RND (version 1.2)

extract.mln.density: Extract Mixture of Lognormal Densities

Description

mln.extraction extracts the parameters of the mixture of two lognormals densities.

Usage

extract.mln.density(initial.values = c(NA, NA, NA, NA, NA), r, y, te, s0,  market.calls, call.strikes, call.weights = 1, market.puts, put.strikes,  put.weights = 1, lambda = 1, hessian.flag = F, cl = list(maxit = 10000))

Arguments

initial.values

initial values for the optimization

r

risk free rate

y

dividend yield

te

time to expiration

s0

current asset value

market.calls

market calls (most expensive to cheapest)

call.strikes

strikes for the calls (smallest to largest)

call.weights

weights to be used for calls

market.puts

market calls (cheapest to most expensive)

put.strikes

strikes for the puts (smallest to largest)

put.weights

weights to be used for puts

lambda

Penalty parameter to enforce the martingale condition

hessian.flag

if F, no hessian is produced

cl

list of parameter values to be passed to the optimization function

Value

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.

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, 383-4

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

Examples

#
# Create some calls and puts based on mln and 
# see if we can extract the correct values.
#


r         = 0.05
y         = 0.02
te        = 60/365
meanlog.1 = 6.8
meanlog.2 = 6.95
sdlog.1   = 0.065
sdlog.2   = 0.055
alpha.1   = 0.4


call.strikes = seq(from = 800, to = 1200, by = 10)
market.calls = price.mln.option(r = r, y = y, te = te, k = call.strikes, 
               alpha.1 = alpha.1, meanlog.1 = meanlog.1, meanlog.2 = meanlog.2, 
                                sdlog.1 = sdlog.1, sdlog.2 = sdlog.2)$call

s0 = price.mln.option(r = r, y = y, te = te, k = call.strikes, alpha.1 = alpha.1, 
                      meanlog.1 = meanlog.1, meanlog.2 = meanlog.2, 
                      sdlog.1 = sdlog.1, sdlog.2 = sdlog.2)$s0
s0
put.strikes  = seq(from = 805, to = 1200, by = 10)
market.puts  = price.mln.option(r = r, y = y, te = te, k = put.strikes, 
                                alpha.1 = alpha.1, meanlog.1 = meanlog.1, 
                                meanlog.2 = meanlog.2, sdlog.1 = sdlog.1, 
                                sdlog.2 = sdlog.2)$put

###
### The extracted values should be close to the actual values.
###

extract.mln.density(r = r, y = y, te = te, s0 = s0, market.calls = market.calls, 
               call.strikes = call.strikes, market.puts = market.puts, 
               put.strikes = put.strikes, lambda = 1, hessian.flag = FALSE)