How do you choose the best numerical method for a given financial problem?

1 Identify the problem type

There are three main types of problems in finance: optimization, simulation, and valuation. Optimization problems involve finding the optimal values of some variables that maximize or minimize a certain objective function, such as portfolio allocation, risk management, or arbitrage. Simulation problems involve generating random scenarios or outcomes based on some assumptions or distributions, such as Monte Carlo methods, bootstrap methods, or stochastic differential equations. Valuation problems involve calculating the present or future value of some financial instruments, such as discounted cash flow, Black-Scholes formula, or binomial tree. Depending on the problem type, you will need different numerical methods that can handle the specific characteristics and challenges of the problem.

2 Evaluate the problem properties

Evaluate the properties of the problem, such as dimensionality, linearity, smoothness, and convexity. Dimensionality refers to the number of variables or parameters involved in the problem. Higher-dimensional problems are more difficult and time-consuming to solve than lower-dimensional problems, and may require more sophisticated numerical methods, such as sparse matrices, grid methods, or dimension reduction techniques. Linearity refers to whether the problem can be expressed as a linear combination of some basis functions or not. Linear problems are easier and faster to solve than nonlinear problems, and can often be solved by direct methods such as matrix inversion, linear programming, or least squares. Nonlinear problems may require iterative methods, such as Newton's method, gradient descent, or quasi-Newton methods.

Smoothness refers to whether the problem has continuous and differentiable functions or not. Smooth problems are more stable and accurate than non-smooth problems, and can often be solved by analytical methods such as Taylor series, Lagrange interpolation, or spline interpolation. Non-smooth problems may require numerical methods that can handle discontinuities, singularities, or jumps, such as finite difference methods, finite element methods, or wavelet methods.

Convexity refers to whether the problem has a single global optimum or not. Convex problems are more tractable and reliable than non-convex problems, and can often be solved by global methods such as simplex method, interior point method, or branch and bound method. Non-convex problems may require local methods that can find suboptimal solutions, such as hill climbing, simulated annealing, or genetic algorithms.

3 Compare the method features

Compare the features of the available methods, such as accuracy, efficiency, stability, and complexity. Accuracy refers to how close the numerical solution is to the true solution of the problem. Higher accuracy is desirable, but it may come at the cost of more computation time or memory. Efficiency refers to how fast the numerical method can produce a solution with a given level of accuracy. Higher efficiency is desirable, but it may come at the cost of more algorithmic sophistication or implementation difficulty.

Stability refers to how sensitive the numerical method is to errors or perturbations in the input data or parameters. Higher stability is desirable, but it may come at the cost of more robustness checks or error control. Complexity refers to how difficult the numerical method is to understand, implement, or modify. Lower complexity is desirable, but it may come at the cost of more generality or flexibility. You should weigh the pros and cons of each method feature and choose the one that best suits your problem and your resources.