extract.am.density: Mixture of Lognormal Extraction for American Options

In RND: Risk Neutral Density Extraction Package

Description

extract.am.density extracts the mixture of three lognormals from American options.

Usage

extract.am.density(initial.values = rep(NA, 10), r, te, s0, market.calls, 
  call.weights = NA, market.puts, put.weights = NA, strikes, lambda = 1, 
  hessian.flag = F, cl = list(maxit = 10000))

Arguments

initial.values	initial values for the optimization
r	risk free rate
te	time to expiration
s0	current asset value
market.calls	market calls (most expensive to cheapest)
call.weights	weights to be used for calls. Set to 1 by default.
market.puts	market calls (cheapest to most expensive)
put.weights	weights to be used for puts. Set to 1 by default.
strikes	strikes (smallest to largest)
lambda	Penalty parameter to enforce the martingale condition
hessian.flag	If FALSE then no Hessian is produced
cl	List of parameter values to be passed to the optimization function

Details

The extracted density is in the form of 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.

For the description of w.1 and w.2 see equations (7) & (8) of Melick and Thomas paper below.

Value

w.1	First weight, a number between 0 and 1, to weigh the option price bounds
w.2	Second weight, a number between 0 and 1, to weigh the option price bounds
u.1	log mean of the first lognormal
u.2	log mean of the second lognormal
u.3	log mean of the third lognormal
sigma.1	log sd of the first lognormal
sigma.2	log sd of the second lognormal
sigma.3	log sd of the third lognormal
p.1	weight assigned to the first density
p.2	weight assigned to the second density
converge.result	Captures the convergence result
hessian	Hessian Matrix

Author(s)

Kam Hamidieh

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

###
### Try with synthetic data first.
###

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

###
### Generate some synthetic American calls & puts
###

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 = 50:150

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 
     }

}


###
### ** IMPORTANT **:  The code that follows may take 1-2 minutes.
###                   Copy and paste onto R console the commands
###                   that follow the greater sign >.
###
### Try the optimization with exact inital values.  
### They should be close the actual initials.
###
#
# > s0      = expected.f0 * exp(-r * te)
# > s0
#
# > extract.am.density(initial.values = theta, r = r, te = te, s0 = s0, 
#                  market.calls = market.calls, market.puts = market.puts, strikes = strikes, 
#                  lambda = 1, hessian.flag = FALSE)
#
# > theta
#
###
### Now try without our the correct initial values...
###
#
# > optim.obj.no.init = extract.am.density( r = r, te = te, s0 = s0, 
#                   market.calls = market.calls, market.puts = market.puts, strikes = strikes, 
#                    lambda = 1, hessian.flag = FALSE)
#
# > optim.obj.no.init
# > theta
#
###
### We do get different values but how do the densities look like?
###
#
###
### plot the two densities side by side
###
#
# > x = 10:190
#
# > y.1 = dmln.am(x = x, p.1 = theta[9], p.2 = theta[10],
#           u.1 = theta[3], u.2 = theta[4], u.3 = theta[5], 
#          sigma.1 = theta[6], sigma.2 = theta[7], sigma.3 = theta[8] )
#
# > o = optim.obj.no.init
#
# > y.2 = dmln.am(x = x, p.1 = o$p.1, p.2 = o$p.2,
#           u.1 = o$u.1, u.2 = o$u.2, u.3 = o$u.3, 
#           sigma.1 = o$sigma.1, sigma.2 = o$sigma.2, sigma.3 = o$sigma.3 )
#
# > matplot(x, cbind(y.1,y.2), main = "Exact = Black, Approx = Red", type="l", lty=1)
#
###
### Densities are close.
###

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