# Short call in binomial option pricing model

I am pretty new at this, so my apologies in advance if the question is too out of place.

I have been reading about portfolio replication models, and stumbled upon this example that I don't quite understand. It goes like this: there's a stock with current price $$S_0 = 100$$, and a call option on it with strike price $$K = 100$$. The up-state and down-state values of the underlying are $$S_u=110$$ and $$S_d = 90$$, respectively. Now the problem states

Let's assume we take a position $$\Delta$$ in shares and short one call option to create our portfolio, which is then comprised by two assets: the underlying and the call option.

If the price goes up, then the shares will be worth $$110\Delta$$ and we will loose 10. Therefore, the value of the portfolio is $$110\Delta \color{red}{-10}$$.

If, on the other hand, the price goes down, the call expires worthless and the value of the portfolio is $$90\Delta$$

My question is: why would I include this call option in my portfolio at all? Clearly I am not profiting from it in any sense, I either would loose money or not gain anything.

• You should have been paid for the call option when you wrote it (went short) – Henry Sep 5 '17 at 8:10
• @Henry Does it mean it should be $110\Delta \color{red}{+10}$? – caverac Sep 5 '17 at 9:01

Under some assumptions, the value of any option can be replicated by continuously trading in the underlying asset. Here, you can think of the $\Delta$ shares that you have as your replicating portfolio, and the amount $\Delta$ is meant to be chosen so that the value of these shares is exactly the value (to you) of the call option you just sold.
You can then price the call option according to how much those $\Delta$ shares cost you. If you chose $\Delta$ appropriately, you break even, and this determines the "market price" for the call option you just sold. This example doesn't give you quite everything you need to calculate that yet, but I imagine whatever you're reading is building up to get you there.