Image from the Guardian article: Mathematical Equation that Caused the Banks to Crash.

**Leverage, Options, and Derivative-Fueled Crashes by Mr. Practical**

Option contracts have been around since antiquity, ever since someone said, “if you do this, then I will do that”. Clever people who knew how to intuitively price options (agreements/bets/contracts), making the “this” worth more than the “that”, got ahead, accumulated wealth. With the advent of liquid markets came the ability to “hedge” options, and with that, the Black-Scholes model. Pricing options, even after the BS model was derived, is obtuse and so they are consistently mis-valued.

Options began trading as securities in the late 1970’s and became part of a group of securities called derivatives: securities or contracts that derive their value from an underlying asset. All derivatives connote leverage in varying degrees with options being the most levered. But unlike most other more vanilla derivatives, options when employed properly serve a real purpose for investors/traders other than leverage: they segment risk into upside and downside and facilitate delineation of return profiles.

But the derivatives market has grown immensely and now dwarfs the cash markets which it overlays. Depending how you measure it, the notional value represented by the derivative markets globally is around a quadrillion dollars, a thousand trillions. Now the tail wags the dog.

What do I mean by that? First, leverage introduces extra risk in the market because asset prices are effectively purchased with debt (i.e., for an up-front payment less than the full value of the underlying, with the remainder to be paid later), driving up prices higher than they normally would be. However, the real problem is that the asset prices act as collateral for the debt that bought them, so when those asset prices fall, the collateral might have to be sold by the lender, driving prices down even further. Second, options introduce a special form of leverage called gamma, and it helped exacerbate the 1987 crash, the early 2000 crashes, and especially the crash of 2008.

An option like a “call” has a probability of being in the money at expiration and having value at termination or out of the money and being worthless. This probability changes as the underlying asset price changes: if the asset price goes down, it has less probability of being in the money and having value and as it rises the probability and value rises. This probability of being in the money is called the “delta”, and the measure of how much it changes over time and price is called “gamma”.

The problem begins because there is a mismatch in the amount of capital supporting sellers versus buyers of options. When you buy an option, you must pay the premium up front. Once the investor does that their maximum loss is set, they can’t lose more than they paid no matter what the market does. The capital is already invested and if the maximum loss occurs it is already incorporated into the volatility of the market. The seller however collects the premium and must put up margin to collateralize the risk. The maximum loss is at least five times this amount (at a 20% margin requirement, for instance), so if half the sellers are positioned with too much risk you can see that the sellers of options will be a problem if the cash asset begins to move against them.

As the cash asset moves against the seller, they realize the probability of loss is rising. Since a good portion of the sellers are likely to have not properly weighed the chance of this loss, and therefore can’t afford it, they must act (buy selling some of the asset they can’t afford). The rate of change of that probability is important: if the rate of change is high the seller will have to act more rapidly than if it was low, and that action will exacerbate the adverse move already occurring. The buyer of the option is making money and taking their time in taking profits.

Thus, if the gamma of an option is high, the embedded leverage or risk is high. So, what determines the level of gamma of an option? It turns out it is the price of the option itself that determines the gamma. This is intuitive as well.

Let’s look at two prices of a put. Say an at the money put has a price of $1. If the market drops $1, the seller of the put is already at break-even: he sold the put at $1 and it is not $1 in the money. Now if that same put is instead trading at $2, then the seller’s break-even is now down $2 in the market. The seller of the second option doesn’t have to hedge their risk when the seller of the first option already has to: the probability of being in the money for the second option is now less than the first so it follows that the original rate of change of the delta is less for the second option than the first. The gamma is less for the more expensive option than the cheap one.

Therefore, the leverage in the market is greater when option prices are cheap than when they are expensive. The options market exacerbates market moves more when options are cheap. Corrections usually start when options are cheap, when sellers of options have driven prices down and have taken much more risk than buyers, and in reality, much more risk than they are able to bear.

Mr. Practical

In real life, "Mr. Practical" is a volatility trader who wishes to remain anonymous.

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Mike "Mish" Shedlock