Toward Real Time Estimation of Black-Scholes Parameters By Steve Rogers, member, EE-Pub Published: December 5, 2005
Abstract For mid to high frequency options trading it is advantageous to arrive at an estimate for the parameters associated with the seminal Black-Scholes equation. Options trading is a very competitive industry and any increased accuracy gained by real-time estimation would be helpful. The Black-Scholes explicit finite difference approach may be modified such that the parameters may be estimated by a large variety of recursive real-time approaches. This paper provides an introduction into the explicit finite difference formulation, a few of the well-known real-time recursive estimation techniques, and then applies them to the resulting Black-Scholes approximation equation.
Article Information Field of Study—Signal Processing Keywords—adaptive-filter, Black-Scholes-equation, Kalman-filte
I. INTRODUCTION The seminal Black-Scholes [1] equation is the most commonly used approach to modeling and evaluating options and derivatives in use today. Most usage is based on closed form estimations of the ‘Greeks’ which are based on the Black-Scholes parameter estimates. The explicit finite difference approach [2-4] is one of the premier mathematical approaches used to solve the Black-Scholes equation. It may be manipulated easily into the standard parameter estimation form, which will be explained later in the paper. A description of the Black-Scholes classical equation follows [2, 4] in Figure 1.
Figure 1 Black-Scholes Classical Equations The parameters described above are slowly time-varying and can be estimated using classical estimation techniques assuming we have values for the partial differential terms.
We may now make estimations of the equation above by using standard approximations [2-4]. The Black-Scholes equations may be discretized following the explicit finite difference approach [2-4] as shown in Figure 2. This figure shows the approximations that enable us to go from the Black-Scholes partial differential equation to a difference equation.
Figure 2 Explicit Finite Difference Development Note that the superscript k in the approximations of Figure 2 refers to time updates, whereas the subscript i refers to asset value increments. Assuming the appropriate values in the explicit finite difference formulation are available, the equation of Figure 2 may be expressed as explained in Figure 3. In order to facilitate parameter estimation the explicit finite difference equation must be reformulated.
Figure 3 Linear-In-Parameters Estimation Model With the system in the form
we may now consider parameter estimation.
II. PARAMETER ESTIMATION Numerous parameter estimation techniques have been developed over the years for the representation used in Figure 3, including fixed filters, adaptive/recursive filters and Kalman filters [5-7]. Fixed filters require a window of time series data between filter parameter updates. In the other approaches the filter parameter updates occur at each time step or introduction of new data. This paper will emphasize the recursive and Kalman filter approaches since we are generally dealing with slowly time-varying parameters to be estimated. A diagram of a typical general adaptive filter is given in Figure 4.
Figure 4 FIR Adaptive Filter
Adaptive filters are filters that automatically change their characteristics to attain the desired response at a given update point in time. Adaptive filters involve one or more input signals and a desired response signal that may or may not be accessible. All adaptive filters contain three modules [5-7] as shown in Figure 4. The filtering structure module forms the output of the filter using measurements of the input signal(s). The structure may be linear or nonlinear. The performance evaluation module assesses the quality of the output with respect to the application requirements. There is normally some balancing needed between what is acceptable and what is mathematically tractable. The adaptation algorithm module uses the performance evaluation criteria or some derivation from it to modify the filter parameters and continually improve its performance. The design requires a significant understanding of the application, even though there is much commonality among most applications. The designer must be able to discriminate his application to select the adaptive filter features to be applied. Most adaptive filters are supervised as illustrated in Figure 5. From the equations at the bottom of the figure we find that the error is formed by subtracting the filter output from the desired signal. The adaptive algorithm causes a change in the filter parameters according to an increment designed based on performance criteria.
Figure 5 Supervised Adaptive Filter Numerous adaptive filter update strategies have been developed including the normalized least mean square algorithm, which is one of the more commonly used. The algorithm is summarized in Figure 6.
Figure 6 Summary of the Normalized LMS Algorithm
III. DISCRETE KALMAN FILTERS The Kalman filter is a special case of the optimal linear filter algorithms. The development is given in numerous texts [6-8] and will not be repeated here. The algorithm consists of five basic equations8 shown in Figure 7.
Figure 7 Basic equations of a Kalman filter A summary of the Kalman filter algorithm is in Figure 8 and a block diagram is shown in Figure 9. The Kalman filter can be used for parameter estimation of the Black-Scholes explicit finite difference formulation shown in Figure 3. In the parameter estimation approach the estimated states are the parameters to be estimated which cause the convergence of the model to the input signal in the same way that the FIR adaptive filter described above does. The application to parameter estimation which may then be used with the Black-Scholes explicit finite difference formulation is described in Figure 10.
Figure 8 Kalman Filter Algorithm
Figure 9 Kalman Filter Block Diagram
Figure 10 Application of Kalman Filter to Parameter Estimation
IV. CONCLUSION The Black-Scholes equation is very important to the financial industry because it is used to model a great number of financial instruments. In some applications it would be useful to have frequent updates of the coefficients in the equation. We have discussed and presented a few possible recursive approaches that would enable real-time parameter
updates. Future research involves testing the approaches using actual data and calculating the ‘Greeks’ from the Black-Scholes equation parameters in real-time.
V. REFERENCES [1] Black, F. and Scholes, M., ‘The Pricing of Options and Corporate Liabilities.’ Journal of Political Economy, v. 81, no. 3 (1973) p. 659-673 [2] Wilmott, P., Quantitative Finance, John Wiley & Sons, 2001, ISBN 0471498629. [3] Hull, J., Options, Futures, and Other Derivatives, 5th Ed., Prentice Hall, 2003, ISBN 81-203-2237-1. [4] Ioffe, B. and Ioffe, M., ‘Application of Finite Difference Method for Pricing Barrier Options,’ http://www.egartech.com/docs/finite_difference_barrier_options.pdf. [5] Moscinski, J. and Ogonowski, Z., Advanced Control with Matlab and Simulink, 1995, Ellis Norwood, ISBN 0-13-309667-X. [6] Haykin, S., Adaptive Filter Theory, 3rd Ed., 1996, Prentice Hall, ISBN 0-13-322760-X. [7] Manolakis, D., etal, Statistical and Adaptive Signal Processing, 2000, McGraw-Hill, ISBN 0-07-1166602. [8] Brown, R., and Hwang, P., Introduction to Random Signals and Applied Kalman Filtering, 3rd Ed., 1997, John Wiley & Sons, ISBN 0-471-12839-2.