mln.objective {RND}

Objective function for the Mixture of Lognormal

Description

mln.objective is the objective function to be minimized in extract.mln.density.

Usage

mln.objective(theta, r, y, te, s0, market.calls, call.strikes, call.weights, 
  market.puts, put.strikes, put.weights, lambda = 1)

Arguments

theta

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

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

obj

value of the objective function

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

#
# The mln objective function should be close to zero.
# The weights are automatically set to 1.
#

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

# This is the current price implied by parameter values:
s0 = 981.8815 

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

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

mln.objective(theta=c(alpha.1,meanlog.1, meanlog.2 , sdlog.1, sdlog.2), 
               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)