extract.gb.density {RND}

Generalized Beta Extraction

Description

extract.gb.density extracts the generalized beta density from market options.

Usage

extract.gb.density(initial.values = c(NA, NA, NA, NA), r, te, y, s0, market.calls, 
  call.strikes, call.weights = 1, market.puts, put.strikes, put.weights = 1, 
  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

y

dividend yield

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

hessian.flag

if F, no hessian is produced

cl

list of parameter values to be passed to the optimization function

Details

This function extracts the generalized beta density implied by the options.

Value

a

extracted power parameter

b

extracted scale paramter

v

extracted first beta paramter

w

extracted second beta parameter

converge.result

Did the result converge?

hessian

Hessian matrix

Author(s)

Kam Hamidieh

References

R.M. Bookstaber and J.B. McDonald (1987) A general distribution for describing security price returns. Journal of Business, 60, 401-424

X. Liu and M.B. Shackleton and S.J. Taylor and X. Xu (2007) Closed-form transformations from risk-neutral to real-world distributions Journal of Business, 60, 401-424

E. Jondeau and S. Poon and M. Rockinger (2007): Financial Modeling Under Non-Gaussian Distributions Springer-Verlag, London

Examples

#
# create some GB based calls and puts
#

r  = 0.03
te = 50/365
k  = seq(from = 800, to = 1200, by = 10)
a  = 10
b  = 1000
v  = 2.85
w  = 2.85
y  = 0.01
s0 = exp((y-r)*te) * b * beta(v + 1/a, w - 1/a)/beta(v,w) 
s0

call.strikes = seq(from = 800, to = 1200, by = 10)
market.calls = price.gb.option(r = r, te = te, y = y, s0 = s0, 
                               k = call.strikes, a = a, b = s0, v = v, w = w)$call

put.strikes  = seq(from = 805, to = 1200, by = 10)
market.puts  = price.gb.option(r = r, te = te, y = y, s0 = s0, 
                               k = put.strikes, a = a, b = s0, v = v, w = w)$put


#
# Extraction...should match the a,b,v,w above. You will also get warning messages.
# Weigths are automatically set to 1.
#

extract.gb.density(r=r, te=te, y = y, s0=s0, market.calls = market.calls, 
              call.strikes = call.strikes, market.puts = market.puts, 
              put.strikes = put.strikes, lambda = 1, hessian.flag = FALSE)