Richardson extrapolation is a valuable numerical technique that can enhance the precision of estimates derived from approximation methods, widely applicable in fields like numerical analysis and financial modeling.
Richardson extrapolation is based on the principle that the error in a numerical estimate can often be expressed as a power series in relation to some parameter, such as step size in numerical differentiation or grid size in numerical integration.
In the context of pricing exotic options, which are financial derivatives with complex payoffs and conditions, Richardson extrapolation can significantly refine pricing accuracy.
Using numerical methods like binomial trees or finite difference methods, the exotic option price is calculated with different time steps. Acknowledge that discretization of the model induces an error in price estimates, which grows with step size.
Combine the two price estimates with weights based on the step size ratio to negate the largest error term. This extrapolation is under the presumption that the error reduces quadratically with a smaller step size.
The extrapolated price is more accurate than the initial estimates as the main source of error is minimized.
The error term in numerical methods often correlates with the square of the step size due to the approximation techniques employed. Key reasons include:
Numerical methods typically involve Taylor series, where truncating after a certain term introduces a local truncation error, often proportional to h^2 .
For second-order methods, halving the step size reduces the error by a factor of four, illustrating that the error is proportional to the square of the step size.
In exotic option pricing, smaller step sizes yield more accurate asset price representations.
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