Options PnL Part 1

First, we'll look at general profit and loss graphs to make sure our audience understand basic algebra (Cartesian planes, slopes, intercepts, etc.). Then we'll take a brief look at futures. And finally, we'll touch on options.

So first, what does your PnL graph look like if you purchased a single contract for the price of 34? If the contract was now worth 34, you would neither make anything or lose anything. If the contract was now worth 35, you would have gained 1. So on and so forth. Drawing this out, we get:


A couple of elementary things to note. First, note that the slope is 1 because we only purchased one contract. For each dollar the contract gains or loses, our PnL will be affected by that much. If we had purchased 3 contracts, then the slope would be 3 since if the contract gains a value of 1 (to become 35) we would have gained 3. Second, notice how much we could possible lose and how much we could possibly gain. If the contract value dropped to zero, the most we could lose is 34, whereas if the contract value went up towards infinity (that is, just kept rising) we could make an unlimited amount of money.

To drive the point home, let's look at selling 4 contracts at the price of 21. If the contract value goes up by 1, we would lose 4. If the contract value goes down by 3, we gain 12. Drawing it out, we get the following:


Don't let the graph fool you. The slope here is -4 since we sold four contracts. Again, what is our maximum loss and maximum gain? If the contract value drops to 0, then we stand to gain 21 * 4 = 84. If the contract value goes up towards infinity, we stand to lose and unlimited amount.

Now, let's consider drawing a graph for future contracts. Would it be much different from the graphs we have above? Not really (you'll see much different graphs soon when we start on options) because buying a future doesn't really give you protection against price movement. Sure, you get to lock in the price early on but if the underlying price moved, you would be susceptible to gain (or lose) purely on the market swings. So, how are options any different?

One thing to note before we get into options is that we pay a price for the right to buy/sell the underlying, whereas futures (and stocks) what you pay is what you get (that is, you can't just abandon a future or stock).

To draw the PnL of an option, we need to specify what option this will be. Without concerning ourselves with what the underlying contract (e.g. corn, oil, gold, etc.), we will just say the graph our PnL at the time of expiration if we were to own a 45 call option. I'll explain later why we specify such things as "at expiration."

Alright, remember what a call is? It is an option that gives us the right to buy at the strike price (which, in this case, is 45). So what happens if the underlying is price goes up to 50? In that case, we just exercise our option and we can buy the underlying (priced at 50) for only 45. In essence, we've just earned 5. On the other hand, if the underlying price drops to 40, we can simply abandon our option. In essence, we wouldn't have to lose anything. So, to summarize, here is our PnL graph:


A few things to note before we move on.

First, some people might ask, "why do we make nothing when the underlying price is equal to the strike price?" Well, if you think about it, you have the option to purchase something for 45 and its also worth 45... so your option is actually worthless. Any market participant has the right to purchase it for 45.

Second, notice that to the right of the kink (at the strike price of 45), the slope is 1. It is as if we own the underlying asset. That is not true when the underlying price is under the strike price, in which case, we don't care since at that point, we would rather abandon the call. (There are extremely rare situations when you would exercise a call where the strike price is higher than the underlying price, but we'll get into that later).

Unfortunately, this graph makes it look like buying calls is a risk free way of making money, since we don't have a loss. Actually, we did omit one vital piece of information: the price we paid to obtain that contract. In our example, let's say the price of the option is 2. That means, we should shift the graph down by 2 units to get the following graph:


Easy enough, right?

Now, consider this: what is the worst case scenario for buying a call option? What is the best case scenario? The worst case scenario is you lose the 2 units you paid purchase this call. The best case scenario is the underlying value shoots off into infinity and we make a killing! Limited downside and unlimited upside!

Now let's look at purchasing a put option. Again, without concerning ourselves with what the underlying contract (e.g. corn, oil, gold, etc.), we will just say the graph our PnL at the time of expiration if we were to own a 45 put option. Using the same logic as above (including actually paying for the put), we come up with this graph:


At the risk of sounding annoying, let me explain in detail what this graph is telling us. Owning the 45 Put gives us the right to sell the underlying at the price of 45. So, at expiration (the last day we have to use the option), if the underlying price is lower than 43, we make on the difference. For instance, if the underlying price is 39, then we can sell it at the price of 45 and (in essence) purchase it back for 39; we would have made 45 - 39 = 6 from the option, less the 2 we paid to obtain the option (total profit is 4). On the other hand, if the underlying price is higher than 45 (like 51), we can sell it at the higher price (of 51) instead of the option's strike price of 45. So, we just abandon the contract, losing the initial amount we paid to get the option.

Okay, we just looked buying calls and puts. Next time, we'll look at selling calls and selling puts.