Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 12

Example in Finance: Black–Scholes Model

 

 

12.1. Introduction: what is an option?

Option is a contract between two parties, buyer and seller, which gives one party the right, but not the obligation, to buy or to sell some asset until some agreed date, while the second party has the obligation to fulfill the contract if requested.

Let us illustrate this notion by using a real-life example. Suppose that you have decided to buy a house and have found somewhere you like. At the moment, you do not have enough money, but you hope to get it in the near future (say, by getting credit or by selling another property). Therefore you make a contract (called an option) with the seller of a house that he will wait for a half of year and will sell the house for an agreed price of, say, $100,000. However, he agrees to wait only under the condition that you pay for this, say, $1,000. Further, we can imagine the following two scenarios:

1.  Within half year, the prices of real estate have increased, and the market price of the house has increased to $120,000. Since the owner has signed a contract with you and is paid for this (by selling the option), he is obliged to sell you the house for $100,000. Thus, in this case, you have a significant profit of

120,000 - 100,000 - 1,000 = $19,000.

2.  Checking the details on that house and talking to the neighbors, you learn that there are many dysfunctional neighbors, and that the house has a very expensive and time-consuming solid-fuel heating ...