
Continuing further from our previous article Payoff Function Examples for Options, in this article we will see how multiple call and put options on the same underlying with the same expiry date but with different strike prices and different positions (long and short) can be added to arrive at the net payoff function chart. This net payoff function chart will then represent the actual payoff diagram.
Remember 
1) You can ONLY add the payoff functions which are on the SAME underlying. Payoff functions on multiple underlying cannot be added.
2) Usually, we add payoff functions with the same expiry date. Adding different expiry date payoffs is a bit complex task and will be explained separately
Mathematics of Options Trading
You are an active options trader  you keep taking positions one after the other as the market moves. So it is quite possible that at any given point, you may have multiple calls and puts in the same underlying with same expiry dates but with different strike prices and some of them are long while some are short positions. The question is  how do you add all these multiple positions together to arrive at the net payoff function chart?Let's begin with an example:
Say you have the following positions in Microsoft Options:
1. Call  Long  Strike Price of $45  Bought at $7
2. Call  Short  Strike Price of $50  Sold at $3
3. Put  Long  Strike Price of $55  Bought at $8
4. Put  Short  Strike Price of $50  Sold at $2
Assume that all the above 4 options have the same expiry date.
Now, let's begin the exercise of coming up with the net payoff function for all these 4 positions which you have:
Step 1  Draw the individual payoff functions for each of the individual option positions, considering ZERO price:
Since we have 4 option positions, we will have to draw 4 payoff functions for them, assuming ZERO price for each. This is how the individual payoff functions will look:
Step 2  Separate the payoff functions for CALLS alone (both LONG CALL options and Short Call options):
Considering ONLY the call options, here is how they will look
There are a total of 2 Call options  one $45 Long call option (TURQUOISE) and another $50 short call option (BLUE):
Step 3  Add all the payoff functions for CALLS alone (both LONG CALL options and Short Call options):
IMPORTANT: Care should be taken for adding the payoff functions for call options  You Must Begin adding them from the LEFT HAND SIDE i.e. start from zero on horizontal axis and move towards right for higher values.
Between 0 and $45, Both the BLUE & TURQUOISE call options are ZERO. Hence, zero + zero = zero. (Horizontal ORANGE line between 0 and $45)
Between $45 and $50, the TURQUOISE call option is increasing (upward slant), while the BLUE payoff function is ZERO. Hence, net sum is same as TURQUOISE payoff function. (Upward slanting ORANGE line between $45 and $50)
Between $50 and above, the TURQUOISE call option continues to increase, while the BLUE call option is decreasing (downward slant). The two slants cancel each other and what you get is a net horizontal line (with zero slope/slant). Hence the net horizontal line above $50. (ORANGE)
This ORANGE colored payoff function is the net payoff function from all the call options. This ORANGE function will be used further in step 6 below
Continue to next steps  Adding multiple Call & Put Options Payoff Functions
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