Thursday, 19 April 2012

Find the value of a European call option using the Black-Scholes formula

Find the value of a European call option using the Black-Scholes formula


The calculator uses an adjusted Black-Scholes model to value European options. (European options are those that can only be exercised at expiry).
The model is adjusted to take into account dividends paid on the underlying security.
The calculator can in fact be used for:

  • stock options
  • index options
  • currency options
  • options on futures.

Input Definitions

Current Share Price
These can be input in any units (e.g. dollars or cents) but the input must be consistent for the share price, strike price (and, if calculating implied volatility, for the option price as well).
Note: as an aid for input, there is the facility to input the share/strike/option prices all as "eighth's". For this, select the button (to the right of the input strike price). With /8 selected, a price of 67 3/8 can be input as 67.3; the program will then automatically convert "67.3" to the decimal 67.375 for use in the calculations. All outputs from the calculator will always be expressed in decimal.

Exercise Price
If calculating the implied volatility, then the actual market price of the option is input. This should be input in the same units as those used for the share price and strike price. If the /8 button has been selected then, the option price must be input expressed as a fractional eighth.

Expected Time To Expiry (years) (aka maturity)
Maturity - period before option expiration date (in this calculator in years)

Risk Free Interest Rate (%)
The simple risk-free interest rate for the period should be input. For example, if valuing a 6 month option, then the 6-month risk-free rate should be used. To input a rate of 6.7%, input "6.7".

Expected Dividend Yield on Share (%)
The standard Black-Scholes model has been "adjusted" to account for dividends payable. (If this is not required then simply input 0 in the dividend yield field). To input a dividend yield of 3.4%, enter "3.4".
Note: if valuing currency options, then the foreign interest rate should be input to this field.

Note: strictly, the Black-Scholes valuation model requires the interest rate to be a continuously compounded rate. So the calculator converts the simple rate input, to a continuously compounded rate. However, some option calculators and examples in books do not make this adjustment, so there is the facility to "switch off" this conversion, by de-selecting the button called cc-int. This might then enable calibration with other option calculators.

Expected Volatility of Share Price (%)
If calculating the theoretical option value, then a forecast of future share price volatility must be input. To input a volatility of 25.5%, input "25.5". (If calculating implied volatility, any figure in the volatility input field will be ignored).

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-Schole'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. Another 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. This assumption relates to the basic Black-Scholes formula. 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.

See the Black-Scholes model page for more details about the Black-Scholes model and to read about how these assumptions relate to real-world scenarios.

Who was Fischer Black

Fischer Sheffey Black (January 11, 1938 - August 30, 1995) was an American economist, best known as one of the authors of the famous Black-Scholes equation.

Black graduated from Harvard College in 1959 and received a Ph.D. in applied mathematics from Harvard University in 1964. He was initially expelled from the PhD program due to his inability to settle on a thesis topic, having switched from physics to mathematics, then to computers and artificial intelligence. Black joined the consultancy Bolt, Beranek and Newman, working on a system for artificial intelligence. He spent a summer developing his ideas at the RAND corporation. He became a student of MIT professor Marvin Minsky, and was later able to submit his research for completion of the Harvard PhD.

n 1973, Black, along with Myron Scholes, published the paper 'The Pricing of Options and Corporate Liabilities' in 'The Journal of Political Economy'.[5] This was his most famous work and included the Black-Scholes equation.

In March 1976, Black proposed that human capital and business have "ups and downs that are largely unpredictable [...] because of 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." A boom is a period when technology matches well with demand. A bust is a period of mismatch. This view made Black an early contributor to real business cycle theory.

Who was Myron Scholes

Myron Samuel Scholes (born July 1, 1941) is a Canadian-born American financial economist who is best known as one of the authors of the Black-Scholes equation. In 1997 he was awarded the Nobel Memorial Prize in Economic Sciences for a method to determine the value of derivatives. The model provides a conceptual framework for valuing options, such as calls or puts, and is referred to as the Black-Scholes model.

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