang, 1992

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Introduction (Cont.)

cannot reach a significant standard differential analysis

Dasgupta, Dispensa & Ghose,1994logistic regressive model

Salchenberger, Cinar, & Lash, 1992linear regression analysis and stepwise polynomial regression model

Gorr, Nagin & Szczypula, 1994 individual merits

Box-Jenkins model Sharda & Patil, 1992

differential analysisCurram & Mingers, 1994

linear regression analysis Bansal, Kauffman & Weitz, 1993

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Introduction (Cont.)

statistical model can be simulated by neural network

linear and non-linear regression model Marquze, Hill, Worthley & Remus, 1991

ARMA(n,n-1) and ARMA(n,n)

Bulsari & Saxen, 1993

neural network and statistical model should complement each other

White, 1989

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‧‧‧

‧‧‧ ‧‧

‧

xx

x

y

y

y

1 1

1

2

2

2

n n

W W

H

H

H

H

xh hy

OutputInput

Vector

3

n

Vector

Fig.2 Architecture of back-propagation neural

networks

Fig.3 Structure of chromosomes

GENE 1

# of GENE 2learning

rate

GENE 6Bias value

GENE 5Connection

Weights

GENE 4 Network

Connectivity

GENE 3Momentum

factor

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Modeling

training cycle, evolution cycle & the steps are briefly described as follows

(1). Initial networksrandomly produce initial networks structure

(2). Training cyclenetworks are conducted through genetic rules and combination of weighted tuning. Training time will be utilized to exchange for the quality of approximation optimal solution until the upper bound of learning numbers can be reached

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Modeling (Cont.)(3). Evolution cycle

level of suitability of various networks for evaluation of fitness function is based on mean square error, and the evolution of networks will be commenced. In addition, based on the survived networks decided by the suitability of various networks, reproduction, crossover and mutation of the survived networks can be treated so as to generate the new generation networks

(4). Return to step (2) to conduct new generation network training until satisfactory learning result or pre-set termination condition is reached

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Fig.4 The architecture of evolution cycle with the nested training cycle for the genetic-based neural network

Increment Iteration Count (i =i + 1)

Neural network Learning (BackPro.)

Evolutes Updating Para. of Network

Networks Crossover &

Mutation

Select Survived Network to next

Generation Networks

Select Most Fit Parents &

Stop

Learning time? Up_Bound ?

Evaluate Population of networks

Yes

Checking the Criteria to Stop ?

No

No

Yes Ranking Population & Store Fittest

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Genetic Descriptions(Genotype)

Neural Network Learning(Behavior)

Neural Network(Phenotype)

Selection Based on(Training Error, Structural

Complexity & Forecast accuracy)

Procedure GeNe Begin

e = 0; initial population Pc(e); fitness Pc(e);

While (termination criterion not reach) e= e + 1; Select Pc(e) from Pc(e-1); Crossover Pc(e); Mutate Pc(e); Fitness Pc(t); End.

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Modeling (Cont.)Construction of two-phase arbitrage model

Phase-IModeling

Construction of genetic-based neural network model while taking in consideration of smile behavior of volatility

Phase-IITiming

the jump grade difference effect of stock price

Strategyconcurrent buy-low & sell-highoptions with the same underlying

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Modeling (Cont.)

Phase-I

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Implied Vol.

S/K

Im p_Vol_X ( S )Im p_Vol_X ( S - b ) Im p_Vol_X ( S + a )

Arb itrageImp_Vol_X ( S + a) >Imp_Vol_Y ( S - b )

Im p_Vol_Y ( S + a )

Imp_Vol_Y ( S )

Imp_Vol_Y ( S - b )

PS. The hanging moon shape is arbitrage space.

Fig.5 Arbitrage model basing on consideration of volatility smile effect

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Modeling (Cont.)the two types (or multiple types) options (call options or put options) constructed from the same underlying

including X commodity and Y commodity for exampleits implied volatility (Imp_Vol_X and Imp_Vol_Y)consideration is given to the upper and lower stock price jump interval that are (X: a1, b1; Y: a2, b2) respectively

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Table. 3 Volatility smile of genetic-based neural network modeling change factor is considered (based

on the example of call option) Supervised genetic-based neural network

premise (input factors) consequence (target factor)Moneyness S/K

Vol. σ

BS Vol. C×(0.398×KS /

)-1

Time_Val C(S, T, E)

– Max(0, S – E)

Intrinsic_Val Max(0, S – E)

Forecast Vol. impσ

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Modeling (Cont.)BS Vol.

Brener & Subrahmanyan, 1988

Forecast_Vol.Manaster & Koehler, 1983

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Phase-II

Modeling (Cont.)

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Modeling (Cont.)

[Theorem 1]For two call options contracts (X & Y) of the same

underlying and it’s issued date and maturity are very close then its underlying price will be set as S. If price of the next transaction is adjusted upwards, then the jump grade will be a1X, a2Y respectively. Also if the price of the next transaction is adjusted downwards, then its jump grade will be b1X, b2Y respectively and arbitrage interval will be Imp_Vol_X(S+a) > Imp_Vol_Y(S-b), and its Imp_Vol is the implied volatility of call options . Based on the same reason the put options can also be inferred to obtain its arbitrage interval

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Modeling (Cont.)[Theorem 2]If underlying in Theorem 1 are stocks (if one lot is 1000 shares), then under the condition that the dividend issue or stock allocation is (1 + l) ×100 (shares), the upper and lower bound interval of stock price shall be adjusted as: upper bound à[S – a(or b)] × [1 + (1 + l)/10]. lower bound à[S + a(or b)] × [1 + (1 + l)/10]

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Empirical Study and Evidence

Table 4 Specified limitation on the minimum jump interval for options commodities and underlying

Minimum jump interval (X, Y: a1, a2; b1, b2)

~less than $5 $5~less than $15 $15~less than $50 $50~ less than $150

Share (S) 0.01

0.05

0.05

0.1

0.1

0.5

0.5

warrant (C) 0.05

Information resource: Taiwan security exchange

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Empirical Study and Evidence (Cont.)

WarrantsChien Hung 07 and Fubon 05

common underlying United Microelectronics, UMC

periods 2000/02/10 ~ 2000/04/06

sampling frequency daily

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Empirical Study and Evidence (Cont.)

New subscription percentage adjustmentN′ = N × (1 + m + n) (2)

New strike price adjustmentK′ = [S′ – (S - K) × N – T – C][N × (1 + m + n)]-1

(3)

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T = N × n × [1 – (1 - t) × 80%] × (face value of each share of each underlying security) × 25%; C = N × m × R × r × d × 365-1; S: closing price of underlying security one day before divestiture; S′: reference price of underlying security on the day of divestiture; R: subscription price per share for cash capital increase; K: strike price before adjustment; K′ strike price after adjustment; N: subscription percentage before adjustment; N′: purchase percentage after adjustment; m: share subscription for cash capital increase; n: percentage of stock allocation without payment. C: payment of cash capital increase loan interest cost by security issue merchant who is holder of equity certificate; r: average interest rate for one-year bond buy back (RP) within security issue merchant within 30 operating days before the day of divestiture; d: number of days from closing day of cash capital increase payment to due date of warrant day; T: Dividend tax for holders of equity certificate of issuing security merchants who participated in divestiture; t: tax exempt percentage for operating business income tax of underlying security company

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Empirical Study and Evidence (Cont.)

in 2000/07/14 the stock allocation without payment of United Microelectronics for underlying security is 120 sharesthe lower bound on top of dividend issue stock price is

Upper bound [stock price - minimum jump interval] * 1.12. Lower bound [stock price + minimum jump interval] * 1.12

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Empirical Study and Evidence(Cont.)

the upper bound of price adjustment [warrant price + minimum jump interval] &

[stock price -minimum jump interval]Lower bound price adjustment [warrant price- minimum jump interval] & [Stock price + minimum interval]is based on the upper and lower jump interval of stock price and warrant to determine the upper and lower bound calculation of continuous jumping warrant price, and is abstracted in Table.6.

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0

0.2

0.4

0.6

0.8

1

1.2

2000/2/1

02000

/2/12

2000/2/1

42000/2/ 1

62000/2/ 1

82000/2/

202000/2/

222000

/2/24

2000/2/2

62000/2

/ 28

2000/3/1

2000/3/ 3

2000 /3/ 5200

0/3/7200

0/3/92000/3

/11

2000/3/ 13

2000/3/ 15

2000/3/ 17

2000/3/1

92000

/3/21

2000/3/2

32000/3/ 2

52000/3/ 2

72000/3/

292000

/3/31

2000/4/2

2000/4/4

2000/4/6

Forecasting Vol. (U&L Bound)

UperBound 富05(C+a)% LowerBound 富05(C-a)% UperBound 建07(C+b)% LowerBound 建07(C-b)%

Arbit rage

Arbit rage

Fig.6 By means of two-phase arbitrage model in the research case, the arbitrage

opportunity interval can be monitored.

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Empirical Study and Evidence(Cont.)

Traditionally, the arbitrage result with BSM as basis is adopted and in respect of issued volatility as condition (refers to Table.7) its total loss are 18,149,722.51(Unit: NT$100,000,000)From Table.7 it can be discovered that it does not guarantee that each arbitrage operation is successfulAnother frequently used arbitrage model basing on BSM is mainly by historical volatility. This research conducts arbitrage operation by means of historical volatility adopted by issuers in their calculation and its result is the same as issued volatility (see Table.8)

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Empirical Study and Evidence(Cont.)The genetic-based neural network model proposed in this research can guarantee successful arbitrage operation and the total payoff profit can be as high as 34,565,821(Unit: NT$100,000,000) that is 1.9045 times of traditional arbitrage model. Its excerpts of its operation process are as Table. 9 and the drawing is as Fig. 6.

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Q & A

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Thanks a lot !!