Equity Options Primer: Facts, Fictions

Investment Analysis Clubs Macro Economic Trends and Risks
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@WendyBG

In this thread (please see next post) I give a fact-based, logical sequence of some of the main features of US listed equity options - the market that US retail investors trade in.

Options are confusing because

  • the instruments themselves are complicated,
  • the math and theory that explains how they are priced using a riskless replicating portfolio gets complicated very fast, and
  • the markets, because they are run at the institutional level by brokers, exchanges, and market makers, have mechanics and market structures that are very opaque to retail investors

Because of the above, even retail investors with substantial options trading experience can have an incomplete understanding of options, and understandably so.

I hope at least someone besides me finds the information helpful.

I find this fact to be the most interesting:
Replicating Portfolios Are Financing Transactions

I welcome constructive, evidence-based comments and discussion that add to, clarify or correct any aspect.

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Trading in US listed equity options is a large and important financial market for institutional and retail investors.

Below are some facts and fictions about the trading and markets of these securities.

TL,DR

  • Market makers (MMs) fill a very large portion of investor US listed equity options trades (instead of investors trading with each other) and therefore their role is central to the economics of options prices and options markets
  • MMs hedge investor trades with a portfolio (which includes underlying stock) that has a gain/loss independent of the investor equity gain/loss, a property called delta-neutral
  • The MM hedging portfolio is called a replicating portfolio, it is a financing transaction that transmits the return/risk of the underlying stock to the investor’s option position
  • The economics of an individual trade between a MM (having a replicating portfolio) and an option investor is in general not zero sum and the aggregate market of such trades is in general not zero sum

While a post on this board cannot provide an exhaustive explanation of options markets, it can provide a basic foundation of understanding:

  • how options are priced,
  • how option pricing depends on the underlying asset,
  • how option pricing enables financial firms (MMs) to have a viable business model in options trading (and hence enable option trading by investors), and
  • how the above determines the economics of individual trades, financial firms’ trading businesses, and entire markets

Fact:

Option Definition

Call/put options grant the buyer the right to buy/sell stock (or ETF) - the underlying - at a set price (strike), before a certain date (expiry, American style).

Fact:

Meaning of Derivative

Options derive their value from the value of the underlying stock, hence the name derivative.

Fact:

Riskless Replicating Portfolio

The theory of asset pricing provides a method to replicate the economics of an option with a portfolio that is riskless (value doesn’t change) with respect to changes in the price of the underlying (all else equal). The change in portfolio value with respect to changes in value of a reference security (such as the underlying stock) is called delta (delta is a mathematical derivative). A replicating portfolio has delta near zero, is called delta-neutral and therefore has little directional equity risk. The theory of asset pricing explains well known options models like Black-Scholes, which is a canonical model in options pricing and trading.

Fact:

Replicating Portfolio Business

The replicating portfolio provides a well-defined and established trading strategy that serves as a foundation for financial firms, including market makers (MMs), to run a sustainably profitable business while minimizing directional equity risk. The business of the MM, in arrangements with options exchanges, is to provide bid-ask quotes and take the opposite position of investors’ options trades: their business of being available to take trades is called providing liquidity.

Fact:

Replicating Portfolios Are Financing Transactions

The replicating portfolio for a long call/put is to borrow dollars/stock and buy/sell the number of shares that makes the portfolio delta-neutral while also selling the call/put to the investor. The borrowing of dollars/stock is the financing portion of the portfolio and the long/short stock position hedges the delta of the short call/put trade facing the investor. In effect, this is a financing transaction. The MM is providing the capital, subject to the MM’s cost of funds, to finance the long/short shares that produce the equity return/risk of the investor-facing call/put. The MM is financing the transfer of equity return/risk - return/risk that derives from the underlying’s stock market - to the investor. It’s only natural that investment banks have a role in this business: banks are in the financing business!!! (The same kind of financing transaction applies to other derivatives, such as equity futures and swaps. These are also trades that finance the transfer of return/risk from the underlying asset to a derivative investor.) Options are said to be leveraged trades - this financing mechanism is how the leverage originates. Very interesting!

Fact:

The Market Maker’s Business and Economics are Explicitly and Necessarily Linked to the Market of the Underlying

The MM’s replicating portfolio is long/short the underlying stock and transfers equity return/risk from the underlying stock to the investor’s call/put position. Thus the MMs’ trading books and the options market as a whole, for each underlying, are directly linked to the market and economics of the underlying stock.

Fact:

Replicating Portfolio Rate of Return

Using a replicating portfolio, a MM can take the opposite side of any investor option trade (buy/sell any option the investor wants to sell/buy in the market) and then also go long the replicating portfolio of the investor’s option position. In financial mathematics, because the trade is riskless, a standard representation is that the replicating portfolio earns the risk-free rate (eg, the Treasury rate). In practice, MMs finance their business at some cost of funds specific to their enterprise and also endeavor to earn a profit above their cost of funds. So, as a reasonable foundation, we can expect the MM to earn its cost of funds plus a spread on its capital used to finance its market making. In practice, there are more risk factors (interest rate risk, dividend risk, etc) that affect returns, but the rate of return that equals cost of funds plus a spread provides the foundation.

Fact:

Replicating Portfolio Returns, For a Given Trade, Do Not Depend on Investor Returns

From above, we know, by using the replicating portfolio when taking investor trades, a MM earns its cost of funds plus a spread. The investor could be trading any number of strategies: a hedge against its portfolio, a speculative strategy, a strategy that mimics a long position in a major index like S&P 500, an option in combination with some other position, etc. In any particular trade with a MM, the investor could have any magnitude gain or loss. However, regardless of the investor’s return, the MM receives the return on the replicating portfolio (cost of funds plus a spread). Interestingly, the MM’s return (from the replicating portfolio) is independent of the investor’s gain/loss.

Fiction:

An Investor’s Gain/Loss is the Market Maker’s Loss/Gain (Each Trade is Zero Sum)

One will see written many places, that options trades are zero sum. This would mean that the MM’s gain/loss is the equal and opposite of the investor’s gain/loss. But this is false as the preceding fact shows. Individual options trades (investor trade with MM, as defined above) are not, in general, zero sum.

Speculation:

Further, because options are explicitly linked to their underlying stock markets and options transfer equity return/risk from the stock market to the option trader (via the replicating portfolio), underlying stock markets might need to be zero sum for options markets to be zero sum (if one wants to argue that options trades and markets are zero sum).

Fact:

Market Maker’s Are the Primary Liquidity Providers in Options Markets

Option market making is a big business and MMs trade a very large volume: the mandate and business model of MMs is to trade as much volume as possible. Also, aspects of the market structure such MM’s obligations to the exchanges to take trades, payment for order flow and the mechanics of how orders are brought to the exchange all serve to increase the MMs’ trading volume.

Fiction:

Investors Often Trade with Each Other, Not Market Makers

Because of the dominance of market makers, explained just above, investors’ opening trades are vastly more likely to be filled by market makers than other investors.

Fact:

For a Single Underlying, the Net Aggregate Book Is Not Typically Perfectly Balanced

Even for a single underlying, options trades have a very large number of specific instruments defined by call/put, strike, expiry and quantity. The chance that all investor trades of calls and puts exactly balance out in delta terms to have delta equal to zero for a meaningful amount of time is very small relative to the alternative case of investor positions netting to non-zero delta.

Fiction:

(This one is a doozy!)

The Net Total of All Investor Trades Perfectly Offset Each Other in Delta Terms, Producing a Perfectly Balanced Book of Open Investor Trades

Because of the prior fact, this fiction is obvious.

Fact:

The Aggregate Economic Outcome of an Options Market is Not Zero Sum

As note above, there will typically be an unbalanced book of open investor trades and MMs will have taken a very substantial volume of these open trades. Because, as explained above, individual trades with MMs will not be zero sum, the aggregate economic outcome of these trades will not be zero sum.

Fiction:

The Aggregate Economic Outcome of an Options Market is Zero Sum

Fiction because of the preceding fact.

References

Retail Options Trading with Wholesalers

Wharton Payment for Order Flow

CBOE Retail Option Flow

Risk Citadel MM of Year

SEC Equity Market Structure

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@mostlylong, I wish there was a super-rec because your awesome post is worth 10 of most ordinary rec’d posts!

I have many questions but this one is first:
You wrote:
" The Aggregate Economic Outcome of an Options Market is Not Zero Sum

As note above, there will typically be an unbalanced book of open investor trades and MMs will have taken a very substantial volume of these open trades. Because, as explained above, individual trades with MMs will not be zero sum, the aggregate economic outcome of these trades will not be zero sum."

Doesn’t this leave the MMs open to risk? Or is their risk hedged by the replicating portfolio?
Wendy

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Dear ML,

The MM seems to have its hands deeply into all aspects. Including creating a market without other traders to some degree. Am I reading that right?

Does that hasten time decay? Looks like it would work against the longs.

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Short answer: unbalanced delta is hedged by MM with replicating portfolio.

Long answer:
For a given underlying stock ticker, if the aggregate investor trades perfectly balance in terms of equity delta (equal amount of long stock and short stock summed over all of the investor option trades), then the MM would be in a great place.

There would be no extra delta, either long or short, for the MM to hedge (the investor trades are perfectly hedging each other).

And the MM would not need to use any balance sheet (capital) to finance hedges.

This is like infinite return on company capital. Of course there are still operational expenses to run the business (rent, salary, software, data feeds, etc).

But, of course, it would be an extreme exception for investor (option) longs to perfectly balance investor (option) shorts (long, short in terms of the underlying stock, via the options).

So there will be some extra, unbalanced delta, either net long or net short the underlying, when totaled over all investor option positions.

This extra net delta can be hedged using the riskless replicating portfolio from the theory of asset pricing (formulas like Black-Scholes).

The MM portfolio will be, for example, for investors that are net long the underlying in aggregate:
[long underlying stock] + [short investor-facing options]

The long stock position could be anything that provides long stock exposure (stock, futures, swaps, options) - whatever is most profitable for the MM given its expertise and business strategy. This position will be adjusted as the trading book changes (called delta hedging) to keep the portfolio delta within the firm’s risk limits.

The short investor-facing options will be just the other side of all such trades.

For example, investor sells a call, MM buys that call in the opening trade. Investor buys a put, MM sells that put. Etc.

The delta hedging creates the riskless (in terms of directional equity risk) portfolio so the MM portfolio value doesn’t change much as the underlying stock fluctuates. And the MM earns its cost of funds plus a spread.

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I think so. MM obligation to the exchange is to provide quotes and then take trades. They want trades, because that is their business.

I believe time decay would mostly depend on moneyness, underlying volatility and maybe interest rates. Probably web search can get you a thorough answer on theta details.

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I apologise for throwing cold water on this thread by also asking a question:

Do you need a college degree in biology to make babies?

No, what you need is to make a deal with the opposite sex. This is not to dismiss Market Makers, they are a necessary feature of option trading. What @mostlylong posted is probably all correct, I have no way of verifying it or interest because it is not part of my trading process.

Trades only happen when there is a match in the buy/sell order books. A penny difference prevents trading. The Market Makers’ job is to promote trades which they do when bid and ask prices are close enough. @mostlylong posted all about it. How long before AI replaces Market Makers?

Option trading should start by defining your portfolio’s objectives. The simplest one could be creating some income. The methods Nassim Nicholas Taleb talked about were wild gambling that leads to Midas riches and/or bankruptcy. He was not very forthcoming and it was only by reading a New Yorker interview with Malcolm Gladwell that I discovered the system (which he stopped using).

KISS?

The Captain

Option pricing is not determined by the Black–Scholes model but by supply and demand. Using the buy-sell midpoint you can calculate the “implied” volatility. In other words, volatility does not set the market price! There is more to pricing than volatility.

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Not really - that should not be an either/or statement.

Thank you for raising this clarification.

Black-Scholes and market-driven (supply-demand) pricing are the same.

Black-Scholes IS market-driven (supply-demand) pricing.

How does Black-Scholes (based on theory of asset pricing) work?

Black-Scholes makes direct use of market inputs, which are themselves determined by supply-demand.

Black-Scholes takes market inputs and calculates an option price that is economically consistent (replicating portfolio) with the markets for those inputs (and because these are “markets”, there is plenty of supply-demand happening).

This is the economic basis for options prices: prices that are economically consistent with related asset markets, like the underlying stock.

Here’s the related fact, from upthread:

What are the market inputs?

Stock price, interest rate, volatility, dividends, etc. All supply-demand determined, for sure.

So Black-Scholes calculates an option price that is ultimately based on supply-demand in related asset markets.

A retail trader may or may not pay much attention to Black-Scholes, depending on their return-risk preferences.

A MM (market maker) may have a very custom and firm-specific pricing model, based on many market inputs, that reflects its return-risk preferences.

A MM can simply be in the lending business by running a replicating portfolio and financing investor trades. The MM’s gain/loss will be independent of the gain/loss of any retail investor counterparty.

One entity can have low return-risk. The other can have higher return-risk.

Both can have a gain, both for a single trade and on average in the long run over many trades.

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I was not avoiding this thread, I wrote a reply but was not too happy with it so I let it ripen a bit. The new reply:

The Black-Scholes model gets a lot of praise and even a Nobel Prize but when I want to trade an option I’m faced with an auction.

When I started learning about the Black-Scholes pricing model it made a lot of sense but back then i knew very little about options. For the past decade or so I’ve traded a lot of options and the Black-Scholes model never entered into it. If I had no use for the model, who did and why? I have no idea. What are the practical applications of the model in the real world?

With the huge market downdraft this quartet I’m in need of rolling up and forward a lot of calls but I didn’t have the tools to help me pick the available trades. With TSLA I was able not only to increase the strike price but to be paid to do it. No such luck with other stocks. I’m half way to adding such a tool to my Call Selector. Last night I pretty much finished the data input side. Now comes the real part, doing the numbers. My trades use option chains with 500 to 2,500 options. Eyeballing is not practical, the computer needs to do the heavy lifting.

Back to Black-Scholes, the theoretical value of the options plays no role in my trading, hard numbers do, Bid, Ask, Strike, Expiration, CAGR , $S$ per day, Discount. These are the numbers that put food on the table (or starve me).

My previous reply below:

o o o o o o o o o o o o o o o o o o o o o .

Do you have an idea of how close that auction price is to the Black-Scholes model calculation? Why calculate an “implied” volatility when the real volatility can be calculated. If they are not identical there is either something wrong with the Black-Scholes model or supply and demand ignores the model. I think the best way to search for the answer is to study the Black-Scholes formula.

Investopedia says:

How the Black-Scholes Model Works

Black-Scholes posits that instruments such as stock shares or futures contracts will have a lognormal distribution of prices following a random walk with constant drift and volatility.

The equation uses this assumption and factors in other important variables to derive the price of a European-style call option.

The Black-Scholes equation requires six variables:

  1. Volatility
  2. The price of the underlying asset
  3. The strike price of the option
  4. The time until the expiration of the option
  5. The risk-free interest rate
  6. The type of option (call or put)

It’s theoretically possible for options sellers to set rational prices with these variables for the options that they’re selling.

The model predicts that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility.

It incorporates the constant price variation of the stock, the time value of money, the option’s strike price, and the time to the option’s expiry when it’s applied to a stock option.

But when I go to trade all I have is an auction, take it or leave it. Is the price in line with my portfolio’s objectives?

Black-Scholes Assumptions

The Black-Scholes model makes certain assumptions:

  • No dividends are paid out during the life of the option.
  • Markets are random because market movements can’t be predicted.
  • There are no transaction costs in buying the option.
  • The risk-free rate and volatility of the underlying asset are known and constant.
  • The returns of the underlying asset are normally distributed.
  • The option is European and can only be exercised at expiration.
  1. The normal distribution is debatable. It does not take into account falling or rising markets
  2. American options can exercised at any time until expiration. As a seller, one can close an option position early to increase the yield
  3. Dividends tend to exercise the option
  4. When an option is exercised before expiration the counter-party is picked at random.

Read All About It!

o o o o o o o o o o o o o o o o o o o o o .

The earlier reply is about theory, the new one is about practice. Do either make sense?

The Captain

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Just to briefly acknowledge your reply and focus on main points.

Nothing you wrote disagrees with what I wrote.

Your main point seems to be that you don’t explicitly use Black-Scholes for trading.

I would expect most retail investors would not, at least not directly and explicitly (but no way to know, and brokers provide many tools these days).

That’s why I wrote:

But none of that changes what I wrote about the economics of options pricing and markets, all explained earlier.

Also unchanged, my response (below), to your statement that prices come from supply and demand (which is exactly what asset pricing models do):

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