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A fuzzy time series-based neural network approach to option price forecasting. Speaker: Prof. Yungho Leu Authors: Yungho Leu, Chien-Pang Lee, Chen-Chia Hung Department of Information Management, National Taiwan University of Science and Technology. Outline. Introduction Main idea
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A fuzzy time series-based neural network approach to option price forecasting Speaker: Prof. Yungho Leu Authors: Yungho Leu, Chien-Pang Lee, Chen-Chia Hung Department of Information Management,National Taiwan University of Science and Technology
Outline Introduction Main idea Fuzzy Time Series The FTSNN Method Option Price Forecasting using FTSNN Results and Performance Conclusion
Introduction Option is an important tool for risk management. The premium, also called the price, of an option is determined by many factors.
Introduction The well-known Black-Scholes model (B-S model) was introduced in 1973 to forecast option price. Many limitations limit the use of the B-S model. We propose a hybrid model, FTSNN, that combines fuzzy time series and neural networks to predict option price.
Introduction In FTSNN, the fuzzy time series is used to select training data set and the neural network is used to build the prediction model. We use FTSNN to predict the option price of TXO. “Taiwan Stock Exchange Stock Price IndexOptions”
Main idea Similarly segments Historical database X1 X1 X2 X2 X3 X3 X4 X4 Find similarly segments X2 X5 X2 X6 X3 X3 X4 X7 X4 X5 X5 X8 X3 X3 X3 X4 X4 X4 X5 X5 X5 X6 X6 X6 . . . . . . . . Xt-5 Xt-5 Xt-5 Xt-4 Xt-4 Xt-4 Xt-3 Xt-3 Xt-3 Xt-2 Xt-2 Xt-2 Xt-4 Xt-4 Xt-4 Xt-3 Xt-3 Xt-3 Xt-2 Xt-2 Xt-2 Xt-1 Xt-1 Xt-1 Xt-3 Xt-3 Xt-3 Xt-3 Xt-2 Xt-2 Xt-2 Xt-2 Xt-1 Xt-1 Xt-1 Xt-1 Xt Xt Xt Xt Use similar segments to train the prediction model Xt-2 Xt-1 Xt ? To predict the next day Xt+1
Main idea Similarly segments X2 X3 X4 X5 X2 X3 X4 X5 X3 X4 X5 X6 X3 X4 X5 X6 Xt-5 Xt-5 Xt-4 Xt-4 Xt-3 Xt-3 Xt-2 Xt-2 Xt-4 Xt-3 Xt-2 Xt-1 Xt-4 Xt-3 Xt-2 Xt-1 Xt-3 Xt-2 Xt-1 Xt Xt-3 Xt-2 Xt-1 Xt Using RBFNN to train a prediction model
Main idea Xt-2 Xt-1 Xt ? To predict the next day Xt+1
Fuzzy Time Series • If F(t) is caused by F(t-1), F(t-2),…,and F(t-n), F(t) is called a one-factor n-order fuzzy time series, and is denoted by F(t-n),…, F(t-2), F(t-1)→F(t).
Fuzzy Time Series • If F1(t) is caused by (F1(t-1), F2(t-1)), (F1(t-2), F2(t-2)),…, (F1(t-n), F2(t-n)), F1(t) is called a two-factor n-order fuzzy time series, which is denoted by(F1(t-n), F2(t-n)),…, (F1(t-2), F2(t-2)), (F1(t-1), F2(t-1))→F1(t).
Fuzzy Logic Relationship (FLR) • Let F1(t)=Xtand F2(t)= Yt , where Xt and Ytare fuzzy variables whose values are possible fuzzy sets of the first factor and the second factor, respectively, on day t. Then, a two-factor n-order fuzzy logic relationship(FLR) can be expressed as: (Xt-n, Yt-n), …, (Xt-2, Yt-2), (Xt-1, Yt-1)→Xt, • where (Xt-n, Yt-n), …, (Xt-2, Yt-2) and (Xt-1, Yt-1), are referred to as the left-hand side (LHS) of the relationship, and Xt is referred to as the right-hand side (RHS) of the relationship..
The FTSNN MethodStep 1: Divide the universe of discourse • The universe of discourse of the first factor is defined as U= [Dmin-D1, Dmax+D2], where Dmin and Dmax are the minimum and maximum of the first factor, respectively; D1 and D2 are two positive real numbers to divide the universe of discourse into n equal length intervals. • The universe of discourse of the second factor is defined as V= [Vmin-V1, Vmax+V2], where Vmin and Vmax are the minimum and maximum of the second factor, respectively; V1 and V2 are two positive real numbers used to divide the universe of discourse of the second factor into m equal length intervals.
FTSNN MethodStep 2:Define Linguistic terms • Linguistic terms Ai, 1 ≤ i ≤n, are defined as fuzzy sets on the intervals of the first factor.
FTSNN MethodStep 2:Define Linguistic terms • linguistic term Bj, 1 ≤ j ≤ m, is defined as a fuzzy set on the intervals of the second factor
The FTSNN MethodStep 3(a): Construct FLRs database • For the historical data on day i, let Xi-n, Yi-ndenote the fuzzy set of F1(i-n) and F2(i-n) of the fuzzy time series. Let Xi denotes the fuzzy set of F1(i). The FLRs database on day i can be represented as follows: (Xi-n, Yi-n), …, (Xi-2, Yi-2), (Xi-1, Yi-1)→Xi.
The FTSNN MethodStep 3(b): Construct the LHS of FLR of the predicting day • The LHS of the FLR on day t can be represented as follows:(Xt-n, Yt-n), …, (Xt-2, Yt-2), (Xt-1, Yt-1).
The FTSNN MethodStep 3(c): Search for similar FLRs • In the above formulae, IXt-nand IYt-n are the subscripts of the fuzzy terms of the first factor and the second factor, respectively, of the LHS of day t’sFLR. Similarly, RXi-n and RYi-n are subscripts of the first factor and the second factor, respectively, of the LHS of day i’sFLR.
Step 3(e) Model Selection • FTSNN uses similar FLRs to build a neural network model. Similar FLRs imply similar trends in the historical data. • How long is the trend ? • We set the order (length) to be 1,2, …, 5 to build five different prediction models. • Then, we choose the best one.
The FTSNN MethodStep 3(e): Model selection • we use the prediction accuracy on day t-1 as the model selection criterion. Error function＝｜Forecasted RHS－Testing RHS｜ • The forecasted RHS denotes the subscript of the forecasted fuzzy term on day t-1, and the testing RHS denotes the actual subscript of the fuzzy term of the RHS on day t-1
The FTSNN MethodStep 4: Forecasting • We feed the LHS of the FLR on the predicting day into the neural network to get the forecasted subscript of the RHS on the predicting day. • We use weighted averageas the defuzzification method. Map the subscript to a value.
The FTSNN MethodStep 4: Forecasting • where M[k] denotes the midpoint value of the fuzzy term k.
Option Price Forecasting using FTSNN • To forecast the price of “Taiwan Stock Exchange Stock Price Index Option (TXO)”. • We choose closing price of TXO asfirst factor and “Taiwan Stock Exchange CapitalizationWeighted Stock Index (TAIEX)” assecond factor.
Option Price Forecasting using FTSNN • Then, we select top five similar FLRs from the FLRs database. In this example, FLR8, FLR5, FLR7, FLR6, FLR4 are selected.
Option Price Forecasting using FTSNN A28 A28 B117 B117 A25 A25 B117 B117 A30 A30 FLR8 A36 A36 B118 B118 A37 A37 B118 B118 A39 A39 FLR5 A39 A39 B119 B119 A28 A28 B117 B117 A25 A25 FLR7 A37 B118 A39 B119 A28 FLR6 A42 B119 A36 B118 A37 FLR4 Training the prediction model
Option Price Forecasting using FTSNN A28 A28 B117 B117 A25 A25 B117 B117 A30 FLR8 A27 Error＝｜27－30｜=3 R code Forecasted fuzzy term Testing the order of FTSNN
Option Price Forecasting using FTSNN • assume that a 2-order neural network model is selected, and the forecasted subscript is 35 on day 11. Substituting 345, 355 and 365 for M[34], M[35], and M[36], respectively. • Note that the actual option price on day 11 is 360 in this example.
Results and PerformanceDataset • The dataset of this paper are the daily transaction data of TXO and TAIEX from January 3, 2005 to December 29, 2006. • Our dataset comprises 30 different strike price from 5,200 to 8,200 and 12 different expiration dates from January 2005 to December 2006.
Results and PerformancePerformance measures • Two different performance measures, mean absolute error (MAE) and root mean square error (RMSE), are used to measure the forecasting accuracy of FTSNN. • where At and Pt denote the real option price and the forecasting option price on day t, respectively.
Conclusion • FTSNN combines a fuzzy time series model and an NN • Fuzzy time series model selects training examples for a RBF NN to build the prediction model. • The performance of FTSNN is better than the existing models.
