The Black-Scholes calculator computes the values for Call and Put Options based on the Black-Scholes equation.
INSTRUCTIONS: Choose units and enter the following:
- (S) Stock or spot price.
- (X) Strike price
- (T) Number of years to maturity
- (v) Percent of volatility
- (r) Risk-free rate
Black-Scholes Call / Put Value (BSV): The calculator returns the call and put value in U.S. dollars (USD). However, these can be automatically converted to other currencies via the pull down menu. Currency values are updated based on the market every two minutes.
The Math / Science
The Black-Scholes equation is based on a partial differential equation that was developed as a model of the financial market. A Wikipedia article on the Black-Scholes equation can be found HERE. This equation is a useful approximation to determine the benefit of purchasing the option and has been tested against two of the algorithms found at espenhaug.org (HERE). However, users should independently confirm their calculations before relying on this or any other equation to make financial decisions.
APPLICATIONS
The Black–Scholes equation, a partial differential equation, gives a theoretical estimate of the price of European-style options over time. The Black-Scholes equation employs the technique of constructing a risk neutral portfolio that replicates the returns of holding an option and produces a closed-form solution for a European option's theoretical price at maturity.
NOTES
OPTIONS TERMINOLOGY
- option - a contract which gives the buyer (the owner and investor) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date. The option specified the following terms:
- the number of stocks, usually in denominations of 100 shares per contract
- the expiration date specifying when the option expires
- the contract style specifying that an investor will exercise an option any time before the maturity date or will exercise the option only on the expiration date
- call option - an option which conveys to the owner the right to buy something at a specific price
- put option - an option which conveys the right of the owner to sell something at a specific price
- strike price - the strike price, also known as the exercise price, is the price at which the underlying contract will be exercise (carried out). The strike price for a call option is the price at which the buyer can buy a security or commodity. The strike price for a put option is the price they can sell a security or commodity.
- riskless rate or risk-free interest rate - the rate specified in the option for a given stable asset in the Black-Scholes model. The model assumes a money market, cash or bond exists that that has a rate of return that is constant.
- time to maturity, T - the option specified time after which the options contract expires.
FORM OF THE EQUATION
The value of a call option for a non-dividend-paying stock exercised after the specified time, T, is given:
Call Value = `N(d_1)*S - N(d_2) * X * e^(-r*T)`
The value of a put option based on put-call parity is given
Put Value = `X*e^(-r*T) - S * "Call Value"` = `N(-d_2) * X * e^(-r*T) - N(-d_1)*S`
where `d_1` and `d_2` are given as:
`d_1` = `1/(v*sqrt(T))*[ln(S/X) + (r + v^2/2) *T]` and
`d_2` = `d_1 - v*sqrt(T)`
For these equations:
- `N( d_1)` and `N(d_2)` are the standard normal cumulative distribution functions for `d_1` and `d_2` respectively
- T is the time to maturity of the option
- S is the spot price of the asset
- X is the strike price
- r is the risk free rate
- v is the volatility of the return from the asset
LIMITATIONS OF THE MODEL
- The Black-Scholes formula has only one parameter that cannot be observed in the market: the average future volatility of the underlying asset. Since the formula is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models.
- One key assumption made by the Black-Scholes model is that volatility is constant. Even if the Black-Scholes model results may not completely accurate, they can still serve as a first approximation. More refined models have been built to address each of the Black-Scholes model limitations.
- Another limitation cited is that the risk-free interest rate is not actually known, and is not constant over time.
Before using the Black-Scholes equation for your own estimations, you should read the following article: http://www.theguardian.com/science/2012/feb/12/black-scholes-equation-credit-crunch. This article takes a look at what caused the 1987 banking debacle and suggests that it was caused by the misuse of the Black-Scholes formula. Pay particular attention to the mention of the bestseller The Black Swan by Nassim Nicholas Taleb. This book looks at the natural phenomena of how extreme events cause even the most robust estimation theory to fail, which is important to remember when doing estimates for investment purposes.
Basic Investor Calculators
- Black-Scholes Equation: Compute the Call and Put Option based on the Black-Scholes equation and the stock or spot price, strike price, number of years to maturity, percent volatility and the risk-free rate.
- 30 Day Yield Equation: Computes the SEC's 30 day yield function for bond funds based on the income in the prior 30 days, accrued expenses in the prior 30 days, outstanding shares and max price per share.
- Investment Return Rate: Computes the return rate based on the beginning and end price and dividends paid.
- Inflation Adjusted Return: Computes the return rate based on the Inflation Rate and the Investment Return.
- Present Value- computes the present value of a fixed annuity.
- Future Value- computes the future value of a fixed annuity.
- Interest Rate for Future Value - computes the annual fixed interest rate required for a present value to accrue to the future value over a number of periods.
- Number of Periods Required - computes the number of periods required to achieve a future value from a present value at a fixed annual interest rate.
- Precious Metal Value - computes the current market value of gold, platinum, silver and palladium based on bullion weight and purity.
- Credit Card Equation - computes the time required to payoff a debt based on an interest rate, initial balance and monthly payment.
- Currency Conversion - computes the current value of a currency amount (e.g., $2,000 US dollars) in Euros, Great Britain Pounds, Canadian Dollars, Yuan, Yen, Rubbles, Swiss Francs, Australian Dollars, South African Rands, Brazial Reals, Mexican Pesos, Indian Rupees and U.S. dollars.
ORIGINAL PUBLICATION
Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model. He coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Prize in Economics for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.