The pricing of structured finance instruments is a major issue in risk management and the more complicated the instruments the more important advanced numerical pricing schemes become. In this article, we concentrate on schemes for financial models which lead to partial differential equations. This covers a wide range of models, such as Black-Scholes, certain models of stochastic volatility and the wide range of one or more factor short-rate models, such as Hull-White and Black-Karasinski. Green´s Functions & Adaptive Integration
Let us start with the easiest case of classical Black-Scholes for the pricing of vanilla options. In this case, the random walk for the underlying equity is assumed to be a geometric Brownian motion. In the absence of transaction costs, it is then possible to find a self-financing trading strategy which leads to a risk-free portfolio of being short one option and long delta shares for hedging. It can be derived then that the value of a European option within this framework satisfies the Black-Scholes equation, a parabolic partial differential equation. To make this Black-Scholes equation well-posed (meaning uniquely solvable with the solution depending continuously on the data), one has to specify an end condition (the pay-off function at maturity) and boundary conditions at zero and at infinity.
The solution of this problem, the fair value of an option, is then given as the payoff-function of the option convoluted by the resolvent kernel of the Black Scholes equation (being the log-normal density in the risk-free measure). This can be utilized for numerical schemes: European options with arbitrary payoffs can be priced by implementing numerical integration schemes like high order Gaussian integration. If one is interested in pricing Bermudan options, one can construct a grid at time t [Bermudan day k], and obtain the option values there by making a time step until t [Bermudan day k+1] and using the same representation as above. At the future time step, typically the Gauss integration points will not be the usual grid points, hence interpolation techniques should be applied. Adaptive Integration can be applied to a wide range of models and to a wide range of derivative instruments.
Upwind Techniques & Streamline Diffusion
The partial differential equations obtained by using one or more-factor short rate models or by modeling convertible bonds considering stochastic equity and stochastic interest rates can be interpreted as convection-diffusion-reaction equations. This type of equation is typically found in applications in continuum mechanics. Its numerical solution using standard discretization methods cause severe problems, resulting in high oscillations in the computed values. It is the drift term which is mainly responsible for these difficulties and which forces us to use specifically developed methods for the numerical solution. These methods have to use so-called upwind strategies, in order to obtain stability meaning, very roughly speaking, that you follow the flow of data to use the most relevant points. In trinomial tree methods, it is the up-branching and down-branching which takes into account the upwinding and leads to nonnegative weights which correspond to stability.
The standard streamline diffusion method, introduced by Hughes and Brooks for the numerical solution of convection, is such an upwind method and is based on the finite element method. It achieves stability by adding artificial diffusion into the direction of the streamlines, which are mainly determined by the drift. In addition to its global stability properties it is a method of higher order of convergence, which yields additional advantages compared with simple upwinding with finite differences.
Graph 2 shows the value of an option on a zero-coupon bond as a function of the short rate r and of the second state variable u of the Hull-White model. This picture demonstrates the stability and robustness properties of the streamline diffusion method.
Inverse Problems & Model Calibration
For the pricing of financial derivatives, the user has to provide input data which describe the random behavior of the underlying process. Typically, volatility is the most critical input. The standard routine in pricing complex structures is, first, to identify model parameters from market prices of liquid and actively traded instruments, and second, use the obtained parameters in advanced pricing schemes like the ones described above.
Calibrating model parameters in our PDE framework means the identification of parameters in parabolic differential equations, a problem which is ill-posed in the sense of Hadamard. This means:
a) For given market data, a solution (model parameters)
need not exist.
b) If it exists, the solution need not be unique.
c) The solution need not depend continuously on the data,
meaning that arbitrarily small perturbations of the data
might lead to arbitrarily large perturbations of the solution.
For the robust solution of ill-posed problems, so called regularization techniques have to be applied to obtain stable and robust algorithms.
Consider the problem of determining a local volatility function (in the sense of Dupire) *(S) from the prices of options with one maturity but with different strikes. As long as there is no noise in the data, output least square approaches work quite well as Graph 3 demonstrates
The solid line is the unknown volatility function and the spot price of the equity is assumed to be one. As expected, there is good identification as long as we are not too deep in the money or to deep out of the money. At the extreme ends, the option prices do not contain much information and therefore the solution identified (dashed line) depends on the starting level (dashed-dotted line).
If you add noise to the input data, and there is always noise in option prices due to bid-offer spreads, the situation becomes nasty without regularization.
With small noise levels of 0.1% and 0.5% output using the least squares approach, lead to oscillating results (Graph 4). But regularization helps (Graph 5):
The dotted curve shows the result for a balanced choice of the regularization parameter, the dashed curve is obtained from under-regularization, the dashed-dotted one for over-regularization. Posteriori techniques, which do not need knowledge on the true solution, for choosing optimal regularization strategies are available.
Conclusion
Advanced numerical techniques which take into account accuracy, speed, stability and robustness are a must in modern financial engineering. Algorithms from engineering applications like computational multiphysics problems can and should be applied to computational finance problems.
This week's Learning Curve was written by Andreas Binder, a managing director at MathConsult and the Industrial Mathematics Competence Center, Heinz Engl is head of the Industrial Mathematics Institute at the University of Linz, Austria and head of the Radon Institute for Computational and Applied Mathematics at the Austrian Academy of Sciences. He is also President of the Austrian Mathematical Society, Andrea Schatz, is a researcher at MathConsult in Linz.
