RND (version 1.2)

mln.am.objective: Objective function for the Mixture of Lognormal of American Options

Description

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

Usage

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

Arguments

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

Value

Details

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.

References

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.

Examples

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.
###