bsm.objective: BSM Objective Function

In RND: Risk Neutral Density Extraction Package

Description

bsm.objective is the objective function to be minimized in extract.bsm.density.

Usage

bsm.objective(s0, r, te, y, market.calls, call.strikes, call.weights = 1, 
  market.puts, put.strikes, put.weights = 1, lambda = 1, theta)

Arguments

s0	current asset value
r	risk free rate
te	time to expiration
y	dividend yield
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 calls
lambda	Penalty parameter to enforce the martingale condition
theta	initial values for the optimization. This must be a vector of length 2: first component is μ, the lognormal mean of the underlying density, and the second component is √{t}σ which is the time scaled volatility parameter of the underlying density.

Details

This function evaluates the weighted squared differences between the market option values and values predicted by the Black-Scholes-Merton option pricing formula.

Value

Objective function evalued at a specific set of values.

Author(s)

Kam Hamidieh

References

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

Examples

r     = 0.05
te    = 60/365
s0    = 1000
sigma = 0.25
y     = 0.01

call.strikes   = seq(from = 500, to = 1500, by = 25)
market.calls   = price.bsm.option(r =r, te = te, s0 = s0, 
                 k = call.strikes, sigma = sigma, y = y)$call

put.strikes    = seq(from = 510, to = 1500, by = 25)
market.puts    = price.bsm.option(r =r, te = te, s0 = s0, 
                 k = put.strikes, sigma = sigma, y = y)$put

###
### perfect initial values under BSM framework
###

mu.0     = log(s0) + ( r - y - 0.5 * sigma^2) * te
zeta.0   = sigma * sqrt(te)
mu.0
zeta.0

###
### The objective function should be *very* small
###

bsm.obj.val = bsm.objective(theta=c(mu.0, zeta.0), r = r, y=y, te = te, s0 = s0, 
              market.calls = market.calls, call.strikes = call.strikes, 
              market.puts  = market.puts,  put.strikes = put.strikes, lambda = 1)
bsm.obj.val

Example output

[1] 6.909194
[1] 0.1013606
[1] 3.761598e-24

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