| extract.am.density {RND} | R Documentation |
Mixture of Lognormal Extraction for American Options
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.
###