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Black-Scholes

The Black-Scholes model is a tool for pricing equity options. The idea behind the Black-Scholes model is that the price of an option is determined implicitly by the price of the underlying stock. The Black-Scholes model is a mathematical model based on the notion that prices of stock follow a stochastic process, i.e. the price of a stock in time t+1 is independent from the price in time t. This is also referred to as random walk.

The Black-Scholes formula consists of three parts: the main equation and two formulas for calculating parameters d1 and d2.

The main equation tells us that the price of a European-style call option with expiration date in time T written on stock S is equal to the price of the stock adjusted for volatility, interest rate, and spread minus present value of the stock delivery price (or a strike price) also adjusted for volatility, interest rate, and spread.

The parameters d1 and d2 are parameters to the φ (Phi) in the first equation; Phi represents a cumulative distribution function of Normal distribution.

The Black-Scholes model can also be used to price put options. If you want to value a put option, you can either calculate it from scratch, or recalculate the Black-Scholes model through the put-call parity. Using the put-call parity approach to calculate put option value given that you know the call option value, you would solve the put-call parity equation for the value of the put option.

Understanding the Black-Scholes model assumptions is very important for the application of the model to real-world scenarios. There are several assumptions underlying the Black-Scholes model.

  1. Constant volatility. The most significant assumption is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. While volatility can be relatively constant in very short term, it is never constant in longer term. Some advanced option valuation models substitute Black-Scholes's constant volatility with stochastic-process generated estimates.
  2. Efficient markets. This assumption of the Black-Scholes model suggests that people cannot consistently predict the direction of the market or an individual stock. The Black-Scholes model assumes stocks move in a manner referred to as a random walk. Random walk means that at any given moment in time, the price of the underlying stock can go up or down with the same probability. The price of a stock in time t+1 is independent from the price in time t.
  3. No dividends. This assumption is that the underlying stock does not pay dividends during the option's life. In the real world, most companies pay dividends to their share holders. The basic Black-Scholes model was later adjusted for dividends, so there is a workaround for this. A common way of adjusting the Black-Scholes model for dividends is to subtract the discounted value of a future dividend from the stock price.
  4. Interest rates constant and known. The same like with the volatility, interest rates are also assumed to be constant in the Black-Scholes model. The Black-Scholes model uses the risk-free rate to represent this constant and known rate. In the real world, there is no such thing as a risk-free rate, but it is possible to use the U.S. Government Treasury Bills 30-day rate since the U. S. government is deemed to be credible enough. However, these treasury rates can change in times of increased volatility.
  5. Lognormally distributed returns. The Black-Scholes model assumes that returns on the underlying stock are normally distributed. This assumption is reasonable in the real world.
  6. European-style options. The Black-Scholes model assumes European-style options which can only be exercised on the expiration date. American-style options can be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility.
  7. No commissions and transaction costs. The Black-Scholes model assumes that there are no fees for buying and selling options and stocks and no barriers to trading.
  8. Liquidity. The Black-Scholes model assumes that markets are perfectly liquid and it is possible to purchase or sell any amount of stock or options or their fractions at any given time.

The Black-Scholes model was developed by Fischer Black and Myron Scholes in 1973. Robert Merton also participated in the model's creation; hence that is why the model is sometimes referred to as the Black-Scholes-Merton model. Black, Sholes, and Merton were awarded the Nobel Prize in Economics for the Black-Scholes model. All three men were college professors working at both the University of Chicago and MIT at the time.

There are known restrictions of the Black-Scholes model when using it in the real world. The major problem is that capital markets often move in ways not consistent with the random walk hypothesis. This also relates to the assumption of the constant volatility. Volatility is not constant in real world. Very short-term options can be valued using the basic Black-Scholes formula because volatility can change only so much in only a few days, but invalidation of these assumptions in longer term in the real world makes the Black-Scholes formula less effective for mid-term and long-term options.

Throughout the years, many other models emerged trying to provide more accurate approach to option valuation. However, with a little generalisation, we can say that probably most of them are enhancements of Black-Scholes. All of them are based on the same valuation principle. The difference between models is mostly how they address assumptions of the Black-Scholes model.

Last Updated: 12th October 2009 12:16
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