What determines an option holder's daily P/L? Part 1
Level 1 - Zero to Options Hero - An easy way to understand your P/L, and an introduction to the Greeks
For a trader of currencies or stocks, it’s quite easy to track your P/L daily. Fear not, for options it is actually not that much more complicated. Let’s dive in!
For a trader who is long a stock, let’s say GE, the P/L of the position from the trade inception is simply the entry stock price compared to the current stock price, plus the foregone cost of carry. Cost of carry is typically a much smaller part of the overall P/L compared to the change in the stock price, but can be thought of as what the alternative yield could have been if you invested the funds used to purchase the stock in US government securities (i.e., 3-month T-bills).1
For a trader who is long a call option on GE, a large part of the P/L of the position is equivalent to the P/L of having a long position in the stock. This is what is called the delta, or delta equivalent.
As an example, a trader who is long 1 call option contract at GE that has a 50% delta would be long the equivalent of 100 * 50% = 50 shares of GE. The 100 here accounts for the contract multiplier, which is 100 in the case of single stock options. We can also express the delta equivalent of the call option position in USD Notional by multiplying the equivalent number of shares by the current price of GE:
50 shares * $107 = USD 5,350 notional delta equivalent
What is Gamma, and how does it affect the delta equivalence?
Now, there is a slight wrinkle when it comes to options in that this delta equivalence is impacted by something known as gamma. Essentially, unlike a long position in a stock, a long position in a call option will get longer the stock as the stock price goes up and less long the stock as the stock price goes down. This is known as long gamma or long convexity, and buyers of options are always long gamma. Most options brokers will tell you the magnitude of your long gamma at any given time on your option position in their app.
To adjust the P/L of an option for the gamma effect, we can use what math folks call a Taylor Expansion to take the gamma into account.
Essentially, since the long option holder has a long gamma position, the P/L of the position will be larger than if the trader was long the delta equivalent of the stock due to the long gamma effect, and using the Taylor Expansion above it is approximately an adjustment of +1/2 * Gamma * (change in stock price) ^ 2 .
To put it in 1 (simple!) equation, it is:
Long Call Option P/L Over 1 Day:
Delta*(StockPrice at Open - StockPrice at Close)
+ 1/2 * Gamma * (StockPrice at Open - StockPrice at Close) ^ 2
That’s it for this post, do let me know if any questions, and in the next part, we will go over another few components that drive the daily P/L of a long call option position!
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To be precise, the cost of carry of being long a stock is the risk-free rate in the currency deducted by the dividend yield earned and also the borrow yield (the rate that traders who want to short the stock pay to borrow the stock in order to go short).