Greeks are Everywhere

The option greeks everyone starts with are delta and gamma. Delta is the sensitivity of the option price with respect to changes in the underlying. Gamma is the change in that delta with respect to changes in the underlying.

If you have a call option that is 25% out-of-the-money (OTM) and the stock doubles in value, you would observe the option graduating from a low delta (when the option is 25% OTM a 1% change in the stock isn’t going to affect the option much) to having a delta near 100%. Then it moves dollar for dollar with the stock.

If the option’s delta changed from approximately 0 to 100% then gamma is self-evident. The option delta (not just the option price) changed as the stock rallied. Sometimes we can even compute a delta without the help of an option model by reasoning about it from the definition of “delta”. Consider this example from Lessons From The .50 Delta Option where we establish that delta is best thought of as a hedge ratio¹:

Stock is trading for $1. It’s a biotech and tomorrow there is a ruling:

• 90% of the time the stock goes to zero
• 10% of the time the stock goes to $10

First, take note, the stock is correctly priced at $1 based on expected value (.90 x $0 + .10 x $10). So here are my questions.

What is the $5 call worth?

• Back to expected value:90% of the time the call expires worthless.10% of the time the call is worth $5

.9 x $0 + .10 x $5 = $.50

The call is worth $.50

Now, what is the delta of the $5 call?

$5 strike call =$.50

Delta = (change in option price) / (change in stock price)

• In the down case, the call goes from $.50 to zero as the stock goes from $1 to zero.
  
  Delta = $.50 / $1.00 = .50
• In the up case, the call goes from $.50 to $5 while the stock goes from $1 to $10
  
  Delta = $4.50 / $9.00 = .50

The call has a .50 delta

Using The Delta As a Hedge Ratio

Let’s suppose you sell the $5 call to a punter for $.50 and to hedge you buy 50 shares of stock. Each option contract corresponds to a 100 share deliverable.

• Down scenario P/L:
  
  Short Call P/L = $.50 x 100 = $50
  Long Stock P/L = -$1.00 x 50 = -$50Total P/L = $0
• Up scenario P/L:
  
  Short Call P/L = -$4.50 x 100 = -$450
  Long Stock P/L = $9.00 x 50 = $450Total P/L = $0

Eureka, it works! If you hedge your option position on a .50 delta your p/l in both cases is zero.

But if you recall, the probability of the $5 call finishing in the money was just 10%. It’s worth restating. In this binary example, the 400% OTM call has a 50% delta despite only having a 10% chance of finishing in the money.

The Concept of Delta Is Not Limited To Options

Futures

Futures have deltas too. If the SPX cash index increases by 1%, the SP500 futures go up 1%. They have a delta of 100%.

But let’s look closer.

The fair value of a future is given by:

Future = Seʳᵗ

where:

S = stock price

r = interest rate

t = time to expiry in years

This formula comes straight from arbitrage pricing theory. If the cash index is trading for $100 and 1-year interest rates are 5% then the future must trade for $105.13

100e^(5% * 1) = $105.13

What if it traded for $103?

• Then you buy the future, short the cash index at $100
• Earn $5.13 interest on the $100 you collect when you short the stocks in the index.
• For simplicity imagine the index doesn’t move all year. It doesn’t matter if it did move since your market risk is hedged — you are short the index in the cash market and long the index via futures.
• At expiration, your short stock position washes with the expiring future which will have decayed to par with the index or $100.
  [Warning: don’t trade this at home. I’m handwaving details. Operationally, the pricing is more intricate but conceptually it works just like this.]
  
• P/L computation: You lost $3 on your futures position (bought for $103 and sold at $100).
  
  You broke even on the cash index (shorted and bought for $100)
  
  You earned $5.13 in interest
  
  Net P/L: $2.13 of riskless profit!

You can walk through the example of selling an overpriced future and buying the cash index. The point is to recognize that the future must be priced as Seʳᵗ to ensure no arbitrage. That’s the definition of fair value.

You may have noticed that a future must have several greeks. Let’s list them:

• Theta: the future decays as time passes. If it was a 1-day future it would only incorporate a single day’s interest in its fair value. In our example, the future was $103 and decayed to $100 over the course of the year as the index was unchanged. The daily theta is exactly worth 1 day’s interest.
• Rho: The future’s fair value changes with interest rates. If the rate was 6% the future would be worth $106.18. So the future has $1.05 of sensitivity per 100 bps change in rates.
• Delta: Yes the future even has a delta with respect to the underlying! Imagine the index doubled from $100 to $200. The new future fair value assuming 5% interest rates would be $210.25.
  
  Invoking “rise over run” from middle school:
  
  delta = change in future / change in index
  delta = (210.25 – 105.13)/ (200 – 100)
  delta = 105%That holds for small moves too. If the index increases by 1%, the future increases by 1.05%
  
• Gamma: 0. There is no gamma. The delta doesn’t change as the stock moves.

Levered ETFs

Levered and inverse ETFs have both delta and gamma! My latest post dives into how we compute them.

✍️The Gamma Of Levered ETFs (8 min read)

This is an evergreen reference that includes:

• the mechanics of levered ETFs
• a simple and elegant expression for their gamma
• an explanation of the asymmetry between long and short ETFs
• insight into why shorting is especially difficult
• the application of gamma to real-world trading strategies
• a warning about levered ETFs
• an appendix that shows how to use deltas to combine related instruments

Bonds

Bonds have delta and gamma. They are called “duration” and “convexity”. The duration is the sensitivity to the bond price with respect to interest rates. Borrowing from my older post Where Does Convexity Come From?:

Consider the present value of a note with the following terms:

Face value: $1000
Coupon: 5%
Schedule: Semi-Annual
Maturity: 10 years

Suppose you buy the bond when prevailing interest rates are 5%. If interest rates go to 0, you will make a 68% return. If interest rates blow out to 10% you will only lose 32%.

It turns out then as interest rates fall, you actually make money at an increasing rate. As rates rise, you lose money at a decreasing rate. So again, your delta with respect to interest rate changes. In bond world, the equivalent of delta is duration. It’s the answer to the question “how much does my bond change in value for a 1% change in rates?”

So where does the curvature in bond payoff come from? The fact that the bond duration changes as interest rates change. This is reminiscent of how the option call delta changed as the stock price rallied.

The red line shows the bond duration when yields are 10%. But as interest rates fall we can see the bond duration increases, making the bonds even more sensitive to rates decline. The payoff curvature is a product of your position becoming increasingly sensitive to rates. Again, contrast with stocks where your position sensitivity to the price stays constant.

Corporations

Companies have all kinds of greeks. A company at the seed stage is pure optionality. Its value is pure extrinsic premium to its assets (or book value). In fact, you can think of any corporation as the premium of the zero strike call.

[A fuller discussion of the Merton model]

Oil drillers are an easy example. If a driller can pull oil out of the ground at a cost of $50 a barrel but oil is trading for $25 it has the option to not drill. The company has theta in the form of cash burn but it still has value because oil could shoot higher than $50 one day. The oil company’s profits will be highly levered to the oil price. With oil bouncing around $20-$30 the stock has a small delta, if oil is $75, the stock will have a high delta. This implies the presence of gamma since the delta is changing.

Games

One of the reasons I like boardgames is they are filled with greeks. There are underlying economic or mathematical sensitivities that are obscured by a theme. Chess has a thin veneer of a war theme stretched over its abstraction. Other games like Settlers of Catan or Bohnanza (a trading game hiding under a bean farming theme) have more pronounced stories but as with any game, when you sit down you are trying to reduce the game to its hidden abstractions and mechanics.

The objective is to use the least resources (whether those are turns/actions, physical resources, money, etc) to maximize the value of your decisions. Mapping those values to a strategy to satisfy the win conditions is similar to investing or building a successful business as an entrepreneur. You allocate constrained resources to generate the highest return, best-risk adjusted return, smallest loss…whatever your objective is.

Games have mine a variety of mechanics (awesome list here) just as there are many types of business models. Both game mechanics and business models ebb and flow in popularity. With games, it’s often just chasing the fashion of a recent hit that has captivated the nerds. With businesses, the popularity of models will oscillate (or be born) in the context of new technology or legal environments.

In both business and games, you are constructing mental accounting frameworks to understand how a dollar or point flows through the system. On the surface, Monopoly is about real estate, but un-skinned it’s a dice game with expected values that derive from probabilities of landing on certain spaces times the payoffs associated with the spaces. The highest value properties in this accounting system are the orange properties (ie Tennessee Ave) and red properties (ie Kentucky). Why? Because the jail space is a sink in an “attractor landscape” while the rents are high enough to kneecap opponents. Throw in cards like “advance to nearest utility”, “advance to St. Charles Place”, and “Illinois Ave” and the chance to land on those spaces over the course of a game more than offsets the Boardwalk haymaker even with the Boardwalk card in the deck.

In deck-building games like Dominion, you are reducing the problem to “create a high-velocity deck of synergistic combos”. Until you recognize this, the opponent who burns their single coin cards looks like a kamikaze pilot. But as the game progresses, the compounding effects of the short, efficient deck creates runaway value. You will give up before the game is over, eager to start again with X-ray vision to see through the theme and into the underlying greeks.

[If the link between games and business raises an antenna, you have to listen to Reid Hoffman explain it to Tyler Cowen!]

Wrapping Up

Option greeks are just an instance of a wider concept — sensitivity to one variable as we hold the rest constant. Being tuned to estimating greeks in business and life is a useful lens for comprehending “how does this work?”. Armed with that knowledge, you can create dashboards that measure the KPIs in whatever you care about, reason about multi-order effects, and serve the ultimate purpose — make better decisions.