Uri,
To work out the formula backwards you could set it up on a spreadsheet like Lotus. Then, all you'll have to do is enter all the parameters including the current price of the option, and let your spreadsheet calculate the parameter of your choice. If, for instance, you want to figure out what the volatility (sigma) is, fill in all other parameters including current stock price, exercise price, t-bill rate and current option place, and your program should calculate the underlying volatility.
As a far as the natural logarithm goes, it is used b/c it assumes *continuous* discounting. I'll illustrate the difference b/w discrete and continous compounding by way of an example:
Say you have a fixed income security that pays a nominal interest rate of 10% compounded semi-annually.
Then, on $1,000 you would earn [(1.05)^2 * 1,000]-1,000 = $102.5
With continuous compounding you would get
[e^0.1 * 1,000]-1,000= $105.17, which is slightly more than what you'd earn with semi-annual compounding.
I believe the rationale for employing continuous compounding in the B-S formula is to maintain a certain level of consistency. Namely, since securities trade continuously, all elements in the formula should be treated as continuous events including the discounting of the T-bill rate. That's what explains the presence of the natural logarithm in the second half of the formula.
I hope this helps... |
