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

 Introduction  Main idea  Fuzzy Time Series  The FTSNN Method  Option Price Forecasting using FTSNN  Results and Performance  Conclusion Outline

 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 Index Options” Introduction

X3X3 X7X7 X5X5 XtXt X1X1 X5X5 X3X3 X t-2 X2X2 X6X6 X4X4 X t-1 X4X4 X8X8 X6X6 ?.... X t-3 X t-5 X t-4 X t-2 X4X4 X2X2 X3X3 X5X5 X t-4 X t-3 X t-1 X t-3 X t-2 XtXt X3X3 X1X1 X2X2 X4X4 X5X5 X3X3 X4X4 X6X6 X t-4 X t-3 X t-1 X t-3 X t-2 XtXt Historical database To predict the next day X t X t-1 X t-3 X t-2 XtXt X t-3 X t-5 X t-4 X t-2 X4X4 X2X2 X3X3 X5X5 X5X5 X3X3 X4X4 X6X6 X t-4 X t-3 X t-1 X t-3 X t-2 XtXt X t-3 X t-5 X t-4 X t-2 Similarly segments Use similar segments to train the prediction model Find similarly segments

X4X4 X2X2 X3X3 X5X5 X5X5 X3X3 X4X4 X6X6 X t-2 X t-4 X t-3 X t-1 X t-3 X t-2 XtXt X t-3 X t-5 X t-4 X t-2 Similarly segments X4X4 X2X2 X3X3 X5X5 X5X5 X3X3 X4X4 X6X6 X t-2 X t-4 X t-3 X t-1 X t-3 X t-2 XtXt X t-3 X t-5 X t-4 X t-2 Using RBFNN to train a prediction model

XtXt X t-2 X t-1 ? To predict the next day X t+1

 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).

 If F 1 (t) is caused by (F 1 (t-1), F 2 (t-1)), (F 1 (t-2), F 2 (t-2)),…, (F 1 (t-n), F 2 (t-n)), F 1 (t) is called a two-factor n-order fuzzy time series, which is denoted by (F 1 (t-n), F 2 (t-n)),…, (F 1 (t-2), F 2 (t-2)), (F 1 (t-1), F 2 (t-1))→F 1 (t).

Fuzzy Logic Relationship (FLR)  Let F 1 (t)=X t and F 2 (t)= Y t, where X t and Y t are 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: (X t-n, Y t-n ), …, (X t-2, Y t-2 ), (X t-1, Y t-1 )→X t,  where (X t-n, Y t-n ), …, (X t-2, Y t-2 ) and (X t-1, Y t-1 ), are referred to as the left-hand side (LHS) of the relationship, and X t is referred to as the right-hand side (RHS) of the relationship..

 The universe of discourse of the first factor is defined as U= [D min -D 1, D max +D 2 ], where D min and D max are the minimum and maximum of the first factor, respectively; D 1 and D 2 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= [V min -V 1, V max +V 2 ], where V min and V max are the minimum and maximum of the second factor, respectively; V 1 and V 2 are two positive real numbers used to divide the universe of discourse of the second factor into m equal length intervals.

FTSNN Method Step 2: Define Linguistic terms  Linguistic terms A i, 1 ≤ i ≤ n, are defined as fuzzy sets on the intervals of the first factor.

FTSNN Method Step 2: Define Linguistic terms  linguistic term B j, 1 ≤ j ≤ m, is defined as a fuzzy set on the intervals of the second factor

 For the historical data on day i, let X i-n, Y i-n denote the fuzzy set of F 1 (i-n) and F 2 (i-n) of the fuzzy time series. Let X i denotes the fuzzy set of F 1 (i). The FLRs database on day i can be represented as follows: (X i-n, Y i-n ), …, (X i-2, Y i-2 ), (X i-1, Y i-1 )→X i.

 The LHS of the FLR on day t can be represented as follows: (X t-n, Y t-n ), …, (X t-2, Y t-2 ), (X t-1, Y t-1 ).

 In the above formulae, IX t-n and IY t-n are the subscripts of the fuzzy terms of the first factor and the second factor, respectively, of the LHS of day t’s FLR. Similarly, RX i-n and RY i-n are subscripts of the first factor and the second factor, respectively, of the LHS of day i’s FLR.

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.

 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

 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 average as the defuzzification method. Map the subscript to a value.

 where M[k] denotes the midpoint value of the fuzzy term k.

 To forecast the price of “Taiwan Stock Exchange Stock Price Index Option (TXO)”.  We choose closing price of TXO as first factor and “Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX)” as second factor.

 Then, we select top five similar FLRs from the FLRs database. In this example, FLR 8, FLR 5, FLR 7, FLR 6, FLR 4 are selected.

A 28 B 117 A 25 B 117 A 30 FLR 8 A 36 B 118 A 37 B 118 A 39 FLR 5 A 39 B 119 A 28 B 117 A 25 FLR 7 A 37 B 118 A 39 B 119 A 28 FLR 6 A 42 B 119 A 36 B 118 A 37 FLR 4 A 28 B 117 A 25 B 117 A 30 A 36 B 118 A 37 B 118 A 39 B 119 A 28 B 117 A 25 Training the prediction model

FLR 8 A 28 B 117 A 25 B 117 A 30 Testing the order of FTSNN A 27 Forecasted fuzzy term A 28 B 117 A 25 B 117 Error ＝｜ 27 － 30 ｜ =3 R code

 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.

 The dataset of this paper are the daily transaction data of TXO and TAIEX from January 3, 2005 to December 29,  Our dataset comprises 30 different strike price from 5,200 to 8,200 and 12 different expiration dates from January 2005 to December 2006.

 Two different performance measures, mean absolute error (MAE) and root mean square error (RMSE), are used to measure the forecasting accuracy of FTSNN.  where A t and P t denote the real option price and the forecasting option price on day t, respectively.

 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.

Thank you for your attentions

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