<p>One reason why you might want to do this, and perhaps is the motivation for this example, is that it gives you a simple example for thinking about option pricing.</p>

<p>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 <em>replicating portfolio</em>, 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.</p>

<p>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.</p>

CC BY-SA 3.0
2017-09-06T09:53:37.923Z
Thanks for the answer! Indeed there're missing pieces in the posting, it is just that I got confused about the benefit of selling the option. But I think I understand it now
caverac
CC BY-SA 3.0
2017-09-05T17:11:16.703Z
2017-09-05T18:16:42.763Z
2017-09-05T18:16:42.763Z
Theoretical Economist
<p>I am pretty new at this, so my apologies in advance if the question is too out of place.</p>
<p>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 <span class="math-container">$S_0 = 100$</span>, and a call option on it with strike price <span class="math-container">$K = 100$</span>. The up-state and down-state values of the underlying are <span class="math-container">$S_u=110$</span> and <span class="math-container">$S_d = 90$</span>, respectively. Now the problem states</p>
<blockquote>
<p>Let's assume we take a position <span class="math-container">$\Delta$</span> in shares and <em>short one call option</em> to create our portfolio, which is then comprised by two assets: the underlying and the call option.</p>
<p>If the price goes up, then the shares will be worth <span class="math-container">$110\Delta$</span> and we will loose 10. Therefore, the value of the portfolio is <span class="math-container">$110\Delta \color{red}{-10}$</span>.</p>
<p>If, on the other hand, the price goes down, the call expires worthless and the value of the portfolio is <span class="math-container">$90\Delta$</span></p>
</blockquote>
<p>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.</p>

CC BY-SA 3.0
2017-09-05T08:10:58.827Z
You should have been paid for the call option when you wrote it (went short)
Henry
CC BY-SA 3.0
2017-09-05T09:01:43.050Z
@Henry Does it mean it should be $110\Delta \color{red}{+10}$?
caverac
<portfolio-theory>
Short call in binomial option pricing model
caverac
# Short call in binomial option pricing model

(Asked by: username_0 on 2017-09-04)

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.

**Tags:** portfolio-theory
> **Comments:**
> **username_1**: You should have been paid for the call option when you wrote it (went short)  
> **username_0**: @Henry Does it mean it should be $110\Delta \color{red}{+10}$?  

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## 1 answers

### Answer by username_2 on 2017-09-05. Score: 2 | Accepted answer

One reason why you might want to do this, and perhaps is the motivation for this example, is that it gives you a simple example for thinking about option pricing.

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.

> **Comments:**
> **username_0**: Thanks for the answer! Indeed there're missing pieces in the posting, it is just that I got confused about the benefit of selling the option. But I think I understand it now  

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