# How do Option Spreads Work?

## A mathematical explanation of why buying option spreads is better than buying naked calls or puts

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When I was first introduced to options, my initial reaction is that this is a tool designed for financial destruction. When buying or selling options, you need to accurately predict three layers of information, the direction of a stock, the magnitude of movement, and the timeframe at which it will happen. This is infinitely more complicated than buying a stock and waiting for it to rise. In spite of the difficulty, it allows people with minimal capital the ability to buy or sell large quantity of stocks.

We will focus on deriving the mathematical relationship of a specific spread known as a bull call spread. The approach used can serve as a framework for other similar spreads.

To understand how spreads work, imagine you are interested in company XYZ because you have belief that its stock price will increase in the future. However, you do not want to directly buy shares of the company because:

1. You do not have enough capital
2. The stock price of XYZ might decrease after you buy shares of the company.

1. Buying a call that is in the money (i.e the strike price has been reached)
2. Selling a call that is out of the money (i.e the strike price has not been reached)

Both contracts have the same expiration date. For a bull call spread, the strike price of the call you sell will always be higher than the strike price of the call you buy. At expiration, there are three possible scenario.

## Scenario 1: Current Price is Greater than Strike Price of Both Calls

The first scenario is where the current stock price of XYZ is greater than the strike price of both the call option you bought and sold. This can be expressed mathematically as:

Since the current price is greater than the strike price of both call options, the holder of these contracts will execute at expiration. Since you purchased a call option, you can execute the contract with the following gain:

where m is the multiplier.

Additionally, since you sold a call, the holder of the contract you sold will execute at expiration incurring the following loss:

Therefore, your net gain can be expressed as:

## Scenario 2: Current Price is Less than Strike Price of Both Calls

The second scenario is where the current stock price of XYZ is less than the strike price of both the call option you bought and sold. This can be expressed mathematically as:

Because the current price is less than the strike price of both call contracts, holders of the contract can not execute it at expiration. The gain and loss is therefore the cost of selling and buying a call contract respectively. The gain at expiration in this situation is the cost of selling a call contract:

and the loss is the cost of buying a call:

Therefore, your net gain can be expressed as:

## Scenario 3: Current Price is Greater than the Strike Price of Call Option Bought but Less than Strike Price of Call Option Sold

The third scenario is where the current stock price of XYZ is greater than the strike price of the call you bought but less than strike price of the call you sold. This can be expressed mathematically as:

Because the current stock price of XYZ is less than the strike price of the call option sold, the holder of this contract can not execute it at expiration. However, since the current price is greater than the strike price of the call option you bought, it be can executed at expiration with the following gain:

The loss in this situation is the net cost in buying and selling a call. When you buy and sell a call option simultaneously, you are starting with a net loss because the cost from buying a call will be higher than the gain from selling a call. This is because the call purchased is already in the money while the call sold is out of the money. The loss in this case is the same as the net profit in the second scenario:

Therefore, your net gain can be expressed as:

Graphing the piecewise function above, we get the following graph:

From the graph, we can see why buying a spread is better than buying or selling a naked call. When you buy a spread, you limit the maximum loss and profit from a position. On the flip side, buying and selling naked calls can in principle incur infinite gain and infinite loss depending on the price of a stock at expiration.

Hope this was helpful! If anyone is interested in buying options, I use Robinhood as my main platform. You can sign up here and Robinhood will give you a free stock upon completion. Feel free to leave a comment if anything is unclear!