{"title": "Sequential Tracking in Pricing Financial Options using Model Based and Neural Network Approaches", "book": "Advances in Neural Information Processing Systems", "page_first": 960, "page_last": 966, "abstract": null, "full_text": "Sequential Tracking in Pricing Financial \nOptions using Model Based and Neural \n\nNetwork Approaches \n\nMahesan Niranjan \n\nCambridge University Engineering Department \n\nCambridge CB2 IPZ, England \n\nniranjan@eng.cam.ac.uk \n\nAbstract \n\nThis paper shows how the prices of option contracts traded in finan(cid:173)\ncial markets can be tracked sequentially by means of the Extended \nKalman Filter algorithm. I consider call and put option pairs with \nidentical strike price and time of maturity as a two output nonlin(cid:173)\near system. The Black-Scholes approach popular in Finance liter(cid:173)\nature and the Radial Basis Functions neural network are used in \nmodelling the nonlinear system generating these observations. I \nshow how both these systems may be identified recursively using \nthe EKF algorithm. I present results of simulations on some FTSE \n100 Index options data and discuss the implications of viewing the \npricing problem in this sequential manner. \n\n1 \n\nINTRODUCTION \n\nData from the financial markets has recently been of much interest to the neural \ncomputing community. The complexity of the underlying macro-economic system \nand how traders react to the flow of information leads to highly nonlinear rela(cid:173)\ntionships between observations. Further, the underlying system is essentially time \nvarying, making any analysis both difficult and interesting. A number of prob(cid:173)\nlems, including forecasting a univariate time series from past observations, rating \ncredit risk, optimal selection of portfolio components and pricing options have been \nthrown at neural networks recently. \n\nThe problem addressed in this paper is that of sequential estimation, applied to \npricing of options contracts. In a nonstationary environment, such as financial \nmarkets, sequential estimation is the natural approach to modelling. This is because \ndata arrives at the modeller sequentially, and there is the need to build and apply the \n\n\fSequential Tracking of Financial Option Prices \n\n961 \n\nbest possible model with available data. At the next point in time, some additional \ndata is available and the task becomes one of optimally updating the model to \naccount for the new data. This can either be done by reestimating the model with \na moving window of data or by sequentially propagating the estimates of model \nparameters and some associated information (such as the error covariance matrices \nin the Kalman filtering framework discussed in this paper). \n\n2 SEQUENTIAL ESTIMATION \n\nSequential estimation of nonlinear models via the Extended Kalman Filter algo(cid:173)\nrithm is well known (e.g. Candy, 1986; Bar-Shalom & Li, 1993). This approach \nhas also been widely applied to the training of Neural Network architectures (e.g. \nKadirkamanathan & Niranjan, 1993; Puskorius & Feldkamp, 1994). In this section, \nI give the necessary equations for a second order EKF, i.e. Taylor series expansion \nof the nonlinear output equations, truncated at order two, for the state space model \nsimplified to the system identification framework considered here. \n\nThe parameter vector or state vector, 0, is assumed to have the following simple \nrandom walk dynamics. \n\n(t(n + 1) = (t(n) + .Y(n) , \n\nwhere .Y( n) is a noise term, known as process noise . .Y( n) is of the same dimension(cid:173)\nality as the number of states used to represent the system. The process noise gives \na random walk freedom to the state dynamics facilitating the tracking behaviour \ndesired in non stationary environments. In using the Kalman filtering framework, \nwe assume the covariance matrix of this noise process, denoted Q, is known. In \npractice, we set Q to some small diagonal matrix. \nThe observations from the system are given by the equation \n\n~(n) = \n\n[({t, U) + w(n), \n\nwhere, the vector ~(n) is the output of the system consisting of the call and put \noption prices at time n. U denotes the input information. In the problem considered \nhere, U consists of the price of the underlying asset and the time to maturity if the \noption. w is known as the measurement noise, covariance matrix of which, denoted \nR, is also assumed to be known. Setting the parameters Rand Q is done by trial \nand error and knowledge about the noise processes. In the estimation framework \nconsidered here, Q and R determine the tracking behaviour of the system. For the \nexperiments reported in this paper, I have set these by trial and error, but more \nsystematic approaches involving multiple models is possible (Niranjan et al, 1994). \nThe prior estimates at time (n + 1), using all the data upto time (n) and the model \nof the dynamical system, or the prediction phase of the Kalman algorithm is given \nby the equations: \n\n~(n + lin) \nP(n + lin) \n\n= \n\n~(n + lin) \n\n~(nln) \nP(nln) + Q(n) \n10 ({t(n + lin)) + 2\" L~i tr (mo(n + 1) P(n + lin)) \n\n1 n8 \n\ni=l \n\nwhere 10 and H~o are the Jacobian and Hessians of the output e; also no = 2. ~i \nare unit vectors in direction i. tr(.) denotes trace of a matrix. The posterior esti-\n\n\f962 \n\nM. Niranjan \n\nmates or the correction phase of the Kalman algorithm are given by the equations: \n\n8(n+1) = l..o(n + l)P(n + 1In)~(n + 1) \n\n1 nil nil \n\nK(n + 1) \n!L(n + 1) \n~(n + lin + 1) \nP(n + lin + 1) \n\n+ 2 L L~i~j (H~o(n + 1) P(n + 1In)illo(n + 1) P(n + lin\u00bb) \n+R \n\ni=1 j=1 \n\nP(n + 1In)l..o(n + 1)8-I (n + 1) \n~(n + 1) -\nl..o(n + l)~(n + lin) \n~(n + lin) + K(n + l)!L(n + 1) \n(I - K(n + l)l..o(n + 1\u00bb P(n + lin) (I - K(n + l)l..o(n + I\u00bb' \n\n+ K(n + 1) R K(n + I)' \n\nHere, K(n + 1) is the Kalman Gain matrix and !L(n + 1) is the innovation signal. \n\n3 BLACK-SCHOLES MODEL \n\nThe Black-Scholes equation for calculating the price of an European style call option \n(Hull, 1993) is \n\nwhere, \n\nIn(8j X) + (r + -%- )0m \n\n2 \n\n(70m \n\nd1 \n\n-\n\n(70m \n\nHere, C is the price of the call option, 8 the underlying asset price, X the strike \nprice of the option at maturity, tm the time to maturity and r is the risk free interest \nrate. (7 is a term known as volatility and may be seen as an instantaneous variance \nof the time variation of the asset price. N(.) is the cumulative normal function. \nFor a derivation of the formula and the assumptions upon which it is based see \nHull, 1993. Readers unfamiliar with financial terms only need to know that all the \nquantities in the above equation, except (7, can be directly observed. (7 is usually \nestimated from a small moving window of data of about 50 trading days. \nThe equivalent formula for the price of a put option is given by \n\nP = -8 N(-dl ) + X e- rt .,. N(-d2 ), \n\nFor recursive estimation of the option prices with this model, I assume that the \ninstantaneous picture given by the Black Scholes model is correct. The state vector \nis two dimensional and consists of the volatility (7 and the interest rate r . The \nJacobian and Hessian required for applying EKF algorithm are \n\n10 \n\n( aC \n\nau \naP \nau \n\nor \nBe ) \naP \nOr \n\nIDo \n\n( a2c \n\nau2 \na2c \nauar \n\nB'e ) ~. = \n\n; \n\nauar \na2c \n8r2 \n\na2p \nau2 \na2p \nauar \n\n( \n\na2p ) \n\nauar \na2p \nar2 \n\nExpressions for the terms in these matrices are given in table 1. \n\n\fSequential Tracking of Financial Option Prices \n\n963 \n\nTable 1: First and Second Derivatives of the Black Scholes Model \n\naC \nau \n\n-\n-\n\naP \nau \n\nS 0m N'(d1 ) \n\naC \nOr \n\naP \nOr \n\nXtmexp( -rtm)N(d2 ) \n\n-Xtmexp( -rtm)N( -d2 ) \n\na2C \n-\nau2 -\n\na2p \nau2 \n\nS~dld2 N'(dd \n\na2C \nar2 \n\na2p \nar2 \n\n-Xtmexp( -rtm) (tmN(d2 ) \n\nXtmexp( -rtm) (tmN( -d2 ) \n\n-\n\n-\n\nd2 'fFN' (d2 ) ) \n\nd2 'fFN' ( -d2 )) \n\na2c \nauar \n\n-\n-\n\na2p \naUaT \n\n-S!lt\", N'(dd \n\n4 NEURAL NETWORK MODELS \n\nThe data driven neural network model considered here is the Radial Basis FUnctions \nNetwork (RBF) . Following Hutchinson et al, I use the following architecture: \n\nwhere U is the two dimensional input data vector consisting of the asset price \nand time to maturity. The asset price S is normalised by the strike price of the \noption X. The time to maturity, trn , is also normalised such that the full lifetime \nof the option gets a value 1.0. These normalisations is the reason for considering \noptions in pairs with the same strike price and time of maturity in this study. The \nnonlinear function 1>(.) is set to 1>( Q) = va and m = 4. With the nonlinear part \nof the network fixed, Kalman filter training of the RBF model is straightforward \n(see Kadirkamanathan & Niranjan, 1993). In the simulations studied in this paper, \nI used two approaches to fix the nonlinear functions. The first was to use the /-l.S \n-J \nand the ~ published in Hutchinson et al. The second was to select the /-l . terms as \nrandom subsets of the training data and set ~ to I. The estimation problem is now \nlinear and hence the Kalman filter equations become much simpler than the EKF \nequations used in the training of the Black-Scholes model. \nIn addition to training by EKF, I also implemented a batch training of the RBF \nmodel in which a moving window of data was used, training on data from (n -\n50) to n days and testing on day (n + 1). Since it is natural to assume that data \ncloser to the test day is more appropriate than data far back in time, I incorporated \na weighting function to weight the errors linearly, in the minimisation process. The \nleast squares solution, with a weighting function, is' given by the modified pseudo \n\n-J \n\n\f964 \n\nM. Niranjan \n\nTable 2: Comparison of the Approximation Errors for Different Methods \n\nStrike Price Trivial RBF Batch RBF Kalman BS Historic BS Kalman \n2925 \n3025 \n3125 \n3225 \n\n0.0180 \n0.0440 \n0.0112 \n0.0349 \n\n0.0173 \n0.0519 \n0.0193 \n0.0595 \n\n0.0790 \n0.0999 \n0.0764 \n0.1116 \n\n0.0632 \n0.1109 \n0.0455 \n0.0819 \n\n0.0845 \n0.1628 \n0.0343 \n0.0885 \n\ninverse \n\nl = (Y' W y)-l y' W t \n\nMatrix W is a diagonal matrix, consisting of the weighting function in its diagonal \nelements, t is the target values of options prices, and I is the vector containing the \nunknown coefficients AI, ... ,Am . The elements of Yare given by Yij = j(Ui ), \nwith j = 1, ... ,m and i = n - 50, ... ,n. \n\n5 SIMULATIONS \n\nThe data set for teh experiments consisted of call and put option contracts on the \nFTSE-100 Index, during the period February 1994 to December 1994. The date of \nmaturity of all contracts was December 1994. Five pairs (Call and Put) of contracts \nat strike prices of 2925, 3025, 3125, 3225, and 3325. \nThe tracking behaviour of the EKF for one of the pairs is shown in Fig. 1 for a \ncall/put pair with strike price 3125. Fig. 2 shows the trajectories of the underlying \nstate vector for four different call/put option pairs. Table 2 shows the squared errors \nin the approximation errors computed over the last 100 days of data (allowing for \nan initial period of convergence of the recursive algorithms) . \n\n6 DISCUSSION \n\nThis paper presents a sequential approach to tracking the price of options contracts. \nThe sequential approach is based on the Extended Kalman Filter algorithm, and I \nshow how it may be used to identify a parametric model of the underlying nonlinear \nsystem. The model based approach of the finance community and the data driven \napproach of neural computing community lead to good estimates of the observed \nprice of the options contracts when estimated in this manner. \n\nIn the state space formulation of the Black-Scholes model, the volatility and inter(cid:173)\nest rate are estimated from the data. I trust the instantaneous picture presented \nby the model based approach, but reestimate the underlying parameters. This is \ndifferent from conventional wisdom, where the risk free interest rate is set to some \nfigure observed in the bond markets. The value of volatility that gives the correct \noptions price through Black Scholes equation is called option implied volatility, and \nis usually different for different options. Option traders often use the differences \nin implied volatility to take trading positions. In the formulation presented here, \nthere is an extra freedom coming in the form what one might call implied interest \nrates. It's difference from the interest rates observed in the markets might explain \ntrader speculation about risk associated with a particular currency. \n\nThe derivatives of the RBF model output with respect to its inputs is easy to \ncompute. Hutchinson et a1 use this to define a highly relevant performance measure \n\n\fSequential Tracking of Financial Option Prices \n\n965 \n\n(a) Estimated Call Option Price \n\n0.25 \\ \n\n0.2 \n~ 0.15 \n0.1 \n\n0.05 \n\n---True \n~~Estimaie \n\nO~--~----L---~----~--~----~----L----L----~--~----~ \n220 \n\n120 \n\n140 \n\n100 \n\n160 \n\n180 \n\n200 \n\n20 \n\n40 \n\n60 \n\n80 \n\n0.1.---.----.----.----.----.----.----.----.----.----,----. . \n\n(b) Estimated Put Option Price \n\n0.08 \n\n0.06 \n\n~ \n\n0.04 \n\n---True \n\n. :Estimate . \n. \n\n. \n\n0.02 \n\n20 \n\n40 \n\n60 \n\n80 \n\n100 \n\n120 \n\ntime \n\n140 \n\n160 \n\n180 \n\n200 \n\n220 \n\nFigure 1: Tracking Black-Scholes Model with EKF; Estimates of Call and Put Prices \n\nsuitable to this particular application, namely the tracking error of a delta neutral \nportfolio. This is an evaluation that is somewhat unfair to the RBF model since \nat the time of training, the network is not shown the derivatives. An interesting \ncombination of the work presented in this paper and Hutchinson et al's performance \nmeasure is to train the neural network to approximate the observed option prices \nand simultaneously force the derivative network to approximate the delta observed \nin the markets. \n\nReferences \n\nBar-Shalom, Y. & Li, X-R. (1993), 'Estimation and Tracking: Principles, Tech(cid:173)\nniques and Software', Artech House, London. \nCandy, J. V. (1986), 'Signal Processing: The Model Based Aproach', McGraw-Hill, \nNew York. \n\nHull, J. (1993), 'Options, Futures and Other Derivative Securities', Prentice Hall, \nNJ. \n\nHutchinson, J. M., Lo, A. W . & Poggio, T. (1994), 'A Nonparametric Approach to \nPricing and Hedging Derivative Securities Via Learning Networks', The Journal of \nFinance, Vol XLIX, No.3., 851-889. \n\n\f966 \n\nM. Niranjan \n\n0.055,....----.-----.----.-----,-----.----.-----,----. \n\nTrajectory of State Vector \n\n: ........ ... . \" \n\no \nx \nlIE \n\n\u00b72925 \n3025 \n3125 . . \n\n+ . ~225. \n\n0.05 \n\n0.045 \n\n0.04 \n\nQ) 0.035 \niii c:: \n~ 0.03 \n$ \nc: \n- 0.025 \n\n0.02 \n\n0.015 \n\n0.01 \n\n. .... .... \", \n\n0.005L.----...J-----'----L...----\"-----'----L...----\"------' \n0.3 \n\n0.18 \n\n0.24 \n\n0.28 \n\n0.14 \n\n0.16 \n\n0.2 \n\n0.22 \n\nVolatility \n\n0.26 \n\nFigure 2: Tracking Black-Scholes Model with EKF; Estimates of Call and Put Prices \nand the Trajectory of the State Vector \n\nKadirkamanathan, V. & Niranjan, M {1993}, 'A Function Estimation Approach to \nSequential Learning with Neural Networks', Neural Computation 5, pp . 954-975. \nLowe, D. {1995}, 'On the use of Nonlocal and Non Positive Definite Basis Functions \nin Radial Basis Function Networks', Proceedings of the lEE Conference on Artificial \nNeural Networks, lEE Conference Publication No. 409, pp 206-211. \n\nNiranjan, M., Cox, I. J., Hingorani, S. {1994}, 'Recursive Estimation of Formants \nin Speech', Proceedings of the International Conference on Acoustics, Speech and \nSignal Processing, ICASSP '94, Adelaide. \nPuskorius, G.V. & Feldkamp, L.A. {1994}, 'Neurocontrol of Nonlinear Dynamical \nSystems with Kalman Filter-Trained Recurrent Networks', IEEE Transactions on \nNeural Networks, 5 {2}, pp 279-297. \n\n\f", "award": [], "sourceid": 1245, "authors": [{"given_name": "Mahesan", "family_name": "Niranjan", "institution": null}]}