Option Pricing Calculator

The option calculator on the home page allows you to define the parameters that you need to calculate and generate theoretical prices and option risk values for European style call and put options. Together with these values you'll notice dynamically drawn payoff graphs that illustrate your theoretical profit and loss as well as your payoff after the option contract has expired.

Black and Scholes

The mathematical formula behind these calculations is called the Black and Scholes option model (1973).

The idea behind pricing an option contract comes down to two things: the expected payoff of the option at expiration and the probability of the option expiring in the money. The formual is described as...

 

Call Option = Call Option Formula

Put Option = Put Option Formula

Where Option d1

And Option d2

S = Stock Price

X = Exercise Price

r = Risk Free Interest Rate

T = Time to Expiration (Years)

N(x) = The Cumulative Normal Distribution Function

Standard Deviation Formula = Standard Deviation

Some things to Consider

Volatility Day Count

When estimating volatility, it is necessary to annualise the input used. Originally it was standard to use a 365 day calendar year, however, these days many option traders are opting to use a 256 trading day calendar as it is stated that volatility can only occur while the markets are open.

Option-Price, however, uses a 365 calendar year.

Dividends

Dividends are inserted as an annual effective yield, rather than discrete values.

Options on Futures

Pricing options on futures is slightly different to stock options. The difference comes down to the forward price.

Stocks use the interest rate component to determine the expected cost of holding the underlying until the expiration date. With a futures contract, the future contracts price IS the forward price, so there is no need to use the interest rate for that purpose.

There is, however, a need for the interest rate component in order to discount the premium of the option.

To get around this with standard Black and Scholes, just enter the interest rate as normal and then use the same value for the Dividend Yield.

This way, the forward price will come back to the spot (futures) price (interest - dividend yield) but the interest rate will still be used as the rate for discounting the option premium.

Profit and Loss Graphs

Notice also that the P&L line seen in the Payoff Graphs displays the actual profit or loss, not the theoretical value of the position. This is why the line always crosses the x axis at the underlying price point.

If you want to see the theoretical value changing across a price grid, then please use the Simulation feature at the bottom of the home page.

Contact Me

If anything is not clear about this site, please don't hesitate to contact me.