RND (version 1.1)

approximate.max: Max Function Approximation

Description

approximate.max gives a smooth approximation to the max function.

Usage

approximate.max(x, y, k = 5)

Arguments

x

the first argument for the max function

y

the second argument fot the max function

k

a tuning parameter. The larger this value, the closer the function output to a true max function.

Value

• approximate maximum of x and y

Details

approximate.max approximates the max of x, and y as follows: $$g(x,y) = \frac{1}{1 + \exp(-k(x-y))}, \ \ \max(x,y) \approx x g(x,y) + y(1 - g(x,y))$$

References

Melick, W. R. and Thomas, C.P. (1997) Recovering an asset's implied pdf from option proces: An application to crude oil during the gulf crisis. Journal of Financial and Quantitative Analysis, 32(1), 91-115

Examples

#
# To see how the max function compares with approximate.max,
# run the following code.
#

i = seq(from = 0, to = 10, by = 0.25)
y = i - 5
max.values = pmax(0,y)
approximate.max.values = approximate.max(0,y,k=5)
matplot(i, cbind(max.values, approximate.max.values), lty = 1, type = "l", 
 col=c("black","red"), main = "Max in Black, Approximate Max in Red")