Portfolio Replication Method

Transactions with options provide the opportunity to achieve higher returns with less investment because buying an option is akin to buying a stock largely using leverage…

There are two methods for valuing options:

• The Portfolio Replication Method
• The Risk-Free World Assumption Method

Today, I will touch on the first one:

The Portfolio Replication Method is based on the finding that by combining debt and equity investment, it is possible to create a portfolio that yields exactly the same probabilistic return as the option.

Let’s assume we have a stock, and there are two expected scenarios in the future: the stock price will either increase by 25% or decrease by 20%. Subjectively, we believe the price will increase, so we are ready to buy a Call option.

With a Call option, you buy the right to purchase this stock at a price fixed today after a certain period.

Thus, if the stock price rises and exceeds the sum of the predetermined price and the price paid for the option, you will make a profit. However, if the future stock price turns out to be lower than the fixed price, the right you bought will become worthless, providing no return, and the price you paid for the option will be your loss.

Now, let’s assume you bought the stock, but instead of paying the full amount from your own funds, you paid part of it with a loan. The amount of this loan, in percentage terms, should be such that if the stock price falls by 20%, selling it will cover at least the loan and interest (remember, we have scenarios of -20% or +25%).

Accordingly, with such a combination, we get the same result as with an option. If the stock price rises, you will sell it at a higher price, repay the loan, and have a profit left over. If the stock price falls, you will sell it at a lower price, repay the loan, but you will incur a loss because you cannot recover the invested part of your own funds – just like with the option, where the amount paid for the option would be your loss.

That is why, in the first case, the option price is equal to the invested equity capital in the second case.

P.S.
Mathematically, this won’t come out exactly if we buy the entire stock in the second case. It is necessary to calculate the portion of the stock that aligns the scales of the first (option) and second (portfolio) cases. (In the example discussed below in the photo, we buy only 0.556 of the stock, which is called the Option Delta, and it is calculated using a simple formula).

P.P.S.
Where did we determine the two alternative stock price changes of minus 20% and plus 25%? These are specific figures, so where do they come from?

First of all, we should mention that this model is very simplified since it only considers two outcomes, while reality is much more complex (I will complicate it in subsequent notes). However, there is a formula for determining these figures, which calculates the historical volatility of the stock (standard deviation from the average).

Source:

Principles of Corporate Finance – by F. Allen, R. A. Brealey, & S. Myers