mln.am.objective: Objective function for the Mixture of Lognormal of American...
In RND: Risk Neutral Density Extraction Package

Description Usage Arguments Details Value Author(s) References Examples

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

1 2	mln.am.objective(theta, s0, r, te, market.calls, call.weights = NA, market.puts, put.weights = NA, strikes, lambda = 1)

`theta`	initial values for the optimization
`s0`	current asset value
`r`	risk free rate
`te`	time to expiration
`market.calls`	market calls (most expensive to cheapest)
`call.weights`	weights to be used for calls
`market.puts`	market calls (cheapest to most expensive)
`put.weights`	weights to be used for calls
`strikes`	strikes for the calls (smallest to largest)
`lambda`	Penalty parameter to enforce the martingale condition

mln is density f(x) = p.1 * f1(x) + p.2 * f2(x) + (1 - p.1 - p.2) * f3(x), where f1, f2, and f3 are lognormal densities with log means u.1,u.2, and u.3 and standard deviations sigma.1, sigma.2, and sigma.3 respectively.

obj

Value of the objective function

Kam Hamidieh

Melick, W. R. and Thomas, C. P. (1997). Recovering an asset's implied pdf from option prices: An application to crude oil during the gulf crisis. Journal of Financial and Quantitative Analysis, 32(1), 91-115.

r       = 0.01
te      = 60/365
w.1     = 0.4
w.2     = 0.25
u.1     = 4.2
u.2     = 4.5
u.3     = 4.8
sigma.1 = 0.30
sigma.2 = 0.20
sigma.3 = 0.15
p.1     = 0.25
p.2     = 0.45
theta   = c(w.1,w.2,u.1,u.2,u.3,sigma.1,sigma.2,sigma.3,p.1,p.2)

p.3 = 1 - p.1 - p.2
p.3
expected.f0   =  sum(c(p.1, p.2, p.3) * exp(c(u.1,u.2,u.3) + 
                    (c(sigma.1, sigma.2, sigma.3)^2)/2) )
expected.f0  
 
strikes = 30:170

market.calls = numeric(length(strikes))
market.puts  = numeric(length(strikes))

for (i in 1:length(strikes))
{

  if ( strikes[i] < expected.f0) {
    market.calls[i] = price.am.option(k = strikes[i], r = r, te = te, w = w.1, u.1 = u.1, 
                      u.2 = u.2, u.3 = u.3, sigma.1 = sigma.1, sigma.2 = sigma.2, 
                      sigma.3 = sigma.3, p.1 = p.1, p.2 = p.2)$call.value

    market.puts[i]  = price.am.option(k = strikes[i], r = r, te = te, w = w.2, u.1 = u.1, 
                      u.2 = u.2, u.3 = u.3, sigma.1 = sigma.1, sigma.2 = sigma.2, 
                      sigma.3 = sigma.3, p.1 = p.1, p.2 = p.2)$put.value 
  }  else {

    market.calls[i] = price.am.option(k = strikes[i], r = r, te = te, w = w.2, u.1 = u.1, 
                      u.2 = u.2, u.3 = u.3, sigma.1 = sigma.1, sigma.2 = sigma.2, 
                      sigma.3 = sigma.3, p.1 = p.1, p.2 = p.2)$call.value

    market.puts[i]  = price.am.option(k = strikes[i], r = r, te = te, w = w.1, u.1 = u.1, 
                      u.2 = u.2, u.3 = u.3, sigma.1 = sigma.1, sigma.2 = sigma.2, 
                      sigma.3 = sigma.3, p.1 = p.1, p.2 = p.2)$put.value 
     }

}

###
### Quickly look at the option values...
###

par(mfrow=c(1,2))
plot(market.calls ~ strikes, type="l")
plot(market.puts  ~ strikes, type="l")
par(mfrow=c(1,1))

###
### ** IMPORTANT **:  The code that follows may take a few seconds.
###                   Copy and paste onto R console the commands
###                   that follow the greater sign >.
###
###
### Next try the objective function.  It should be zero.
### Note: Let weights be the defaults values of 1.
###
#
# > s0      = expected.f0 * exp(-r * te)
# > s0
#
# > mln.am.objective(theta, s0 =s0, r = r, te = te, market.calls = market.calls,  
#                 market.puts = market.puts, strikes = strikes, lambda = 1)
#
###
### Now directly try the optimization with perfect initial values.
###
#
#
# > optim.obj.with.synthetic.data = optim(theta, mln.am.objective, s0 = s0, r=r, te=te, 
#                 market.calls = market.calls, market.puts = market.puts, strikes = strikes, 
#                 lambda = 1, hessian = FALSE , control=list(maxit=10000) )
#
# > optim.obj.with.synthetic.data
#
# > theta
#
###
### It does take a few seconds but the optim converges to exact theta values.
###