Currently, Richardson extrapolation is a commonly used technique in financial applications.  We will look at its effect on option pricing.  Benefits include that Repeated Richardson Extrapolation (RRE) can be repeatedly used until a desired accuracy is reached. 

                Richardson Extrapolation has been used in the past with Geske-Johnson (uses arithmetic time steps) and modified Geske Johnson (uses geometric time steps) formulas.  Other time stepping methods that have been experimented with include…

 harmonic sequence: {1, 2, 3, 4, 5, 6, 7, 8, 9, …}

double harmonic (Deuflhard) sequence: {2, 4, 6, 8, 10, 12, 14, 16, …}

Burlisch sequence: {2, 4, 6, 8, 12, 16, 24, 32, …}

Romberg sequence: {2, 4, 8, 16, 32, 64…}

 The method is a fairly straight-forward Richardson extrapolation to eliminate the first term in the error series.

F(h) = a₀ + a₁h^γ1 + a₂h^γ2 + O(h^γ3)

F(q⁻¹h) = a₀ + a₁(q⁻¹h) ^γ1 + a₂(q⁻¹h) ^γ2 + O(h^γ3)

F(q⁻²h) = a₀ + a₁(q⁻²h) ^γ1 + a₂(q⁻²h) ^γ2 + O(h^γ3)

 F(0) = F(h) +(A/C)(F(h) − F(q⁻¹h)) – (B/C) (F(q⁻¹h) − F(q⁻²h))

where

A = q^−2γ2 − q^2γ1 + q^−γ1 − q^γ2,

B = q^−γ2 − q^γ1 ,

C = q^−2γ2(q^−γ1 – 1) − q^−2γ1(q^−γ2 – 1) + q^−γ2 − q^γ1

This produces results such as … F(0) = P(1, 2, 4) = P(4) +(5/3)(P(4) − P(2)) –(1/3)(P(2) − P(1)) where P(1, 2, 4) is the approximated value of the option and 1,2,4 are the steps used.

However, we cannot use repeated Richardson’s with this older method.  Cox-Rubinstein-Ross (CRR) and Breen’s model allow us to use repeated Richardon’s.  Through experimentation, Breen’s model has found to be more accurate than CRR based on RMS values.  This gives us the power to have any degree of accuracy we want.  We need a method to predict the interval of true option values for these methods.  Breen AB model uses arithmetic time steps to get the approximation formula while the modified Breen AB model uses geometric time steps.  We can use repeated Richardson Extrapolation for predicting intervals of American-Options values.   

 The full algorithm can be found here:  http://www.rstapleton.com/papers/retpao.pdf

It works in such a way that we can set up the following table

hi           A(i, 0)         A(i,1)          A(i,2)          A(i,3)      …

h1          A(1,0)        A(1,1)         A(1,2)         A(1,3)

h2          A(2,0)        A(2,1)         A(2,2)

h3          A(3,0)        A(3,1)         

h4          A(4,0)   

.

.

Schmidt proved that |A(i,m+1) – F(0)| <= | A(i,m+1) - A(i,m)| where F(0) is the true value of the option.  This implies that with each application of RRE, accuracy is gained, even when i is sufficiently large.

 Thus, Richardson extrapolation allows us to specify the accuracy of our approximation of the option price, and the Schmidt inequality can then be used to predict tight bounds on the value.  Mathematical techniques are very important for making financial decisions.  Richardson extrapolation could have a wide impact in financial areas because of its accuracy.  This method for predicting option values has been in use for less than 10 years.

References:

http://www.rstapleton.com/papers/retpao.pdf

http://www.dma.unive.it/wpdma/2006wp147.pdf

http://www.cse.uiuc.edu/iem/integration/richrdsn/

http://en.wikipedia.org/wiki/Richardson_extrapolation