dew {RND}

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

Details

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

Value

density value at x

Author(s)

Kam Hamidieh

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")