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?

## Speculation

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:

today

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?

## 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*.

## Short Selling

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.