mln.objective: Objective function for the Mixture of Lognormal

In RND: Risk Neutral Density Extraction Package

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)

Example output

[1] 1.180987e-11

RND documentation built on May 1, 2019, 10:52 p.m.