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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
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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