optionValues {FMStable}

Values of Options over Finite Moment Log Stable Distributions

Description

Computes values of European-style call and put options over assets whose future price is expected to follow a finite moment log stable distribution.

Usage

putFMstable(strike, paramObj, rel.tol=1.e-10)
callFMstable(strike, paramObj, rel.tol=1.e-10)
optionsFMstable(strike, paramObj, rel.tol=1.e-10)

Arguments

strike

The strike price for an option.

paramObj

An object of class stableParameters which describes a maximally skew stable distribution. This is the distribution which describes possible prices for the underlying security at the time of expiry.

rel.tol

The relative tolerance used for numerical integration for finding option values.

Value

optionsFMstable returns a list containing the values of put options and the values of call options.

Note

When comparing option values based on finite moment log stable distributions with ones based on log normal distributions, remember that the interest rate and holding cost have been ignored here.

Rather than using functions putFMstable and callFMstable for options that are extremely in-the-money (i.e. the options are almost certain to be exercised), the values of such options can be computed more accurately by first computing the value of the out-of-the-money option and then using the relationship spot + put = call + strike. This is done by function optionsFMstable.

See Also

An example of how an object of class stableParameters may be created is by setParam. Procedures for dealing with the log normal model for options pricing include BSOptionValue.

Examples

paramObj <- setMomentsFMstable(mean=10, sd=1.5, alpha=1.8)
putFMstable(c(10,7), paramObj)
callFMstable(c(10,7), paramObj)
optionsFMstable(8:12, paramObj)
# Note that call - put = mean - strike

# Values of some extreme put options
paramObj <- setMomentsFMstable(mean=1, sd=1.5, alpha=0.02)
putFMstable(1.e-200, paramObj)
putFMstable(1.e-100, paramObj)
pFMstable(1.e-100, paramObj)
putFMstable(1.e-50, paramObj)

# Asymptotic behaviour
logmlogx <- seq(from=2, to=6, length=30)
logx <- -exp(logmlogx)
x <- exp(logx)
plot(logmlogx , putFMstable(x, paramObj)/(x*pFMstable(x, paramObj)), type="l")

# Work out the values of some options using FMstable model
spot <- 20
strikes <- c(15,18:20, 20:24, 18:20, 20:23)
isCall <- rep(c(FALSE,TRUE,FALSE,TRUE), c(4,5,3,4))
expiry <- rep(c(.2, .5), c(9,7))
# Distributions for 0.2 and 0.5 of a year given distribution describing
#   multiplicative change in price over a year:
annual <- fitGivenQuantile(mean=1, sd=.2, prob=2.e-4, value=.01)
timep2 <- iidcombine(.2, annual)
timep5 <- iidcombine(.5, annual)
imp.vols <- prices <- rep(NA, length(strikes))
use <- isCall & expiry==.2
prices[use] <- callFMstable(strikes[use]/spot, timep2) * spot
use <- !isCall & expiry==.2
prices[use] <- putFMstable(strikes[use]/spot, timep2) * spot
use <- isCall & expiry==.5
prices[use] <- callFMstable(strikes[use]/spot, timep5) * spot
use <- !isCall & expiry==.5
prices[use] <- putFMstable(strikes[use]/spot, timep5) * spot
# Compute implied volatilities.
imp.vols[isCall] <- ImpliedVol(spot=spot, strike=strikes[isCall],
  expiry=expiry[isCall], price=prices[isCall], Call=TRUE)
imp.vols[!isCall] <- ImpliedVol(spot=spot, strike=strikes[!isCall],
  expiry=expiry[!isCall], price=prices[!isCall], Call=FALSE)

# List values of options
cbind(strikes, expiry, isCall, prices, imp.vols)

# Can the distribution be recovered from the values of the options?
discrepancy <- function(alpha, cv){
  annual.fit <- setMomentsFMstable(mean=1, sd=cv, alpha=alpha)
  timep2.fit <- iidcombine(.2, annual.fit)
  timep5.fit <- iidcombine(.5, annual.fit)
  prices.fit <- rep(NA, length(strikes))
  use <- isCall & expiry==.2
  prices.fit[use] <- callFMstable(strikes[use]/spot, timep2.fit) * spot
  use <- !isCall & expiry==.2
  prices.fit[use] <- putFMstable(strikes[use]/spot, timep2.fit) * spot
  use <- isCall & expiry==.5
  prices.fit[use] <- callFMstable(strikes[use]/spot, timep5.fit) * spot
  use <- !isCall & expiry==.5
  prices.fit[use] <- putFMstable(strikes[use]/spot, timep5.fit) * spot
  return(sum((prices.fit - prices)^2))
}
# Search on scales of log(2-alpha) and log(cv)
d <- function(param) discrepancy(2-exp(param[1]), exp(param[2]))
system.time(result <- nlm(d, p=c(-2,-1.5)))
# Estimated alpha
2-exp(result$estimate[1])
# Estimated cv
exp(result$estimate[2])

# Searching just for best alpha
d <- function(param) discrepancy(param, .2)
system.time(result <- optimize(d, lower=1.6, upper=1.98))
# Estimated alpha
result$minimum