A Volatility Skews- based Options Arbitrage Model via Artificial Intelligence

Transcription

1 A Volatility Skews- based Options Arbitrage Model via Artificial Intelligence Department & Graduate School of Business Administration College of Management National Changhua University of Education Shinn-Wen Wang 1

2 outline Motivation Introduction Empirical Study and Evidence Conclusions 2

3 Motivation Observations Black-Scholes formula real world considerations six unreasonable assumptions implied volatility skew jump-grade (or the ranking system) Object Ga-Neural Modeling jump-grade considerations implied volatility skew Easy to extend model 3

4 Motivation (Cont.) BSM -r T C = S N( d1) ke N( d2) (1) d 1 = ln( S / K) + r T 1 + σ σ T 2 T d 2 = d 1 σ T C :fair value of options; S :spot price of underlying; K: strike price; r : instantaneously risk free rate; T: maturity; : underlying return of instantaneously standard deviation; ln(.): natural-log; 4 N (.): accumulated properties of standardize normal distribution

5 Implied vol Imp. Vol. Polynomial 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 Log-Return Described statistics mean S.D. kurtosis skewness

6 Introduction Volatility skew analysis tree solutions CRR Cox, Ross & Rubinstein, 1979 local volatility Derman & Kani, 1994; Dupire, 1994; Rubinstein, 1994 the implied trinomial tree Derman, Kani & Chriss,

7 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,

8 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 analysis Gavridis, 1998; Moody & Wu,

9 Introduction (Cont.) Neural Networks neural network is better than nontraditional statistical model multiple differential analysis Yoon & Swales, 1991 multiple regression analysis Kimoto, Asakawa, Yoda & Takeoka, 1990 Logistic regression model and linear differential analysis Tam & Kiang,

10 Introduction (Cont.) cannot reach a significant standard differential analysis Dasgupta, Dispensa & Ghose,1994 logistic regressive model Salchenberger, Cinar, & Lash, 1992 linear regression analysis and stepwise polynomial regression model Gorr, Nagin & Szczypula, 1994 individual merits Box-Jenkins model Sharda & Patil, 1992 differential analysis Curram & Mingers, 1994 linear regression analysis Bansal, Kauffman & Weitz,

11 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,

12 Input Vector x x x 1 H2 1 2 n H1 H H 3 n y y y 2 n Output Vector W xh W hy Fig.2 Architecture of back-propagation neural networks GENE 1 # of GENE 2 learning rate GENE 3 Momentum factor GENE 4 Network Connectivity GENE 5 Connection Weights GENE 6 Bias value Fig.3 Structure of chromosomes 12

13 Modeling training cycle, evolution cycle & the steps are briefly described as follows (1). Initial networks randomly produce initial networks structure (2). Training cycle networks 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 13

14 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 14

15 Increment Iteration Count (i =i + 1) Evolutes Updating Para. of Network Networks Crossover & Mutation Neural network Learning (BackPro.) Select Most Fit Parents & Checking the Criteria to Stop? No No Select Survived Network to next Generation Networks Stop Yes Learning time? Up_Bound? Yes Evaluate Population of networks Ranking Population & Store Fittest Fig.4 The architecture of evolution cycle with the nested training cycle for the 15 genetic-based neural network

16 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. Genetic Descriptions (Genotype) Neural Network (Phenotype) Neural Network Learning (Behavior) Selection Based on (Training Error, Structural Complexity & Forecast accuracy) 16

17 Modeling (Cont.) Construction of two-phase arbitrage model Phase-I Modeling Phase-II Construction of genetic-based neural network model while taking in consideration of smile behavior of volatility Timing Strategy the jump grade difference effect of stock price concurrent buy-low & sell-high options with the same underlying 17

18 Modeling (Cont.) Phase-I 18

19 Imp_Vol_X ( S - b ) Imp_Vol_X ( S ) Im p_vol_x ( S + a ) Imp_Vol_Y ( S - b ) Imp_Vol_Y ( S + a ) Imp_Vol_Y ( S ) Arbitrage PS. The hanging moon shape is arbitrage space. Fig.5 Arbitrage model basing on consideration of volatility smile effect 19

20 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 example its implied volatility (Imp_Vol_X and Imp_Vol_Y) consideration is given to the upper and lower stock price jump interval that are (X: a 1, b 1 ; Y: a 2, b 2 ) respectively 20

21 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) Moneyness Vol. BS Vol. S/K σ C (0.398 S / Time_Val C(S, T, E) K Max(0, S E) Intrinsic_Val Max(0, S E) ) -1 consequence (target factor) Forecast Vol. σ imp 21

22 Modeling (Cont.) BS Vol. Brener & Subrahmanyan, 1988 Forecast_Vol. Manaster & Koehler,

23 Modeling (Cont.) Phase-II 23

24 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 a 1X, a respectively. 2Y Also if the price of the next transaction is adjusted downwards, then its jump grade will be b 1X, b 2Y 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 24

25 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] 25

26 Empirical Study and Evidence Table 4 Specified limitation on the minimum jump interval for options commodities and underlying Minimum jump interval (X, Y: a 1, a 2 ; b 1, b 2 ) ~less than $5 $5~less than $15 $15~less than $50 $50~ less than $ Share (S) warrant (C) 0.05 Information resource: Taiwan security exchange 26

27 Empirical Study and Evidence (Cont.) Warrants Chien Hung 07 and Fubon 05 common underlying United Microelectronics, UMC periods 2000/02/10 ~ 2000/04/06 sampling frequency daily 27

28 Empirical Study and Evidence (Cont.) New subscription percentage adjustment N = N (1 + m + n) (2) New strike price adjustment K = [S (S - K) N T C][N (1 + m + n)] -1 (3) 28

29 T = N n [1 (1 - t) 80%] (face value of each share of each underlying security) 25%; C = N m R r d ; 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 29

30 Empirical Study and Evidence (Cont.) in 2000/07/14 the stock allocation without payment of United Microelectronics for underlying security is 120 shares the lower bound on top of dividend issue stock price is Upper bound [stock price - minimum jump interval] * Lower bound [stock price + minimum jump interval] *

31 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. 31

32 Arbitrage Arbitrage 2000/2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /3/1 2000/3/ /3/ /3/7 2000/3/9 2000/3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /4/2 2000/4/4 2000/4/6 Fig.6 By means of two-phase arbitrage model in the research case, the arbitrage opportunity interval can be monitored. 32

33 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 successful Another 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) 33

34 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 times of traditional arbitrage model. Its excerpts of its operation process are as Table. 9 and the drawing is as Fig

35 Q & A 35

36 Thanks a lot!! 36