Liquidity Provision Risk Premium and the Cost of Leverage

How I look at option valuation and why I have a problem with Black Scholes.

Mar 25, 2023

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Have you had enough of macro? I certainly have. Let’s do something else for a change. :)

This article is short and sweet, but it is also extremely information dense. I recommend reading it when you have a peaceful moment on your computer (as opposed to reading it on the phone) and it will help to open the underlying Excel model which I have attached for premium subscribers. For free subscribers, I have added screenshots throughout to facilitate following my thoughts.

I have been trading options for years. However, my approach is very different from what you may have seen from others. I do not care at all about chart analysis. I do not care about gamma squeezes. I do not care about 0DTE. I only care about my appetite for volatility and leverage. My plan is to slowly introduce my approach to you over time with articles building on top of one another. This is the first one. Subscribe for more content like this below.

Black-Scholes (BS) is an option pricing model developed by Fischer Black and Myron Scholes in the 1960s/70s. By providing a sound theoretical foundation for derivatives valuation, it was a game changer for option trading and both received the Nobel price in 1997. It is only a model that is supposed to represent reality, but its impact has been so powerful that has become reality for most people. The implied volatility that you occasionally see in the option chain is calculated based on BS for example.

The BS formula assumes that asset returns are normally distributed random variables, i.e. they can be entirely explained by two parameters: the drift μ and the volatility σ.

BS assumes that the drift μ of the underlying asset is equal to the risk-free rate RFR. This can be illustrated through a simple comparison of the results of BS and a simulation exercise that I have illustrated below. BS (orange highlight) delivers pretty much the same result as the simulation (yellow).

In this article, I do not want to venture into the mathematical reasoning why BS is set up the way it is (risk-neutral pricing etc.). I want to look at the big picture to understand how representative this model really is. Everyone knows that the drift of the stock market is NOT equal to the RFR after all. In fact, the S&P as a representation of the stock market has exhibited a remarkably consistent total return drift of 10-12% over the past century.

Deduct 3% dividend yield from that suggests that the relevant drift for the underlying asset price should be 7-9%, no? How can we ignore that and believe we can value options properly?

By not incorporating the market drift/cost of equity as an input parameter in the option valuation, BS misses a key input factor. It feels like trying to value a company without making an assumption on the revenues.

Okay, let’s do incorporate the actual market drift then. What if I replace the drift of 4% by 8% in the simulation?

Look at the cells highlighted in red at the bottom right. It now appears that the call should actually be worth more than suggested by the BS valuation, while it appears the put is worth less. How does that make sense?

Does that mean that markets structurally overvalue puts and undervalue calls? Can we just short the former and long the latter?

It’s very enticing to think about it this way, but I don’t think it is the right way to think about it. We should not think along the lines of “markets price options based on BS”. Markets do not price something based on a model. They do it based on market forces. The model is just our effort to make sense of them.

The deltas highlighted in red in the screenshot above do not represent a mispricing. They just need interpretation. This is is about leverage and what I call the liquidity provision risk premium.

First, before I get into the explanation what I mean by that, I need to formalize my problem resulting from implementing a more reasonable assumption for market drift: If the expected put payout is $4.71, why would anyone pay $6.01 for that today? And if the expected call payout is $12.54, why would anyone sell that for $9.94 today?

Let’s look at both instruments separately:

Put

In my opinion, the best way to think about the difference between expected put payout and the put premium (quoted in consistency with BS) is to view it as an insurance premium to be protected against share declines. From a micro perspective, just like a plain insurance carrier, the put seller needs to have a positive expected return to be compensated for the skewed payout profile he agrees to. From a macro perspective, since he stands ready to BTFD, he provides liquidity to the market, which is generally compensated by the market with a positive expected return. I call this positive expected return the liquidity provision risk premium LPRP.

What is the expected return of the LPRP? The best way to assess this is to take the perspective of the put seller. What is his expected return? Return is gain divided by capital. His total committed capital is $100 (the strike at which he agrees to buy the underlying asset) minus $6.01 (the premium he gets), so it is $93.99. His expected return is the difference between the put premium and expected put payout, which is $1.30. So his expected return is $1.30 / $93.99 = ~1.4% (highlighted in blue below):

Initially, I was tempted calling this the volatility risk premium because the put seller is compensated for shorting volatility. However, the volatility risk premium is already used in option trading to describe the phenomenon that implied volatility tends to be higher than realized volatility, which refers to another topic. In fact, I believe that both the liquidity provision risk premium and the volatility risk premium can be aggregated together into what I call the expected put selling premium:

Call

As I have indicated in the article below, buying calls is equivalent to buying stocks with leverage. The market maker becomes your lender and stock custodian (because they delta hedge your trade by buying shares) and theta is the interest you pay to them.

Fallacy Alarm

The Great Pandemic Era Option Bubble

This article explains the last 3y from an angle you might not have heard before and the implications are profound in my opinion. While it is quite technical to follow, I encourage you to work your way through even if option trading might not be your forte or particular area of interest. I will try to convey my message as clear and digestible as possible…

2 years ago · 11 likes · 2 comments · Fallacy Alarm

So, can option prices tell us the amount of leverage hidden in the call contract? It turns out they can. The call buyer pays $9.94 and his expected payout a year later is $12.54. This means that his expected return is 26%. With 8% market drift, this translates into 3.3x leverage (again highlighted in blue below):

The reason for this leverage is that he only has to use a fraction of the total cost of the shares to fully participate in their performance as long as it is positive. On the other hand, if the shares have a negative performance, he will be wiped out. The price of the call (determined by market forces, not by BS) is determined by the markets’ price of this leverage, i.e. the implied interest.

The role of interest rates

Buying a call is like buying shares on credit without explicitly paying interest. The higher the interest rate, the higher the price of the call must be to ensure there is not arbitrage vs. outright buying the stock on credit. Buying a put is like selling a stock short, but you don’t get to earn interest on the cash proceeds. The higher the interest rate, the lower the price of the put must be to ensure there is no arbitrage vs. outright shorting the stock.

It therefore appears obvious that the expected put seller return and the implied leverage in the call are a function of interest rates. Taking the 1Y Treasury Yield as the representation for the RFR, the RFR has risen by 4% over the past year.

In the model excerpt below, I have modified RFR from 4% to 0%. As a result, the expected put seller return jumps from 1.4% to 3.7% and the implied call leverage jumps from 3.3x to 7.0x.

Increasing interest rates makes both put selling and call buying less attractive. Both are leveraged long bets on the underlying asset. Remember how I said in the Great Pandemic Era Option Bubble article that it first pumped stocks then crashed stocks and that it is over now? Falling and surging rates played a crucial role in this process.

The best way to think about option prices is to think about the liquidity provision risk premium and the cost of leverage.
The price of a put is not the present value of its expected payout discounted at the RFR. Puts are structurally priced higher than their expected payout. This is however not mispricing. It is a feature baked into efficient markets. It causes a positive expected return for put sellers to compensate them for providing liquidity. And put buyers pay for taking liquidity.
Calls are priced structurally lower vs. their expected payout. Their expected return is based on the price of leverage. The price of leverage can be low/attractive or high/unattractive.
The liquidity provision risk premium and the cost of leverage are functions of expected market volatility and the interest rate environment.
Markets reward the provision of liquidity with a positive expected return. Buy and hold also implicitly provides liquidity by not selling into weakness. Put selling explicitly provides liquidity into weakness. Providing liquidity is the only way to systematically earn a profit in the stock market.

I would love to hear feedback from you on this article because I have so far not been able to find anyone who has framed option valuation the way I do.