price.am.option {RND}

Price American Options on Mixtures of Lognormals

Description

price.am.option gives the price of a call and a put option at a set strike when the risk neutral density is a mixture of three lognormals.

Usage

price.am.option(k, r, te, w, u.1, u.2, u.3, sigma.1, sigma.2, sigma.3, p.1, p.2)

Arguments

k

Strike

r

risk free rate

te

time to expiration

w

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 lognoral

u.3

log mean of the second lognoral

sigma.1

log sd of the first lognormal

sigma.2

log mean of the second lognormal

sigma.3

log mean of the third lognormal

p.1

weight assigned to the first density

p.2

weight assigned to the second density

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.

Note: Different weight values, w, need to be assigned to whether the call or put is in the money or not. See equations (7) & (8) of Melick and Thomas paper below.

Value

call.value

American call value

put.value

American put value

expected.f0

Expected mean value of asset at expiration

prob.f0.gr.k

Probability asset values is greater than strike

prob.f0.ls.k

Probability asset value is less than strike

expected.f0.f0.gr.k

Expected value of asset given asset exceeds strike

expected.f0.f0.ls.k

Expected value of asset given asset is less than strike

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

###
###  Set a few parameters and create some
###  American options.  
###

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))