# impliedVolatility: Computations Regarding Value of Options for Log Normal... In FMStable: Finite Moment Stable Distributions

## Description

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

## Usage

 ```1 2 3 4 5``` ```BSOptionValue(spot, strike, expiry, volatility, intRate=0, carryCost=0, Call=TRUE) ImpliedVol(spot, strike, expiry, price, intRate=0, carryCost=0, Call=TRUE, ImpliedVolLowerBound=.01, ImpliedVolUpperBound=1, tol=1.e-9) lnorm.param(mean, sd) ```

## Arguments

 `spot` The current price of a security. `strike` The strike price for an option. `expiry` The time when an option may be exercised. (We are only dealing with European options which have a single date on which they may be exercised.) `volatility` The volatility of the price of a security per unit time. This is the standard deviation of the logarithm of price. `price` The price for an option. This is used as an input parameter when computing the implied volatility. `intRate` The interest rate. `carryCost` The carrying cost for a security. This may be negative when a security is expected to pay a dividend. `Call` Logical: Whether the option for which a `price` is given is a call option. `ImpliedVolLowerBound` Lower bound used when searching for the inplied volatility. `ImpliedVolUpperBound` Upper bound used when searching for the inplied volatility. `tol` Tolerance specifying accuracy of search for implied volatility. `mean` The mean of a quantity which has a lognormal distribution. `sd` The standard deviation of a quantity which has a lognormal distribution.

## Details

The lognormal distribution is the limit of finite moment log stable distributions as `alpha` tends to 2. The function `lnorm.param` finds the mean and standard deviation of a lognormal distribution on the log scale given the mean and standard deviation on the raw scale. The function `BSOptionValue` finds the value of a European call or put option. The function `ImpliedVol` allows computation of the implied volatility, which is the volatility on the logarithmic scale which matches the value of an option to a specified price.

## Value

`impVol` returns the implied volatility when the value of options is computed using a finite moment log stable distribution. `approx.impVol` returns an approximation to the implied volatility. `lnorm.param` returns the mean and standard deviation of the underlying normal distribution.

Option prices computed using the log normal model can be compared to those computed for the finite moment log stable model using `putFMstable` and `callFMstable`.
 ```1 2 3``` ```lnorm.param(mean=5, sd=.8) BSOptionValue(spot=4, strike=c(4, 4.5), expiry=.5, volatility=.15) ImpliedVol(spot=4, strike=c(4, 4.5), expiry=.5, price=c(.18,.025)) ```