price.mln.option {RND}

Price Options on Mixture of Lognormals

Description

mln.option.price gives the price of a call and a put option at a strike when the risk neutral density is a mixture of two lognormals.

Usage

price.mln.option(r, te, y, k, alpha.1, meanlog.1, meanlog.2, sdlog.1, sdlog.2)

Arguments

r

risk free rate

te

time to expiration

y

dividend yield

k

strike

alpha.1

proportion of the first lognormal. Second one is 1 - alpha.1

meanlog.1

mean of the log of the first lognormal

meanlog.2

mean of the log of the second lognormal

sdlog.1

standard deviation of the log of the first lognormal

sdlog.2

standard deviation of the log of the second lognormal

Details

mln is the density f(x) = alpha.1 * g(x) + (1 - alpha.1) * h(x), where g and h are densities of two lognormals with parameters (mean.log.1, sdlog.1) and (mean.log.2, sdlog.2) respectively.

Value

call

call price

put

put price

s0

current value of the asset as implied by the mixture distribution

Author(s)

Kam Hamidieh

References

F. Gianluca and A. Roncoroni (2008) Implementing Models in Quantitative Finance: Methods and Cases

B. Bahra (1996): Probability distribution of future asset prices implied by option prices. Bank of England Quarterly Bulletin, August 1996, 299-311

P. Soderlind and L.E.O. Svensson (1997) New techniques to extract market expectations from financial instruments. Journal of Monetary Economics, 40, 383-429

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

Examples

#
# Try out a range of options
#

r  = 0.05
te = 60/365
k  = 700:1300
y  = 0.02
meanlog.1 = 6.80
meanlog.2 = 6.95
sdlog.1   = 0.065
sdlog.2   = 0.055
alpha.1   = 0.4


mln.prices = price.mln.option(r = r, y = y, te = te, k = k, alpha.1 = alpha.1, 
  meanlog.1 = meanlog.1, meanlog.2 = meanlog.2, sdlog.1 = sdlog.1, sdlog.2 = sdlog.2)

par(mfrow=c(1,2))
plot(mln.prices$call ~ k)
plot(mln.prices$put  ~ k)
par(mfrow=c(1,1))