RND (version 1.2)

dshimko: Density Implied by Shimko Method

Description

dshimko is the probability density function implied by the Shimko method.

Usage

dshimko(r, te, s0, k, y, a0, a1, a2)

Arguments

r

risk free rate

te

time to expiration

s0

current asset value

k

strike at which volatility to be computed

y

dividend yield

a0

constant term in the quadratic polynomial

a1

coefficient term of k in the quadratic polynomial

a2

coefficient term of k squared in the quadratic polynomial

Value

Details

The implied volatility is modeled as: $\sigma(k) = a_0 + a_1 k + a_2 k^2$

References

D. Shimko (1993) Bounds of probability. Risk, 6, 33-47

E. Jondeau and S. Poon and M. Rockinger (2007): Financial Modeling Under Non-Gaussian Distributions Springer-Verlag, London

Examples

#
# a0, a1, a2 values come from Shimko's paper.
#

r     =  0.05
y     =  0.02
a0    =  0.892
a1    = -0.00387
a2    =  0.00000445
te    =  60/365
s0    =  400
k     =  seq(from = 250, to = 500, by = 1)
sigma =  0.15

#
# Does it look like a proper density and intergate to one?
#

dx = dshimko(r = r, te = te, s0 = s0, k = k, y = y, a0 = a0, a1 = a1, a2 = a2)
plot(dx ~ k, type="l")

#
# sum(dx) should be about 1 since dx is a density.
#

sum(dx)