# NAME

Math::Business::BlackScholes - Black-Scholes option price model functions

# SYNOPSIS

```
use Math::Business::BlackScholes
qw/call_price call_put_prices implied_volatility_call/;
my $volatility=implied_volatility_call(
$current_market_price, $option_price_in, $strike_price_in,
$remaining_term_in, $interest_rate, $fractional_yield
);
my $call=call_price(
$current_market_price, $volatility, $strike_price,
$remaining_term, $interest_rate, $fractional_yield
);
$volatility=Math::Business::BlackScholes::historical_volatility(
\@closing_prices, 251
);
my $put=Math::Business::BlackScholes::put_price(
$current_market_price, $volatility, $strike_price,
$remaining_term, $interest_rate
); # $fractional_yield defaults to 0.0
my ($c, $p)=call_put_prices(
$current_market_price, $volatility, $strike_price,
$remaining_term, $interest_rate, $fractional_yield
);
my $call_discrete_div=call_price(
$current_market_price, $volatility, $strike_price,
$remaining_term, $interest_rate,
{ 0.3 => 0.35, 0.55 => 0.35 }
);
```

# DESCRIPTION

Estimates the fair market price of a European stock option according to the Black-Scholes model.

call_price() returns the price of a call option. put_price() returns the value of a put option. call_put_prices() returns a 2-element array whose first element is the price of a call option, and whose second element is the price of the put option with the same parameters; it is expected to be computationally more efficient than calling call_price() and put_price() sequentially with the same arguments. Each of these routines accepts the same set of parameters:

`$current_market_price`

is the price for which the underlying security is currently trading. `$volatility`

is the standard deviation of the probability distribution of the natural logarithm of the stock price one year in the future. `$strike_price`

is the strike price of the option. `$remaining_term`

is the time remaining until the option expires, in years. `$interest_rate`

is the risk-free interest rate (per year) as a fraction. `$fractional_yield`

is the fraction of the stock price that the stock yields in dividends per year; it is assumed to be zero if unspecified.

## Discrete Dividends

If `$fractional_yield`

is specified as a number, then the actual timing of the ex-dividend dates relative to the current time and the option expiration time can affect the option price by as much as the value of a single dividend.

`$fractional_yield`

may instead be specified as a hashref, each key of which is the remaining amount of time before the dividend is paid, in years, and each value of which is the dividend amount. This produces more accurate results when the dividends that will be assigned during the term of the option are reliably predictable. The ex-dividend date of each dividend so represented is assumed to occur within the *remaining* term of the option, even if the dividend is paid after the term expires.

## Determining Parameter Values

`$volatility`

and `$fractional_yield`

are traditionally estimated based on historical data. `$interest_rate`

is traditionally equal to the current T-bill rate. The model assumes that these parameters are stable over the term of the option.

`$volatility`

(a.k.a. *sigma*) is sometimes expressed as a percentage, which is misleading because it's not a ratio. If you have it as a percentage, then you'll need to divide it by 100 before passing it to this module. Ditto for `$interest_rate`

and `$fractional_yield`

.

Two ways to estimate `$volatility`

are provided. historical_volatility() takes an arrayref of at least 10 (preferably 100 or more) consecutive daily closing prices of the underlying security, in either chronological or reverse chronological order. It then multiplies the variance of the log of day-to-day returns by the number of trading days per year specified by the second argument (or 250 by default). The square-root of this yearly variance is returned.

implied_volatility_call() computes the implied volatility based on the known trading price of a "reference" call option on the same underlying security with a different strike price and/or term, using the Newton-Raphson method, or the bisection method if it fails to converge otherwise. It's invoked like call_price(), except that the second argument is taken as the price of the call option, and the volatility is returned. You can override the default option price tolerance of 1e-4 by passing an additional argument beyond `$fractional_yield`

. If called in an array context, the second element of the return value is an estimate of the error magnitude, and the third element is the number of iterations required to obtain the result. The error magnitude may be quite large unless you use a reference option whose price exceeds its intrinsic value by an amount larger than or comparable to the absolute difference of the market price and the strike price, and it is undefined if the price of the reference option is less than what would be calculated with zero volatility. If the price of the reference option is greater than what would be calculated with infinite volatility, then both the result and the error estimate are undefined. An exception is thrown if it fails to converge within `$Math::Business::BlackScholes::max_iter`

(100 by default) iterations. An analogous implied_volatility_put() is also available.

## American Options

Whereas a European stock option may be exercised only when it expires, an American option may be exercised any time prior to its expiration. The price of an American option can be approximated as the maximum price of similar European options over all possible remaining terms not greater than the remaining term of the American option. This maximum usually occurs at the end of the remaining term, or just before or just after the final ex-dividend date within the remaining term.

## Negative Market Value

An underlying security with a negative market value is assumed to be a short. Buying a short is equivalent to selling the security, so a call option on a short is equivalent to a put option. This is somewhat confusing, and arguably a warning ought to be generated if it gets invoked.

# DIAGNOSTICS

Attempting to evaluate an option with a negative term will result in a croak(), because that's meaningless. Passing suspicious arguments (*e.g.* a negative interest rate) will result in descriptive warning messages. To disable such messages, try this:

```
{
local($SIG{__WARN__})=sub{};
$value=call_price( ... );
}
```

# CAVEATS

This module requires

`Math::CDF`

.The fractional computational error of call_price() and put_price() is usually negligible. However, while the computational error of second result of call_put_prices() is typically small in comparison to the current market price, it might be significant in comparison to the result itself. That's probably unimportant for most purposes.

historical_volatility() tends to produce misleading results because the behavior of the underlying security is most likely not truly log-normal. In particular, the price varies predictably after a dividend is distributed, and the daily variance is expected to be greater after financial announcements are made. Also, a large number of data points are required to obtain statistically meaningful results, but having a large number of data points implies that the results are outdated.

The author categorically disclaims any liability for this module.

# BUGS

The length of the namespace component "BlackScholes" is said to cause unspecified portability problems for DOS and other 8.3 filesystems, but the consensus of the Perl community was that it is more important to have a descriptive name.

You can't passed a blessed reference with the

`0+`

(numeric conversion) operator overloaded (see overload) as a numerical`$fractional_yield`

. Instead, convert it into a numeric scalar before calling these functions.

# SEE ALSO

# AUTHOR

Anders Johnson <*anders@ieee.org*>

# ACKNOWLEDGMENTS

Thanks to Richard Solberg for helping to debug the implied volatility functions.