The current price of Apple stock is $131.74. You are offered a chance to enter into the following agreement:

• If tomorrow Apple stock increases to $133.22, you receive nothing;
• If tomorrow Apple stock decreases to $129.88, you receive $1.12. you receive $1.12.

Assuming that you know that starting with the initial stock price of $131.74, Apple stock has a 60% chance to increase to 133.22 and only a 40% chance to decrease to $129.88, how much are you willing to pay for this agreement? Or, in other words, what is a fair price to charge for this "contract"?

The above scenario is an example of a financial option called a "put" stock option, which gives the owner the right, but not the obligation, to sell an asset to another person at a particular time (or times) in the future. In the above case, you are guaranteed that you can sell Apple for $131, regardless of stock value. How to fairly price such options, as well as other types of financial derivatives, is an interesting aspect of mathematical finance and stochastic calculus, a field that has its roots in a French mathematical thesis from 1900, resulted in the 1997 Nobel prize in economics, and provides a concrete use for such oddities of real analysis as continuous functions that are nowhere differentiable.

By the end of this talk you should be able to fairly price the above Apple stock option, as well as understand and price more complicated financial derivatives, within a universe where time discretely ticks by.