extract.mln.density: Extract Mixture of Lognormal Densities
In RND: Risk Neutral Density Extraction Package

Description Usage Arguments Details Value Author(s) References Examples

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

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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))

`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

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.

`alpha.1`	extracted proportion of the first lognormal. Second one is 1 - `alpha.1`
`meanlog.1`	extracted mean of the log of the first lognormal
`meanlog.2`	extracted mean of the log of the second lognormal
`sdlog.1`	extracted standard deviation of the log of the first lognormal
`sdlog.2`	extracted standard deviation of the log of the second lognormal
`converge.result`	Did the result converge?
`hessian`	Hessian matrix

Kam Hamidieh

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

#
# 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)