The Black-scholes Model: The Key To Valuing Option Contracts
Arguably the most important and widely used options pricing model in use today
, the Black-Scholes Model for option pricing was developed by three influential finance theorists: Fischer Black, Myron Scholes and Robert Merton. (An option is a derivative instrument that provides a right, without obligation, to buy or sell an asset under specified conditions.)
Taking neither a Keynesian nor Monetarist view on monetary policy, Black believed that,"In the U.S. economy, much of the public debt is in the form of Treasury bills. Each week, some of these bills mature, and new bills are sold. If the Federal Reserve System tries to inject money into the private sector, the private sector will simply turn around and exchange its money for Treasury bills at the next auction. If the Federal Reserve withdraws money, the private sector will allow some of its Treasury bills to mature without replacing them." Black purported further that the rise and fall of markets and the availability of human capital can be quite capricious due to the "... basic uncertainty about what people will want in the future and about what the economy will be able to produce in the future. If future tastes and technology were known, profits and wages would grow smoothly and surely over time."
This thinking gave rise to the Black's classic paper entitled, "The Pricing of Options and Corporate Liabilities", published in a 1973 issue of the Journal of Political Economy, which introduced what would be known as the Black-Scholes formula for option pricing.
The Black-Scholes formula provided a groundbreaking, theoretical securities valuation model for trading markets that have reached equilibrium (e.g., there are no arbitrage opportunities). In 1973, Robert Merton developed his own formula, known as the Merton Model. This model is a generalized version of the Black-Scholes formula, intended for the treatment of a company's equity as an option on its assets. Later, Black, Scholes and Merton worked jointly on a landmark project for valuing financial instruments.
As a result of this work, the Black-Scholes and Merton Models together became the winner of the Nobel Prize in Economic Sciences, for their contributions in transforming market theory into real-world predictors of securities' risk and return. (Unfortunately, Fischer Black passed away just two years before Scholes and Merton won the 1997 Nobel Prize in Economic Sciences for the paper.) However, the Black-Scholes formula later spawned the development of many other option valuation models - such as the binomial model.
* What does the Black-Scholes Model do?
The Black-Scholes Model provides a formula for calculating the theoretical fair value of an option contract, where an option is a derivative whose value is based on some underlying asset.
Specifically, the model calculates the theoretical value of a European call option on a stock that does not pay any dividends. (N.B.: This was the original claim of the model, but it has since been discovered that dividends can be incorporated into the model).
In addition to calculating the theoretical or fair value for both call and put options, the Black-Scholes Model also calculates Option Greek values such as delta, gamma, theta, vega, and rho. These values tell option traders how the theoretical value of the option will change given certain changes in variables. The Greeks are an invaluable tool in portfolio hedging.
* Model Inputs
From the formula the Black Scholes model requires six (6) factors, or inputs, in order to value an option contract. These six inputs are:
- Underlying Price (price of the stock)
- Exercise Price (Strike Price)
- Time to Expiration (in years)
- Risk Free Interest Rate (rate of return)
- Dividend Yield
- Volatility
Of this list of inputs, the first five (5) are known and can be found easily. Volatility, however, is not known, and therefore this input must be estimated, after careful analysis.
* Black-Scholes Volatility
Volatility, the most important factor in pricing options, is a measure of risk for a security. It indicates the predictability (or unpredictability) of a stock price from day to day. As such, the more an its value changes over a period of time, the more volatile the asset is said to be.
Traders can use either historical volatility of the asset as the estimation for future volatility, or use a forecasting method for the volatility input (such as the Generalized Autoregressive Conditional Heteroscedasticity, or GARCH, Model). Calculating historical volatility involves downloading the price series for the underlying assets and generating the standard deviation output over a given period of time.
If you perform a web search for "historical volatility calculator excel", you can see an example of how this is done.
* Implied Volatility
By using the equation in reverse, traders are able to calculate what's known as the implied volatility. That is, by entering the market price of the option and all other known parameters, the implied volatility tells a trader how much volatility to expect from the asset, given the current share price and current option price.
* Assumptions of the BS Model
1) No Dividends
When the original Black-Scholes Model was introduced, it didn't include dividends in its calculations. Tell me, who could find a company not paying discrete dividends to its shareholders? Do you know any shareholders who would opt-in for no return on their investments these days? I think not. So, that assumption is not valid. We know now that dividends can easily be incorporated into the existing BS Model by adjusting the underlying price input. You can do this in two ways: deduct the present value of all expected discrete dividends from the current stock price (before entering into the model), or deduct the estimated dividend yield from the risk free interest rate during the calculations.
2) European Options
A European option means the option cannot be exercised prior to the expiration date of the option contract. Expiration for European options takes place at expiration of the option only.
American style options are more flexible, allowing for the option to be exercised any time prior to the expiration date. Expert traders can calculate the money value of American options relative to timing, and sometimes find it best to exercise options early. American options tend to be more valuable due to this flexibility as traders can exercise a call option on a stock, in order to be eligible for a dividend payment.
American options are generally priced using another pricing model called the Binomial Option Model.
3) Efficient Markets
Efficient, basically means that there is no directional bias present in the price of the security; any information available to the market is already priced into the security.
4) Frictionless Markets
Friction refers to the presence of transaction costs, such as brokerage and clearing fees. The pure Black-Scholes Model was developed without consideration for brokerage and other transaction costs.
5) Constant Interest Rates
The Black-Scholes Model assumes that interest rates are constant and known for the duration of the options life. In reality, interest rates are subject to change at any time.
6) Asset Returns are Lognormally Distributed
Calculating the volatility of options prices is an important factor in determining how much asset returns will move over time. Distributions that follow an even price path are said to be normally distributed. A normal distribution's price return curve will be a bell shape that is symmetrical around the current price. Typically, the probability of an asset being higher or lower than the current price on the next day is unknown, and hence having a 50/50 probability to be above or below the mean.
It is generally accepted, however, that stocks (and many other assets) tend to have an upward drift - due partly to the positive expectation that most equities will increase in value over the long term, and also because stock prices have a price floor of zero. The upward bias in the returns of asset prices results in a distribution that is lognormal, when the periodic rate of return is normal. A lognormally distributed curve is non-symmetrical and has a positive skew to the upside.
* Geometric Brownian Motion
While we maintain that asset returns are lognormally distributed, it must be mentioned that the price path of a security is said to follow a Geometric Brownian Motion - or GBM. GBM theory is based on Brownian motion, referred to as "particle theory": the random change of motion in the universe. GBM is most commonly used in finance for characterizing price series data. As described in Wikipedia, a Geometric Brownian Motion is a "continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion." More simply put, GBM states that a variable's value changes in one unit of time by an amount is normally-distributed around the mean.
by: Tyler Peterson
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