Sequential Tracking in Pricing Financial 
Options using Model Based and Neural 
Network Approaches 
Mahesan Niranjan 
Cambridge University Engineering Department 
Cambridge CB2 1PZ, England 
niranjan(eng.cam.ac.uk 
Abstract 
This paper shows how the prices of option contracts traded in finan- 
cial markets can be tracked sequentially by means of the Extended 
Kalman Filter algorithm. I consider call and put option pairs with 
identical strike price and time of maturity as a two output nonlin- 
ear system. The Black-Scholes approach popular in Finance liter- 
ature and the Radial Basis Functions neural network are used in 
modelling the nonlinear system generating these observations. I 
show how both these systems may be identified recursively using 
the EKF algorithm. I present results of simulations on some FTSE 
100 Index options data and discuss the implications of viewing the 
pricing problem in this sequential manner. 
1 INTRODUCTION 
Data from the financial markets has recently been of much interest to the neural 
computing community. The complexity of the underlying macro-economic system 
and how traders react to the flow of information leads to highly nonlinear rela- 
tionships between observations. Further, the underlying system is essentially time 
varying, making any analysis both difficult and interesting. A number of prob- 
lems, including forecasting a univariate time series from past observations, rating 
credit risk, optimal selection of portfolio components and pricing options have been 
thrown at neural networks recently. 
The problem addressed in this paper is that of sequential estimation, applied to 
pricing of options contracts. In a nonstationary environment, such as financial 
markets, sequential estimation is the natural approach to modelling. This is because 
data arrives at the modeller sequentially, and there is the need to build and apply the 
Sequential Tracking of Financial Option Prices 961 
best possible model with available data. At the next point in time, some additional 
data is available and the task becomes one of optimally updating the model to 
account for the new data. This can either be done by reestimating the model with 
a moving window of data or by sequentially propagating the estimates of model 
parameters and some associated information (such as the error covariance matrices 
in the Kalman filtering flamework discussed in this paper). 
2 SEQUENTIAL ESTIMATION 
Sequential estimation of nonlinear models via the Extended Kalman Filter algo- 
rithm is well known (e.g. Candy, 1986; Bar-Shalom & Li, 1993). This approach 
has also been widely applied to the training of Neural Network axchitectures (e.g. 
Kadirkamanathan & Niranjan, 1993; Puskorius & Feldkamp, 1994). In this section, 
I give the necessary equations for a second order EKF, i.e. Taylor series expansion 
of the nonlinear output equations, truncated at order two, for the state space model 
simplified to the system identification framework considered here. 
The parameter vector or state vector, 0, is assumed to have the following simple 
random walk dynamics. 
0_(n+l) = 0_(n) + u(n), 
where u(n) is a noise term, known as process noise. u(n) is of the same dimension- 
ality as the number of states used to represent the system. The process noise gives 
a random walk freedom to the state dynamics facilitating the tracking behaviour 
desired in nonstationary environments. In using the Kalman filtering framework, 
we assume the covariance matrix of this noise process, denoted Q, is known. In 
practice, we set Q to some small diagonal matrix. 
The observations from the system are given by the equation 
_() = _f(0, _V) + w(), 
where, the vector z(n) is the output of the system consisting of the call and put 
option prices at time n. U denotes the input information. In the problem considered 
here, U consists of the price of the underlying asset and the time to maturity if the 
option. w is known as the measurement noise, covaxiance matrix of which, denoted 
R, is also assumed to be known. Setting the paxameters R and Q is done by trial 
and error and knowledge about the noise processes. In the estimation framework 
considered here, Q and R determine the tracking behaviour of the system. For the 
experiments reported in this paper, I have set these by trial and error, but more 
systematic approaches involving multiple models is possible (Niranjan eta/, 1994). 
The prior estimates at time (n + 1), using all the data upto time (n) and the model 
of the dynamical system, or the prediction phase of the Kalman algorithm is given 
by the equations: 
(n4- l[n) = (n[n) 
/5(n 4- l[n) = /5(n[n) 4- Q(n) 
i(n + lln) = + lln)) + 
1 no _ ( H----O(n + 
Zeitr 1) JS(n l[n)) 
i----1 
where J-0 and H0 are the Jacobian and Hessians of the output z_; also no = 2. e i 
are unit vectors in direction i. tr(.) denotes trace of a matrix. The posterior esti- 
962 M. Niranjan 
mates or the correction phase of the Kalman algorithm are given by the equations: 
S(n q- 1) - J_0(n q- 1)16(n q- l[n)J_(n q- 1) 
1 o o (Ho(n+ + 1) fi(n l[n)) 
+  ZZe-ie-J -- 1) P(n 11n)Ho(n+ + 
i=1 j=l 
+R 
K(n + 1) = P(n + lln)J_o(n + 1)s-l(n + 1) 
v(. + 1) = _(. + 1) - J_o(- + 1)(. + 11-) 
(- + 11- + 1) = (. + 11-) + :(- + 1)v(. + 1) 
P(- + 11- + 1) = ( - :(. + 1)J_o(. + 1)) P(. + 11-) ( - :(- + 1)J_o(- + 1))' 
+ K(. + 1) R K(. + 1/ 
Here, K(n + 1) is the Kalman Gain matrix and v(n + 1) is the innovation signal. 
3 BLACK-SCHOLES MODEL 
The Black-Scholes equation for calculating the price of an European style call option 
(Hull, 1993) is 
where, 
C = S .]V'(dl) -- X e-r ;m .]V'(d2) ' 
ln(S/X) + (r +-5-)tv/ 
dl = 
d2 = dl - rr tv/ 
Here, C is the price of the call option, $ the underlying asset price, X the strike 
price of the option at maturity, t, the time to maturity and r is the risk free interest 
rate. a is a term known as volatility and may be seen as an instantaneous variance 
of the time variation of the asset price. Af(.) is the cumulative normal function. 
For a derivation of the formula and the assumptions upon which it is based see 
Hull, 1993. Readers unfamiliar with financial terms only need to know that all the 
quantities in the above equation, except a, can be directly observed. a is usually 
estimated from a small moving window of data of about 50 trading days. 
The equivalent formula for the price of a put option is given by 
P = -S.Af(-dl) + X e- ' tm Af(-d2), 
For recursive estimation of the option prices with this model, ! assume that the 
instantaneous picture given by the Black $choles model is correct. The state vector 
is two dimensional and consists of the volatility a and the interest rate r. The 
Jacobian and Hessian required for applying EKF algorithm are 
Oa Or Oa 2 OaOr Oa  OaOr 
!o = ; o = ; IL = 
OP OP OC OC Op Op 
Ocr Or oqaOr Or 2 oqaOr Or 2 
Expressions for the terms in these matrices are given in table 1. 
Sequential Tracking of Financial Option Prices 963 
Table 1: First and Second Derivatives of the Black Scholes Model 
4 NEURAL NETWORK MODELS 
The data driven neural network model considered here is the Radial Basis Functions 
Network (RBF). Following Hutchinson et al, I use the following architecture: 
j----1 
where U is the two dimensional input data vector consisting of the asset price 
and time to maturity. The asset price $ is normalised by the strike price of the 
option X. The time to maturity, t,, is also normalised such that the full lifetime 
of the option gets a value 1.0. These normalisations is the reason for considering 
options in pairs with the same strike price and time of maturity in this study. The 
nonlinear function b(.) is set to b(a) = v  and m = 4. With the nonlinear part 
of the network fixed, Kalman filter training of the RBF model is straightforward 
(see Kadirkamanathan & Niranjan, 1993). In the simulations studied in this paper, 
I used two approaches to fix the nonlinear functions. The first was to use the js 
and the E published in Hutchinson et al. The second was to select the/_j terms as 
random subsets of the training data and set E to I. The estimation problem is now 
linear and hence the Kalman filter equations become much simpler than the EKF 
equations used in the training of the Black-Scholes model. 
In addition to training by EKF, I also implemented a batch training of the RBF 
model in which a moving window of data was used, training on data from (n - 
50) to n days and testing on day (n + 1). Since it is natural to assume that data 
closer to the test day is more appropriate than data far back in time, ! incorporated 
a weighting function to weight the errors linearly, in 'the minimisation process. The 
least squares solution, with a weighting function, is' given by the modified pseudo 
964 M. Niranjan 
Table 2: Comparison of the Approximation Errors for Different Methods 
Strike Price Trivial RBF Batch RBF Kalman BS Historic BS Kalman 
2925 0.0790 0.0632 0.0173 0.0845 0.0180 
3025 0.0999 0.1109 0.0519 0.1628 0.0440 
3125 0.0764 0.0455 0.0193 0.0343 0.0112 
3225 0.1116 0.0819 0.0595 0.0885 0.0349 
inverse 
l = (Y'WY) - Y'Wt 
Matrix W is a diagonal matrix, consisting of the weighting function in its diagonal 
elements, t is the target values of options prices, and / is the vector containing the 
unknown coefficients A, ...,A,. The elements of Y are given by yij - bj(Ui), 
with j - 1, ..., m and i - n - 50, ..., n. 
5 SIMULATIONS 
The data set for teh experiments consisted of call and put option contracts on the 
FTSE-100 Index, during the period February 1994 to December 1994. The date of 
maturity of all contracts was December 1994. Five pairs (Call and Put) of contracts 
at strike prices of 2925, 3025, 3125, 3225, and 3325. 
The tracking behaviour of the EKF for one of the pairs is shown in Fig. 1 for a 
call/put pair with strike price 3125. Fig. 2 shows the trajectories of the underlying 
state vector for four different call/put option pairs. Table 2 shows the squared errors 
in the approximation errors computed over the last 100 days of data (allowing for 
an initial period of convergence of the recursive algorithms). 
6 DISCUSSION 
This paper presents a sequential approach to tracking the price of options contracts. 
The sequential approach is based on the Extended Kalman Filter algorithm, and I 
show how it may be used to identify a parametric model of the underlying nonlinear 
system. The model based approach of the finance community and the data driven 
approach of neural computing community lead to good estimates of the observed 
price of the options contracts when estimated in this manner. 
In the state space formulation of the Black-Scholes model, the volatility and inter- 
est rate are estimated from the data. I trust the instantaneous picture presented 
by the model based approach, but reestimate the underlying parameters. This is 
different from conventional wisdom, where the risk free interest rate is set to some 
figure observed in the bond markets. The value of volatility that gives the correct 
options price through Black Scholes equation is called option implied volatility, and 
is usually different for different options. Option traders often use the differences 
in implied volatility to take trading positions. In the formulation presented here, 
there is an extra freedom coming in the form what one might call implied interest 
rates. It's difference from the interest rates observed in the markets might explain 
trader speculation about risk associated with a particular currency. 
The derivatives of the RBF model output with respect to its inputs is easy to 
compute. Hutchinson et al use this to define a highly relevant performance measure 
Sequential Tracking of Financial Option Prices 965 
(a) Estimated Call Option Price 
0.25 . :- ..... : ......................... :-- 
 : i i , :True 
0.2 -- :Estimate 
0.1 .,,,.,.^ _ 'A : : : : 
0.05 ' ' ' " 
20 40 60 80 100 120 140 160 180 200 220 
0.1 
0.08 
0.06 
0.04 
0.02 
0 
(b) Estimated Put Option Price 
I I I I I I it 
 ' :True . 
 .-.... :Estimate 
,-,,,./., /jyt d . 'v : : : ' : : 
........ "'--" .... ' i ........ -'- 
20 40 60 80 100 120 140 160 180 200 220 
time 
Figure 1: Tracking Black-Scholes Model with EKF; Estimates of Call and Put Prices 
suitable to this particular application, namely the tracking error of a delta neutral 
portfolio. This is an evaluation that is somewhat unfair to the RBF model since 
at the time of training, the network is not shown the derivatives. An interesting 
combination of the work presented in this paper and Hutchinson et a/'s performance 
measure is to train the neural network to approximate the observed option prices 
and simultaneously force the derivative network to approximate the delta observed 
in the markets. 
References 
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niques and Software', Artech House, London. 
Candy, J. V. (1986), 'Signal Processing: The Model Based Aproach', McGraw-Hill, 
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Hull, J. (1993), 'Options, Futures and Other Derivative Securities', Prentice Hall, 
NJ. 
Hutchinson, J. M., Lo, A. W. & Poggio, T. (1994), 'A Nonparametric Approach to 
Pricing and Hedging Derivative Securities Via Learning Networks', The Journa/of 
Finance, Vol XLIX, No. 3., 851-889. 
966 M. Niranjan 
0.055 
Trajectory of State Vector 
I I ! ! 
0.05 
0.045 
0.04 
e 0.035 
 0.03 
- 0.025 
0.02 
0.015 
0.01 
.2925 
3025 
3125 
3225 ' 
x 
............. +-+. i( x.O. 
+ xO 
+ (.xO 
.......... + .----!x o ..... 
+  x o 
++  x o 
: ++ x o  
 .'t.......:. ....... . .......... . ........ + ......... '!' ..  ....... 
I I I I I 
O. 16 0.18 0.2 0.22 0.24 0,26 0.28 
Volatili 
0.005 
0.14 0.3 
Figure 2: Tracking Black-Scholes Model with EKF; Estimates of Call and Put Prices 
and the Trajectory of the State Vector 
Kadirkamanathan, V. & Niranjan, M (1993), 'A Function Estimation Approach to 
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