<aside> ℹ️ About

• This homework will help a student understand why selling covered calls or cash-secured puts is not income. In fact, the student will see that options are always volatility trades — you will appreciate just how deeply insightful the concept of put-call parity is.
• Prerequisites:
  • basic probability including conditional probability (although you do not need Bayes Thereom, just logic. You will need to do a lot of reasoning the same way you need to turn any word problem into math statements)
  • Excel or your preferred computational tool

First, a self-indulgent remark…

Good news

The purpose of this exercise

The origin of this exercise

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Scenario Description

<aside> ⚙️ Assumptions

• A stock is trading for $100.

• The risk-free rate and dividends are zero so we can ignore cost of carry or hurdle concerns.

• Options on the stock expire after 20 periods. You can think of them as days

• How the stock moves

  In each period the stock can only go up or down

  You may want to refresh yourself on coin flips or binomial distribution as encapsulated by this picture:

  

  Probabilities and Magnitude

  • P(up) = probability stock increases in value = 50%
  • P(down) = 50%
  • Magnitude: The stock always moves $1.00 </aside>

<aside> 📊 Understanding the distribution

1. For any single period, the stock is 50% to go up $1 and 50% to go down $1. What’s the expected value of the stock after the first period? After 20 periods? The stock starts at $100

   • 1 period
   • 20 periods

2. After 20 periods, how many possible outcomes are there?

   • Hint
   • Answer to the number of outcomes
   • Answer to the number of paths
   • Visual answer

3. What is the probability of each outcome? This is the distribution.

   • Hint
   • Answer to the distribution </aside>

<aside> 🏒 Pricing Options

In this discrete distribution, we know there are 21 possible terminal prices for the stock after 20 periods. By knowing all the possible expiration prices and their probabilities we can price an option for any strike by simply weighting all of the possible payoffs.

<aside> 🔑 Help

• General hint
• In case you need to refer to the distribution because you didn’t compute it yourself </aside>

1. What is the fair value of the 103-strike call at the beginning of the 20 periods when the stock is $100?
   • Answer

The delta of an option predicts how much the price of the option will change for a $1 change in the stock. In practice, this is an output of an option model such as Black-Scholes and like the formula in general is sensitive to the option’s moneyness, implied volatility, and time to expiry.

But we were able to price the 103-call arithmetically using a binomial tree. We were able to do this because we simplified the stock process into something discrete — either the stock will go up or down $1 each day. We do this because the intuition from such a model is deeply instructive without needing to wade into continuous models and calculus.

Likewise, we can compute delta arithmetically, just as we priced the option.

Here’s your chance.

1. What’s the delta of the 103 call?
   • Hint
   • Answer </aside>

<aside> 💰 Computing P/Ls

Imagine we sell the 103-strike call as part of a covered call strategy.

1. Compute the P/L at expiration for each of the 21 terminal prices assuming you own 1 share of the stock (assume you purchased it for $100) and sold the call at its fair value you computed above.
   • Hints
   • Answer

<aside> ⏸️ A pause for some words

Assuming you are not a masochist volatility trader, if you sell covered calls you hold a portfolio of long shares, short calls until expiration and maybe you sell calls again for the next term.

What’s going through your brain is “I’m going to own the shares anyway, I might as well receive some premium income, and if I get called away on the shares I’ll be happy anyway.”

To be blunt — this logic is mysterious to me since the whole point of owning equities is for the upside but that’s not really the point of this particular exercise.

Your interest in covered calls is an opportunity to understand how options are intimately about volatility and why you should not use them unless you have a view on volatility that is differentiated from the market’s consensus.

Stay with this exercise and it will all make sense…

</aside>

We’re doing basic p/l calculations still.

• If you own a share of stock and it increases by $1, you enjoy a $1 profit and vice versa.
• If you have a covered call strategy, and the stock increases by $1 you make $1 profit on your shares lose on your short calls. If you are short a 25% delta call, it will gain $.25 of value on that same $1 move.

  • Share position: +$1

  • Call position: -$.25

    Net P/L: +$.75

    [If the shares go down in value you will lose $.75 (-$1 on the shares and +$.25 on the short call position.]

The key insight: at the inception of this portfolio you have a 75% exposure to the stock instead of 100%. You make or lose $.75 instead of $1 when the stock moves $1.

If you are a professional options trader explicitly playing the abstract game of “trading volatility” you’d hedge the entire delta…you would not want to have exposure to direction because you have no opinion on the company as an investment.

But the covered call seller clipping some coupon that “averages down their purchase price” or some marketing b.s. like that is now implicitly a volatility trader that just has a long bias.

The difference between this investor and the volatility trader is the investor is not going to continuously rebalance their share position to maintain a constant exposure to the stock.

[the other difference is the investor doesn’t realize they are now a “vol trader” but that’s the point of this whole post — to make this crystal clear]

The volatility trader tries to maintain zero exposure.

The covered call seller who sells a .25 delta call initiates a 75% exposure. However, the exposure is going to bounce around between 0 and 100% depending on how far the stock is from the strike price, how volatile the stock is, and how much time remains.

This is easy to understand at the extremes. If you are short the 103 call and long a share of stock trading for $200 you no longer have any marginal exposure to the price. If the stock goes to $199 or $201 you are unaffected. The owner of the call you are short is the one exposed to the stock. You can think of them as owning the shares that are in your account.

Similarly, if the stock is trading for $10, you own 100% of the exposure. The 103 call you are short is worthless because it’s so far out-of-the-money.

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<aside> ⚖️ Portfolio Comparisons

</aside>

<aside> 🚦 Checkpoint

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<aside> 💡 Discussion on how volatility impacts your covered call option trade

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