Thread by @AgustinLebron3

1/ Options 101 (a thread):

When you teach options, they put you in jail if you don't start with this:

A call/put is the right (but not the obligation) to buy/sell a security for a given price at (or by) a certain time. Boring! Let's do an example:

2/ If I own the Mar 700 call in $TSLA then that means anytime before ~5pm on Mar 19 2021, I can exercise my right to buy TSLA shares.

I hand over $700/share and I get some TSLA stock. First let's note a few things:

3/ US equity multipliers are typically 100 (barring corporate actions). So when you buy 1 options contract (call or put), you're actually buying the right to 100 shares.

Remind me to tell you the story of when I messed up the DAX multiplier. I was *sure* was going to get fired.

4/ That call I bought is an "American-style" call which means I can exercise it anytime before expiry. European-style options only allow you to exercise *at* expiry.

5/ Since an American option gives you more rights, they can't ever be worth less than their European equivalent. But usually they're worth the same. Why?

6/ Because exercising early means you give up being able to "look into the future".

Say $TSLA is $750 on Mar 16th, so you exercise that 700 call on Mar 16th (i.e. you buy stock for $700).

But then on Mar 17th Elon announces it was all a big fraud and stock goes to zero.

7/ Don't you wish you would have waited till the very last moment to exercise?

Because then you just wouldn't have exercised your call at all, and you wouldn't be out the $700/share!

8/ So when *are* A options worth more than E options?

For calls, it's when the option is going to lose the right to something at some point. For example when stocks pay divs.

For puts, it's when money having money is worth a lot (High interest environments. Remember those?).

9/ Ok let's talk about moments. When you buy/sell a share of stock (for edge), you're making a bet on its EV.

Buying means you think the EV of the stock (some date in the future) will be higher than the price you paid.

EV means (loosely) the first moment of the distribution.

10/ For options, you're making a bet on both the first moment (EV) and the second moment (volatility). You're also making bets on the higher moments, but we'll ignore those for now.

The important point is: trading a simple vanilla option is actually trading a basket of risks.

11/ Why does this matter?

Because a lot of the IMO poor thinking about options comes from the fact that people don't disentangle, or don't specify well enough, which of the two bets (the EV bet or the vol bet) they're trying to make.

12/ If you think $TSLA is going up, buy the stock. It's cheaper and simpler.

"Ah, but I want leverage so I'll buy the call instead." you say.

Fine, but you're paying for that leverage.

13/ Options MMs *love* selling options to people who want to bet on the stock. They just shove that EV risk off by hedging.

To the MM you bought your call from, they just stood in the middle of a simple equity trade. And made money selling you some vol you didn't think about.

14/ Ok, that's enough for today. Need to do some real work. Tune in tomorrow when we start talking about vol.

WTF is "implied vol" anyway, and why does it matter so much?

15/ Ok we're back for day 2 of Options 101. I'm overwhelmed by the positive response so far, and I'll try to deserve the attention.

Today we're going to talk about vol, but we're going start by talking about not-vol.

16/ You see, puts and calls are really the same thing. Assuming the underlying stock is "normal" (no big divs, no corp action, easy-to-borrow, etc), there is *no* difference between the two.

I'm going to prove it:

17/ Say $TSLA is at $740. You to buy the Mar 700 call for C, and sell the Mar 700 put, receiving P. Your total cash outlay is C - P.

What happens at expiry? If TSLA > 700, exercise the call. You pay $700 and get 1 share. Put expires.

What's your position if TSLA < 700? Wait...

18/ The same thing! Your call is now worthless, but whoever you sold the put to will exercise it.

They sell TSLA @ 700, so when you're "assigned the put", you pay 700 and get 1 share.

NO MATTER where TSLA is at expiry, you will own 1 share of stock and have paid $700 in cash.

19/ So in order for there to *not* be free money lying around, the value of your stuff today must = the (time-adjusted) value of your stuff at expiry.

C - P = Stock_px - Strike_price + (small details I'm ignoring for now)

Ok so what?

20/ Well, it means you can always transform a call into a put and vice versa with a simple stock trade.

Buying a call is just buying a put + buying some stock. Calls and puts are the same thing (modulo a stock trade).

But why does this matter?

21/ Well, remember yesterday?

If you care about EV of the stock, trade stock.

Options, really, are for trading *vol*.

Buying calls = buying puts = buying vol. Same for selling.

Ok, so what's this "vol" thing?

22/ Well, we could define it a few hundred different ways. And if you want to get into the math, knock yourself out.

I'm going to skip all that.

Having taught options a few times I've learned that the math can obscure the intuition. Trust me a bit here...

23/ So let's define a magical (invertible) function that transforms prices into "vols" and back again.

Call the vol -> px function "BS(vol)".

Call the px -> vol function "BS_inv(px)".

They're inverses. So BS(BS_inv(px)) = px.

24/ BS and BS_inv are functions that are parameterized by the details of the option in question: und price, strike price, time to expiry, American vs Euro, call/put, a couple of interest rates, etc.

And if you're following along, you should be screaming the following question:

25/ Why would we go through this rigamarole? Why not deal with prices of options directly?

Cuz options prices suck. They're hard to think about.

Also, in trading we really care about *changes* in things, not levels.

And change in option prices are *really* hard to think about.

26/ Options prices are all over the map.

The $TSLA Mar 700 call costs ~$105. Say tomorrow it costs $100.

Yes it went down 5% but that's not the most useful way to think about the change in *value* of the option.

27/ What if I want to compare TSLA options to NKLA options? Or Mar options to Jun options? Or the 700 call to the 750 call?

There are too many parameters, it's just too complicated to get good intuitions about. Intuitions matter.

28/ The purpose of BS and BS_inv are to *renormalize* prices of options into a measuring stick that we *can* get some intuitions about.

That's all "implied vol" is: a renormalization. Physicists rejoice!

29/ Famously, the BS (or Black-Scholes) vol is the "wrong number inserted into the wrong formula to get you the right price".

But look, no one on Earth thinks the BS assumptions of stock dynamics are accurate.

30/ But millions of BS vols get calculated every second because they get a very important job done:

They let you *think* about options. And that's no small feat.

Well, this got longer than I expected again, so let's leave vol curves and surfaces for tomorrow.

See you then!

31/ We're back with day 3 of Options 101. Today we'll talk about vol curves! Finally the good stuff, right?

Maybe. We'll see how long this gets. :)

32/ As a reminder, so far we've established that if you're going to trade options, you're doing it because you want to trade vol.

Or more accurately, you have opinions about the underlying distribution *other* than its mean.

33/ Imagine a world where stocks actually *do* follow that mythical log-normal distribution assumed by Black-Scholes. What would you see?

You'd see that every option's price would transform (using BS_inv from yesterday) into the same implied vol.

34/ So if I plotted implied vol vs strike, both puts and calls would be straight horizontal lines on top of each other.

Of course, real life doesn't look like that. If you plot IV vs
strike, you (almost always) get a smirk shape. Let's look at an example: $TSLA Mar 2021 options.

35/ I thought about using my own internal tools to create this plot.

But I realized that embedded in those tools are all sorts of important knowledge and decisions that matter *a ton* for how to look at options. So I took the data from a public source (in this case @Barchart).

36/ Here it is:

37/ Ew! So gross! This data has problems, but that's ok. All data has problems.

You just need to do the work to fix the problems.

38/ Problem 1: what's the deal with those zeros?

Those are strikes that, while they're *listed*, don't have real markets. So we need to strip those strikes out.

39/ Ah much better.

Problem 2: why are the put IVs almost always higher than the call IVs? Remember, our put-call-parity lesson from yesterday tells us that they should be the same.

40/ Well, remember those fiddly bits at the end of the equation C - P = Stock_px - strike_px + (stuff I'm ignoring)?

The stuff I ignored (a) matters, and (b) is so much harder to get right than you can possibly imagine.

41/ If you've ever built systems to calculate lots of IVs very quickly, you'll learn that getting that parity thing right will cause you endless grief.

So let's do a simple adjustment.

42/ Ok, getting better.

43/ Let's note some more things:

The puts look smoother on the left side, the calls smoother on the right. This makes sense. Remember, puts = calls so if you're going to trade an option, trade the out of the money option. Why?

44/

- The prices are lower. You don't have all the in-the-money-ness to deal with.
- The prices are less sensitive to stock moves, so when trading you don't have to worry about getting picked off to stock as much.
- We've all agreed on this. Liquidity begets liquidity.

45/ What else? Well, downside vol is clearly higher than upside. What this means is that *relative to log-normal*, the probability mass is thicker on the downside.

This also makes sense. Big down moves are more frequent and larger than big up moves.

46/ Wait, does this mean that I should trade options if I have a directional opinion about the stock?

No! Aren't you paying attention? 😆

Higher downside vol is just an artifact of the fact that log-normal isn't right. The empirical dist is different, but the mean is the same.

47/ Ok, back to that bumpiness, does that mean I can buy the cheap ones and sell the rich ones?

You can try, but again you'll see that it's a data artifact. There isn't *one* vol per option. There's a vol representing the bid price and one representing the offer.

48/ What's the "right" vol to use? Mid? That's entirely up to you and your modeling assumptions.

That's where the $ lie, so you're not going to get that from some public data source.

49/ I think that's enough for today.

I'm always open to comments and suggestions and requests on where to take this next...

50/ Options factoid: spreads tell you the probabilities of things. How?

Imagine I buy the 50-51 call spread. That means buying the 50 strike call and selling the 51 strike call.

51/ What's the least this structure can be worth?

0 obviously, when both options expire out of the money. I.e. when stock < 50 at expiration.

The most it can be worth is 1. That happens when stock >= 51.

52/ Let's ignore (for now) when stock ends up between 50 and 51.

Now if the world thinks the stock is basically guaranteed to close below 50, you should expect to pay almost nothing for this spread.

And if the world thinks > 51 is guaranteed, you should expect to pay 1 for it.

53/ Those prices are starting to look a lot like probabilities, aren't they? 🤔

In fact they are. if we define C(K) as the price of a call with strike price K, then (C(K+e)-C(K))/e is (when e is small):

The probability stock expires above K!

54/ Pretty awesome huh? You can recover the market-implied CDF (and hence the PDF) by "differentiating" the prices of the options.

Exercise for the student: show the same is true for put spreads.

55/ More factoids like this? Or something different? What say you #fintwit?

Ok everyone, you've passed Options 101. Time to start on Options 201!

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