One of the main methods required in option pricing is the Black-Scholes framework. This theory is very appealing and somewhat convincing as we will see in the following.
The Setting for Option Pricing
Today, there are numerous types of options and all of them can be priced using Black-Scholes. Lets assume that we want to estimate a fair price of the following option:
We have the option to buy ten Google Stock for 600$ in one year from now. Suppose that today, the Google stock is worth 600$, too. What is a fair value for that option?
If you would like to speculate that Google stocks go up, how much would be fair for you to pay? Asking your stomach, there is a 60% chance that the stock will be worth 750$ in one year and there is a 40% chance that the stock is worth 450$ in one year. Then, the situation would be the following:
Stock = 600$, Option value = ?
in one year
(Scenario 1, 40% chance)
Stock = 450$, we are allowed to buy the stock for 600$,
i.e. we would not exercise the option.
(Scenario 2, 60% chance)
Stock = 750$, we are allowed to buy the stock for 600$,
we buy the option, sell it immediately and pocket 10 · 150$.
So, you have a 40% chance for zero return and a 60% chance for a return of 10 · 150$. That means you as a speculator would buy the option if someone would sell it to us for 0.6 · 10 · 150$ = 900$ or less. But, is this really a fair price?
This is where Black and Scholes come into play. They got the idea to buy some of the Stock to get to a fair price. That means they buy “Hedge” many stocks. In order to determine, what the Hedge is, they think that it would be fair if you could create a portfolio which has the same value in both scenarios and this value should be equal to the portfolio value today. This situation is the following:
Hedge * Stock today + Option price = Hedge * Stock in one year (Scenario 1) + Option price
Hedge * Stock today + Option price = Hedge * Stock in one year (Scenario 2) + Option price
This leaves us with two equations and two unknowns:
Hedge * 600$ + Option price = Hedge * 450$ + 0$
Hedge * 600$ + Option price = Hedge * 750$ + 10 · 150$
Using school maths, we can easily rearrange this to
Hedge = 10 · -150$ / 300$ = -5
which further means
Option price = Hedge * 450$ + 0$ – Hedge * 600$ = -5 * -150$ = 750$.
So, we got two prices for the option 900$ or less and 750$. The explanation for the 750$ has an important advantage: It is an explanation without a stomach feeling for the probabilities. It is based on the so-called hedge argument.
But, how do we buy -5 Google stocks? This can be done if you find someone who owns Google stock and you borrow 5 Google stocks for one year and sell them immediately. Effectively, this leaves us with -5 Google Stock in your portfolio.
Conclusion and Outlook
It is obvious that these option price models have issues and must be improved:
- Two possible stock prices (450$ and 750$) in one year are clearly unrealistic and insufficient.
- We neglected interest rates, which should be included.
- Borrowing stocks, buying and selling stock and so forth have associated costs and must be taken into account.
As a conclusion, this example shows two things:
- We can determine an option price using probabilities from our stomach and use them from a speculation perspective.
- Using a simple hedge argument, we can determine an option price without any probabilities and determine a fair price.