Special thanks to Daniel Rock for reviewing this before publication and helping address some of my questions.
I guess we could call this a primer following my prior posts in options strategies, mostly because at least a few of you follow me for options and vol-related content, and I’ve been lacking on that front lately.
The prior posts:
The thread that sparked this post was an exchange I had with @darjohn25 (an excellent follow) and a few others:
Mind you, option strategy naming in general makes no sense, and at no point in this post am I going to even attempt to be consistent with what I call things. That said, I will spell out the construction accordingly (e.g. time/tenor/strike) so you can call it whatever you like.
Also, for the time being, I’ve discontinued my Twitter presence. I haven’t deleted my account (unless Twitter decides to), but I’ve been exhausted by the perpetual onslaught of harassment and abuse. It’s very sad in ways, since it’s been a persistent theme of my online experience in trading communities, but also in many ways predictable due to age, gender, and my opinions. Come back date TBD.
Last caveat — I have no machinations this post will be error-free. In fact, you should consider everything following this warning a work of fiction and performance art.
Options are an interesting form of derivatives, which are assets whose prices are defined in relation to another asset. Derivatives include such fun things as swaps, options, futures, and an ever-wider assortment of crazy financial innovations coming from the minds of crypto folk like Dave White:
Many view options as a form of leverage, and this is prima facie correct too. We can separate leverage into two types:
1) Recourse leverage - Car loans are a form of recourse leverage, in which you can lose not only your initial collateral, but more. Essentially if you have a car loan, in general it is backed by the value of the car itself (if you don’t pay, the lender can take your car away). However, cars also depreciate really rapidly, and if the lender repossesses the car, sells it, and the balance doesn’t cover the loan, they can go after your other assets as well.
2) Non-recourse leverage - Non-recourse leverage, on the other hand, is backed solely by the value of the collateral. This means if you have a non-recourse loan on let’s say a house, the lender can - if you do not repay the loan - repossess the house. But if the market is bad and the house sells for less than the initial loan value — tough luck to the lender. The rest of your assets are fully protected.
Options of course are the second type of leverage, non-recourse. If you are a long an option contract, you pay the premium, which for the lifetime of the option acts as your collateral. No matter what happens to the underlying, you are guaranteed from the moment of purchase of an option to never lose more than the premium of your option and this makes it enticing. We can compare an option in many ways to just normal leverage, which can be done (at a much lower ratio, usually) using futures or portfolio margin. In the latter cases, our risk of loss is unbounded — even if we designate some level of max loss (for instance, a stop) before closing the position, there is no guarantee that our stop will hit, and our broker can liquidate all assets in the account via margin call (and potentially assets outside the account, but this is more rare).
Conversely, with an option, you can get (for the lifetime of the option) identical exposure to the normal leverage methods, with much less risk of ruin due to the non-recourse nature of the option (this of course means sizing correctly/not YOLO-ing).
In fact, the major benefit for most traders of the non-recourse nature of the option for leverage (especially for low volatility assets, like the S&P 500 and its associated ETPs/derivatives) is removing the element of path dependency. By nature, the market spot price of an asset is stochastic — while there is some element of the behavior that is deterministic (e.g. stocks go up over time, usually), at any given point there’s only a probability of whether the price will go up or down. There are many gotchas (we call those “edges”) which help anneal hat probability distribution in your favor; or conversely, many forms of trading which use certain derivatives of the spot price behavior (e.g. betting on volatility spreads, statistical arbitrage) which tend to be more deterministic than price itself.
@TheRobotJames has a good thread on path dependency and stop losses:
The dangerous nature of recourse leverage tends to be because of what many fancy statistics folks call ergodicity, but me, a simpleton, calls risk-of-ruin. Essentially, for a realistic portfolio, there is a certain max amount of loss you can handle before either your broker margin calls you, you start looking up ways to fake your own death, or if you’re very lucky, the government bails you out. This is compounded by the idea of mark-to-market — the way we (meaning your broker) evaluates your current loss or gain is based on available market prices. For stocks and even options, the underlying is fairly liquid, so you have nearly continuous mark-to-market risk. This means that no matter if your end result is profitable, at any point in the trade you risk complete ruin if the market moves against you. We can construct a simple example here:
Let’s imagine I’m betting on a stock Z currently at $50. I believe in a week there will be good news about its current drug trial, and the stock price will conservatively hit $65. I decide to open a $100,000 position using my $50,000 portfolio, borrowing $50,000 from my broker to use as margin.
The issue of course here is by levering the position 2x, all profit and loss similarly (and continuously) is evaluated at 2x. So if the stock price halves during the week (maybe Biden choked on a pretzel or he decided to nationalize all drug companies but then walked it back), I lose my entire position — even if it turns out Z hits $65 when I expected it to!
This is the danger of recourse leverage. Of course, if I got very very unlucky, I could lose even more than my entire position (let’s say a gap down to $10), depending on my broker’s rules.
Conversely, I could also express my view next week with options, buying the $50 strike option for $1.35 expiring in a week. If I instead by $100,000 worth of those (which, by simple terrible math equates to 20 contracts), I’ve outlaid $27 in premium ($2700 total, plus transaction fees). No matter if the stock round-trips to $0 and to $100 in that week, my total loss max is $2700, not $50,000. My gain, if correct, is still $30,000, minus the premium paid (so $27,300).
But in the market, there’s no such thing as a free lunch (I’ve been reminded many times that cleverness tends to be punished, not rewarded). We can observe here that it seems almost too good to be true — why would you ever not buy an option instead of levering the underlying? Your risk is substantially lower, due to the non-recourse structure.
It’s simple. In options-land, you are paying for volatility itself.
When we say an option costs $1.35 (in the above example), the next logical question is — why does it cost $1.35? In my example, I clearly possessed insider information, and if the unlucky sap who sold me the option had the same information, they should’ve clearly charged me more.
The answer comes from Black-Scholes and the dynamic hedging argument of option pricing. For a vanilla option (a call or a put), the price you pay for an option isn’t defined by simply the distance or max value of the option itself, but by the implied price required for the option seller to perfectly hedge the option by buying and selling the underlying. This makes a lot of very fun, non-real world assumptions (e.g. infinite liquidity), which gives rise to interesting phenomena like options degeneracy depending on how it gets hedged.
What this means in English is even though at face value, it should seem like the seller made a bad trade (since the option premium they sold me for $1.35 is worth $15 at expiry), it doesn’t have to be, especially if they could continuously hedge their position on the way up.
This is where the intuitive relationship with volatility comes in — the cost of hedging, even assuming infinite liquidity, is directly related to how volatile the underlying is. Even if we subscribe to the dynamic hedging argument of Black-Scholes (that we can rehedge every infinitesimal move based on the option delta), this implies that the cost of the option should be the position-weighted (delta) sum of all infinitesimal stock moves to expiry. Or, for humans, we call this future realized volatility.
This is because we don’t live in a perfect world like the below example.
Let’s say we observe the same stock Z at $50, with the same end condition ($65 in a week). If I knew that once the stock hit $65 it would never go below it, the ideal hedging strategy would be for me to set a limit order at exactly $65.00 for 100 shares. This is intuitive — if I know it would never dip below, I would essentially have a costless hedge minus interest rates paid, given our agreement (the call option) to sell you the shares. By that same vein, if everyone had the same information, we can use elementary game theory to show that the market price of the option should be more or less exactly equal to the interest cost of the position.
However, in the real world, stocks do not move like that. Because stocks move up and down on all timeframes, we cannot statically hedge like that — we must adjust our hedged position according to the probability they end up in the money. This incurs a specific cost, which is proportional to the future realized volatility of the asset.
In options world, two things are paramount - volatility and time. As discussed in my prior posts, in many ways volatility is time — an asset that is volatile, much like the Einsteinian concept of time dilation, experiences time much more rapidly than a non-volatile asset. The reason of course is both volatility and time tell us about the future possibilities of the asset’s price range — as time increases, we have longer opportunity for price to move and therefore the price at expiry covers a wider range. Similarly, a more volatile asset will cover a wider price range, even with time held constant. In the world of markets, clock time is a fairly meaningless and naive concept — the reason for its wide acceptance is more of convention, and to give all participants a uniform scale in which to operate in (and more importantly, create agreements/derivatives for). It’s very hard for example to speak in volatility time, simply because volatility is not a constant — it itself changes as a function of time for a given asset.
However, understanding volatility time is critical to understanding why options operate the way they do. If we treat volatility as the ‘true’ time to expiry of an option, it starts to make intuitive sense why the Greeks behave like they do:
As volatility increases, we can nearly equivalently treat it as the time remaining on an option contract also increases. This should be intuitive — volatility measures how much the price can move in a given unit time, which is what we actually care about. If volatility is zero and you know it to be zero indefinitely, the value of an option is trivial (it’s equivalent to the discounted price of the forward).
Similarly, as volatility decreases, we can equivalently treat it as the time remaining on the option contract decreasing as well. The logic is identical to the above.
This gives us an operant space in which without math we can infer the behavior of the Greeks. If you remember my previous series, the major “first-order” derivatives are:
The change of the option price as the stock price changes (we call this delta)
The change of the option price as volatility changes (we call this vega)
The change of the option price as the risk free interest rate changes (we call this rho)
The change of the option price as the time left to maturity changes (we call this theta).
With a bit of math, you can see some beautiful symmetry in the behavior of the second-order derivatives due to a property called Schwarz’s theorem, which is used in the vector calculus to form the Hessian matrix of a function. That said, let’s stay away from math and figure out the empirical basis for this.
Let’s rehash two arguments:
1) Delta is, at a rough approximation, equivalent to the probability that an option expires in the money. This makes a ton of non-real-world assumptions about the nature of the markets, but is good enough for reasoning.
2) Increasing volatility is equivalent to increasing time remaining on an option; decreasing volatility, the opposite.
We can observe the following to be trivially true:
1) For an option with more time remaining, there’s more chance for an ITM option to go out of the money, and vice versa. This is why a an at-the-money option represents 50, not 100 delta — assuming equivalent chance of the stock going up or down in price, we anticipate there’s a 50% chance we land OTM, 50% ITM.
In options speak, this implies that a wider price range is covered on the spectrum for 0 to 100 delta as time to expiry increases.
2) As time winds down, we anticipate (much like the closing umbrella analogy) a smaller price range is available for us to realistically move to (assuming whatever distribution we please). This is why as time ticks down, delta increases and decreases more for smaller spot price moves (we call this gamma, or the change in this behavior dGamma/dTime color).
3) As described above, changes in volatility equate to changes in time, so plugging in 1 and 2 we can expect:
As volatility increases, OTM options should gain delta (because they have more “time” to become ITM, and ITM options should lose delta.
If vol is time, we should see the impact highest at near-ATM options (much as gamma increases as we get close to expiry) -
As vol increases, we should see delta decrease most for options near-ATM but ITM (much like if we increases time remaining to expiry). Similarly, we should see delta increase for options near-ATM but OTM.
As vol decreases, we should observe the opposite behavior.
Much as we see as time remaining ticks down gamma becomes a dominant force near-ATM (leading to the monthly OPEX pin slip effect on SPY, for example), we should see volatility has a negative relationship with gamma (or changes in delta due to changes in price). As volatility increases, the impact to delta for a small change in spot price should decrease (much as how gamma decreases with time remaining increases). Unlike above, this should behave bidirectionally — both OTM and ITM near-ATM options should change delta less in high volatility as spot price moves.
And this behavior is precisely what we observe in options. It’s a bit nonsense to observe options as anything but levered instruments of volatility, because that’s what they are. You are paying for volatility and time, but those are also essentially the same thing.
So why does this matter? When you’re constructing an options position, you need to have an outlook on volatility itself. While many directional traders can use options advantageously, this tends to come at a high price — volatility, it turns out, can be cheap, or very, very expensive.
The hardest part of trading is not knowing the future. It would be very nice if we knew things that were about to happen, and frankly it’d be more in line with most backtest results I’ve seen.
The issue of course is predicting volatility if one were to trade options. This is of course non-trivial, but surprisingly simpler in many ways than predicting price itself. There’s a few reasons for this:
1) Benoit Mandelbrot, among others, noticed early on that markets exhibit volatility clustering — that is, high volatility begets future high volatility, and low volatility begets similarly low volatility. This is not hypothetical; it is quite well known and understood that volatility is “sticky” in this respect (in certain markets like Bitcoin, volatility is even more sticky).
2) Despite the clustering, volatility is heavily mean reverting over time — we observe a long-term historical “average” volatility.
It’s hard to explain both rules succinctly, but several models try to account for this, since of course predicting volatility is by definition required to properly price an option.
At the time of an option transaction, the cost of an option must reflect a forward projection of volatility — the volatility the seller and buyer expect to observe by the time the option expires. However, by virtue of sensitivity to the spot price (after all, the payout at expiry is simply the difference (or 0) between spot and strike price), an option’s price explicitly reflects realized volatility during the contract’s lifetime. We can define two “types” of volatility:
1) Implied volatility - this is the volatility reflected in pricing the option, and is the future expectation of volatility over the lifetime of the option. It is highly sensitive both to market projections and supply and demand for a given option.
2) Realized volatility - this is the volatility that actually occurs over the lifetime of the option.
When we speak of vega or its derivatives (volga, vomma, vanna), we are talking about changes in the implied volatility of the option. Conversely, when we speak of gamma and its derivatives, we are largely concerned with how the option price reacts to realized volatility (changes in the stock price). Or in a more brief, practical list:
Gamma - reflects realized volatility. Sensitive to supply/demand (order flow) of underlying stock.
Vega - reflects implied volatility. Sensitive to supply/demand (order flow) of option, coupled with projections of future underlying movement.
The relationship between these two underpins one of the major tenets of options strategy - implied - realized volatility (also known as IV-RV). In general, it’s accepted that implied volatility tends to be larger than the corresponding realized volatility historically, due to the variance risk premium. In English, this means option sellers can charge more than the volatility that actually occurs, simply because they’re exposed to the full range of potential price paths (while the buyer, conversely, is not due to the non-recourse nature of options).
However, this rule of thumb certainly is not stable, and largely depends on the supply and demand of options over time. Many have commented in recent years that this “sure bet” doesn’t work anymore (or at least not surely), given the proliferation of structured options overlay strategies like the infamous JPM collar.
These regularly timed and well known equity collaring (or “yield generating” strategies) work to accomplish one of two aims:
1) Regular yield generation — this is accomplished by either selling a call while holding the underlying (buy-write) or a cash-secured put (put-write) and rolling it at a regular basis.
2) Downside protection — this is accomplished usually by selling a call or call spread while holding the underlying (mimicking a covered call strategy) to fund a put or put spread (collaring). The rationale here is to produce a “costless” hedge that protects in market volatility.
The issue with this approach of course is both the public nature of the roll (given it is usually an ETF strategy with a public prospectus) and the depressive nature it has on implied volatilities. We can observe the collapse of IV-RV in the S&P 500 and associated products through the returns of SPX versus the associated buy-write and put-write indices (graph borrowed from IPS Strategic Capital, also recommend following their head trader, Pat Hennessy):
While performance of the buy and put-write (and even iron condor variant) tended to be better than index returns pre-2008, the performance has suffered in recent years largely due to option mispricing. This is largely due to the supply/demand reflexive effect of implied volatility — while implied volatility does reflect future projections of volatility to some degree, the primary mechanism for determining implied volatility is simple supply and demand. When options are highly in demand to long (that is, non-market makers are buying the contracts), this necessarily increases the implied volatility (the spot price of the option increases). When options are highly shorted (e.g. what these structured products are causing through insensitive rolling) or not in high demand, this necessarily decreases the implied volatility (the spot price of the option decreases). This does not necessarily project to future performance in an intuitive way — while implied volatility is considered a leading indicator of future realized volatility, it is de facto separate.
When implied volatility is richer than future realized volatility, in the absence of dynamic hedging (for example, selling a covered call or cash secured put) we anticipate a simple way to determine if we made money selling (buying) an option:
1) If we sold an option on stock Z at $50 for $1.35 and at expiry Z is now $51, we made money in the trade ($1.35 - $1, or $0.35).
2) If we bought an option on stock Z at $50 for $1.35 and at expiry Z is now $55, we made money in the trade ($5 - $1.35, or $3.65).
Given that the final payoff of the option is the spot minus strike, we can intuitively observe the payoff of realized volatility — the future spot price of the underlying is essentially the beginning price (or in Black-Scholes, S_0) plus the summation of all realized price moves over the lifetime of the option. This implies the importance of IV - RV — in a regime where IV > RV, an unhedged seller will on average make a profit selling options; conversely, when IV < RV, an unhedged seller will on average lose money selling options.
Two forces are largely explanatory for the unprofitability of statically hedged systematic options overlays. The rise of these structured strategies and increasing inflows has necessarily increased the amount of contracts written (options shorted), which in aggregate led to enough imbalance to depress implied volatility on SPX options. Perhaps more impactfully, the rise of the short volatility trade and the expression of short volatility via SPX options (e.g. shorting puts) also has led to a depressive effect on SPX implied volatilities.
However, it’s still a bit naive to consider implied volatility in a vacuum — a single digit implied volatility is not necessarily “cheap”, and an implied volatility in the hundreds is not necessarily “rich”. There are in essence two ways to think of options profits (for a singular vanilla option) without a directional outlook:
1) Naively holding to expiry — in this regard, as described above, we make money when IV < RV. To remove directional bias, we can consider the ATM straddle - buying a put and a call ATM. In this construction, we have a pure play on aggregate variance — all we care about is how far at expiry the underlying has moved compared to what we paid for it.
2) Outlook on volatility — if we anticipate a change in implied volatility in our favor, even without an outlook on RV, we can still profit. Given the supply/demand nature of IV, this is identical to projecting future supply/demand for the option, which may be in certain ways simpler than projecting directionality of the underlying.
IV and RV have an intuitive relationship. It’s very rare (but of course possible) to observe large realized volatility without an associated change in IV. This is much less common than the reverse (high IV without future realized volatility), however. But in general, IV and RV tend to move in the same direction, and IV tends to — in cases of volatility shocks and jumps, such as the Gamestop squeeze in January 2021 — stay elevated substantially longer than realized volatility.
In the next post, I’m going to delve more deeply into how to use options to construct outlooks on the evolution of IV and RV, as well as second order phenomena like the behavior of spot and volatility changes (the spot-vol correlation) and the behavior of the volatility smile.
Respectfully yours,
Lily
"While performance of the buy and put-write (and even iron condor variant) tended to be better than index returns pre-2008, the performance has suffered in recent years largely due to option mispricing."
Why is it caused by option mispricing? I always found it strange that the sharpe ratio of selling ATM puts pre-2003 (and to an exent 2003-2009) plainly exceeded the sharpe ratio of being invested in the market. I would say that if anything, the current regime we are in, where IV is depressed and there is easy availability of short vol products, is more efficient than the previous ones. Even with IV behaving like it has for the last decade, it's tough to improve your sharpe ratio much by buying puts.