Important Updates for Investors
Carla Pasternak's Premiere Issue of High-Yield International Just
Released
Income expert Carla Pasternak's debut issue of High-Yield
International covers a Taiwanese manufacturer yielding 9.5%... a
rare Mexican monopoly yielding 13.4%... and other top-performing
investments yielding up to 19.0%.
Government's Biofuel Timetable Could Spell +15,900% Growth
+15,900% growth might seem far-fetched... but it's not. In fact, it
is mandated by law. And I've identified the ONLY stock positioned to
capture this growth.
The
Silver Lining to a Falling Dollar
Despite the U.S. national debt, there is a silver lining for income
investors. This massive spending, combined with movement out of U.S.
Treasuries, is going to take its toll on the dollar, and
international income investors could reap the rewards in the form of
higher dividends. |
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| Black-Scholes
Model |
What It Is:
The Black-Scholes model is a formula used to assign prices to option contracts.
The model is named after Fischer Black and Myron Scholes, who developed it in
1973. Robert Merton also participated in the model’s creation, and this is why
the model is sometimes referred to as the Black-Scholes-Merton model. All three
men were college professors working at both the University of Chicago and MIT at
the time.
The Black-Scholes formula is:
C0 = S0N(d1) - Xe-rTN(d2)
Where:
d1 = [ln(S0/X) + (r + σ2/2)T]/ σ √T
And:
d2 = d1 - σ √T
And where:
C0 = current option value
S0 = current stock price
N(d) = the probability that a random draw from a standard normal distribution
will be less than (d).
X = exercise price
e = 2.71828, the base of the natural log function
r = risk-free interest rate (annualized continuously compounded rate on a safe
asset with the same maturity as the expiration of the option; usually the money
market rate for a maturity equal to the option’s maturity.)
T = time to option’s maturity, in years
ln = natural logarithm function
σ = standard deviation of the annualized continuously compounded rate of
return on the stock
Clearly, computers have greatly eased and extended the use of the Black-Scholes
model.
How It Works/Example:
Let’s assume you would like to know the value of an option to purchase one
share of XYZ Company stock for $95. The current price of the shares is $100, and
the option expires in three months (one-quarter year). Assuming that the stock pays no dividends, the standard deviation of the
stock’s returns is 50% per year, and the risk-free rate is 10% per year, we
can calculate that the value of the option, even though it is out of the money
right now, is as follows:
d1 = [ln($100/$95) + (.10 + .52/2).25]/ .5√.25 = .43
d2 = .43 - .5√.25 = .18
N(.43) = .6664
N(.18) = .5714
Thus, the value of the call options is:
C0 = 100 x .6664 - 95e-.10 x .25 x .5714 = 66.64 - 52.94 = $13.70
Note that this formula values a call option. The Black-Scholes model can be used
to price other derivatives, including puts. To value a put option (P),
use the value of the call option to solve for the value of the put option, as
follows:
P = C0 + PV(X) - S0 = C0 + Xe-rT - S0 = $13.70 + $95e-.10 x .25 - $100 = $6.35
The basic mission of the Black-Scholes model is to calculate the probability
that an option will expire in the money. To do this, the model looks beyond the
simple fact that the value of a call option increases when the underlying stock
price increases or when the exercise price decreases. Rather, the model assigns
value to an option by considering several other factors, including the
volatility of XYZ Company stock, the time left until the option expires, and
interest rates. For example, if XYZ Company stock is considerably volatile,
there is more potential for the option to go in the money before it expires.
Also, the longer the investor has to exercise the option, the greater the chance
that an option will go in the money and the lower the present value of the
exercise price. Higher interest rates raise the price of the option because
they lower the present value of the exercise price.
It is important to note that the Black-Scholes model is geared toward European
options. American options, which allow the owner to exercise at any point up to
and including the expiration date, command higher prices than European options,
which allow the owner to exercise only on the expiration date. This is because
the American options essentially allow the investor several chances to capture
profits, whereas the European options allow the investor only one chance to
capture profits.
Why It Matters:
Empirical studies show that the Black-Scholes model is very predictive, meaning
that it generates option prices that are very close to the actual price at which
the options trade. However, various studies show that the model tends to
overvalue deep out-of-the-money calls and undervalue deep in-the-money calls. It
also tends to misprice options that involve high-dividend stocks. Several of the
model’s assumptions also make it less than 100% accurate. First, the model
assumes that the risk-free rate and the stock’s volatility are constant.
Second, it assumes that stock prices are continuous and that large changes (such
as those seen after a merger announcement) don’t occur. Third, the model
assumes a stock pays no dividends until after expiration. Fourth, analysts can
only estimate a stock’s volatility instead of directly observing it, as they
can for the other inputs. Obviously, analysts have developed variations of the
Black-Scholes model to account for these limitations.
Ultimately, however, the Black-Scholes model represents a major contribution to
the efficiency of the options and stock markets, and it is still one of the most
widely used financial tools on Wall Street. Besides providing a dependable way
to price options, it helps investors understand how sensitive an option’s
price is to stock price movements. This in turn helps investors maximize the
efficiency of their portfolios by giving them a way to calculate hedge ratios
and more effectively implement portfolio insurance.
Despite the tremendous efficiencies created by the Black-Scholes model, many
financial theorists claim the model’s introduction indirectly increased the
volatility of the stock and options markets by encouraging more trading (as
investors sought to constantly fine-tune their hedge positions). Others claim
the model actually steadies the markets because of its ability to measure
equilibrium pricing relationships. When these relationships are violated,
arbitrageurs are usually the first to discover and exploit mispriced options.
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