RND (version 1.2)

dew: Edgeworth Density

Description

dew is the probability density function implied by the Edgeworth expansion method.

Usage

dew(x, r, y, te, s0, sigma, skew, kurt)

Arguments

x

value at which the denisty is to be evaluated

r

risk free rate

y

dividend yield

te

time to expiration

s0

current asset value

sigma

volatility

skew

normalized skewness

kurt

normalized kurtosis

Value

density value at x

Details

This density function attempts to capture deviations from lognormal density by using Edgeworth expansions.

References

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

R. Jarrow and A. Rudd (1982) Approximate valuation for arbitrary stochastic processes. Journal of Finanical Economics, 10, 347-369

C.J. Corrado and T. Su (1996) S&P 500 index option tests of Jarrow and Rudd's approximate option valuation formula. Journal of Futures Markets, 6, 611-629

Examples

#
# Look at a true lognorma density & related dew
#
r       = 0.05
y       = 0.03
s0      = 1000
sigma   = 0.25
te      = 100/365
strikes = seq(from=600, to = 1400, by = 1)
v       = sqrt(exp(sigma^2 * te) - 1)
ln.skew = 3 * v + v^3
ln.kurt = 16 * v^2 + 15 * v^4 + 6 * v^6 + v^8

skew.4 = ln.skew * 1.50
kurt.4 = ln.kurt * 1.50

skew.5 = ln.skew * 0.50
kurt.5 = ln.kurt * 2.00

ew.density.4   = dew(x=strikes, r=r, y=y, te=te, s0=s0, sigma=sigma, 
                     skew=skew.4, kurt=kurt.4)
ew.density.5   = dew(x=strikes, r=r, y=y, te=te, s0=s0, sigma=sigma, 
                     skew=skew.5, kurt=kurt.5)
bsm.density    = dlnorm(x = strikes, meanlog = log(s0) + (r - y - (sigma^2)/2)*te, 
                 sdlog = sigma*sqrt(te), log = FALSE)

matplot(strikes, cbind(bsm.density, ew.density.4, ew.density.5), type="l", 
        lty=c(1,1,1), col=c("black","red","blue"), 
        main="Black = BSM,  Red = EW 1.5 Times,  Blue = EW 0.50 & 2")