The Greeks: Measuring Risk

Once a trader understands the basics of options trading, they should look more closely at the factors which dictate the movement of an option’s price. The Greeks are measurements of the different facets of risk that are required to quantify when pricing an option. They are used as tools for traders to make well-informed decisions about what and when to trade. They also function as indicators for how different risk variants including token price, time, interest rates and volatility affect the price of an option.

The primary Greeks are the most important to understand as a trader, these are Delta, Gamma, Vega, Theta. There are also a number of minor greeks which more advanced traders may utilize to model the risk profile of certain options positions which will be addressed later in this guide.

Chapter 2: The Greeks: Measuring Risk will cover

• Volatility: IV vs RV
• Delta
• Gamma
• Vega
• Theta
• The minor greeks
• Using the greeks to your advantage
  

The importance of volatility in options trading

Volatility in terms of the options market refers to the fluctuations in the price of an underlying asset. Before the Greeks, there is another important concept that any options trader must understand. Volatility plays a significant role in options pricing and as an extension, options trading. Generally, the more volatile an asset is, the more risky it is considered to be as an investment. There are two types of volatility to be aware of; implied volatility (IV) and historical/realized volatility (RV).

Implied volatility

Specific to options markets, implied volatility refers to the prediction made by market participants about the fluctuations in the price of an asset in the future. IV gives us a glimpse into the sentiment and expectations of the broader market about how much a tokens price will move as it trades. It is also a key parameter in an options pricing model, such as the Black Scholes pricing model, to give the market price of the option.

Implied volatility is a dynamic figure that changes with activity in the options marketplace and is expressed as a percentage of the token price, indicating a one standard deviation move over the course of a year. Usually, as IV increases so too does the options price, assuming other elements stay the same. Therefore, if IV increases after a trade is made it is usually good for the option buyer but bad for the seller.

Idea: An option with an implied volatility of 50% is saying that the underlying asset is expected to trade within a 50% range (high to low) within the next year.

Realized (historical) volatility

Unlike IV which is forward looking, realized volatility is a measure of how much the price changes over a given period of time. Sometimes referred to as historical volatility, RV provides a retroactive view on the previous tendency of an asset to move away from its mean over time. When someone says, “crypto is a volatile asset class” or “bonds are stable and safe yielding assets”, they are referring to historical volatility.

As we move further in this Lyra Learn blog series, we will explore strategies which utilize the discrepancy between realized historic volatility and the markets perception of future price movements in IV to get edge and make strategic trades.

The Greeks

There are many inputs into the price of an option: the price of the asset, its implied volatility, the time to expiration, amongst other things. But how does the price of an option change as these variables change? If volatility goes up 2%, how much does my call go up? If the asset increases by $10, how much does my put go down? The Greeks offer an answer to these questions by measuring the sensitivity of an option’s price to changes in the input parameters. There are many different Greeks, but we will focus on just five of them. As previously mentioned we will cover the five main greeks in detail: Delta, Vega, Theta, Gamma, Rho.

Delta

In other words, how much does the price of the option go up or down as the price of the token goes up and down? Delta measures how much the price of the option will change as a result of a $1 change in the price of the asset. The Delta of an option varies over the life of that option, depending on the underlying token price and the amount of time left until expiration. Another way to think about delta is a rough estimate of the probability of the asset finishing in the money at expiration.

Call options have a positive relationship with delta because their value increases as the underlying token increases. An option premium is expected to increase as the price of the underlying token increases. If a call has a 0.50 (or ‘50' delta) and the asset moves up $1.00, the option will increase $0.50 in value.

Vega

It refers to the change in the option’s price for a 1% change in implied volatility. Thus, whenever volatility goes up, the price of the option goes up and when volatility drops, the price of the option will also fall. As a rule of thumb, longer dated, at the money options have the highest vega, thus their premiums are the most sensitive to volatility changes. Nearest to expiry options are impacted the most by shifts in IV since they have less time value in their extrinsic value.

For example if Alice buys a call with an IV of 40% and $2.00 of vega. If the IV increases to 41%, Alice’s option value will increase by $2.00. Vega risk is extremely important to assess and understand when trading options, the next section illustrates why.

I bought a call and the asset price increased, why am I losing money?

More often than not, the culprit is vega. A common misconception is that if you buy a call and the price of the asset goes up, you will make money (vice versa with a put). There are often times where traders buy a call for an asset with a relatively high IV, and are taken aback when they lose money if the asset price grinds up slowly. For example, if a trader buys an option with 100% IV (implying ~5% per move per day) and the asset moves up 0.50% for 3 days in a row, they could end up losing money if the market decides that the asset's IV should really be 80%. This would result in a reduction in the option's value of approximately 20 * optionVega. If this drop is larger than the effect of the spot price increase, the trader will lose money.

Theta

Specifically, it describes how much the value of an option changes each day as expiration nears. Options tend to lose value as the expiration date nears, so Theta is always a negative number for net long options and positive on net short options. As an option gets closer to maturity, Theta accelerates due to gamma becoming large. In other words, the option convexity (its bread and butter) increases as maturity approaches, and more convexity leads to more "theta rent" needs to be paid. In fact, in relative terms, the ratio of Gamma / Theta is constant through time, so each unit of $ gamma "costs" the same theta no matter what maturity is. Near term ATM options pay the most theta and a higher IV will increase the Theta of an option.

The theta is typically quoted as the dollar amount which an option will lose if time is fast-forwarded one day. For example, if the theta of an option is $5.00, the option will be worth $5.00 less in one day's time, assuming all else equal.

Gamma

Gamma is the Greek which clearly identifies the power of options. Other products like leveraged perpetuals and futures can offer enhanced first order exposure to an asset, but it is the second derivative properties of options which give them their non-linearity or convexity. Options with gamma increase in value at an increasing rate if the underlying moves in the desired direction.

To illustrate, let’s look at an example.

You purchase a 2000 strike ETH call option has a 50 delta (0.50) with the spot price of the underlying equal to $2000.

Lets say the spot price increases by $1 and the delta increases from 0.5 to 0.52.

The option will actually increase in value by $0.51. Why? Because the ‘average delta’ of the option over a $1 move is between 0.5 and 0.52, which is 0.51. The option price will increase by the average delta, not the “left point” delta. A more mathematically correct explanation is that dC = delta * dS + 0.5 * gamma * dS^2, i.e. change in option price not only comes from delta, but also from half gamma times the squared move.

Now, why is gamma so important? Suppose you delta hedge this option, so you short 0.5 ETH and make yourself delta neutral. Now, because of gamma, no matter if ETH goes up or down, you will earn a profit. Why? If ETH goes up $1, your short loses $0.5 exactly (since it is linear), but the option wins $0.51 as explained above. If ETH goes down, short wins $0.5 and option loses $0.49 (same reason as before, delta goes to 0.48, average delta is 0.49). So in both up-move and down-move scenarios, the option holder earns money! This is exactly due to gamma, and this is precisely why options are valuable. This "free roll" that holders have has a cost - this cost is theta.

Minor greeks

While Delta, Vega, Theta and Gamma are the most commonly mentioned Greeks, there are more variants to options risk. There are a number of second and third order Greeks that are less common - known as minor greeks. While not useful to the average trader, they are still good concepts to learn for the serious trader.

• Rho: The rate at which the price of an option changes relative to the risk-free interest rate
• Charm: Delta decay over time
• Lambda: The sensitivity of the price of a token is to a 1% change in IV
• Epsilon: How sensitive the value of an option is to a change in the yield of an underlying token
• Vomma: Measures how sensitive Vega is to a change in volatility
• Vera: How sensitive Rho is to volatility
• Speed: How sensitive Gamma is to a change in the underlying token
• Zomma: How sensitive Gamma is to changes in volatility
• Color: How sensitive Gamma is to time passing
• Ultima: How sensitive Vomma is changes in volatility

Using the Greeks to trade options

As we previously discussed, the Greeks are an important facet in understanding the different elements of risk in trading options. They also provide a valuable way to quantify the risk of your options portfolio as a trader as well as make informed decisions about the value of certain options trades. The Greeks help to take the guesswork out of options trading but it can become a very complicated system to understand, even for experienced traders. The Greeks are constantly fluctuating and do not work in isolation, meaning that the movement in one Greek will likely affect the other. To manage this complexity, a trader can rely on a framework like the one provided below.

A plus means that an increase in this factor will work in your favor, whereby a minus means that this factor is unfavorable to your position. For example, Delta is positive for long calls meaning that the position will gain value for an increase in the tokens price. In other words, a long call is a bullish strategy or a short put has a positive Theta meaning that time decay works in favor of this position.

However, it is important to remember that the Greeks are calculated using theoretical models that operate on assumptions, and are therefore only as accurate as the models used to quantify them. While greeks can be used to estimate the future value of option prices, there is no guarantee that they’ll hold true.

About Lyra

Lyra is an open protocol for trading options built on Ethereum. Lyra allows traders to buy and sell options that are accurately priced with the first market-based, skew adjusted pricing model. Lyra also quantifies the risks incurred by liquidity providers and actively hedges them, encouraging more liquidity to enter the protocol.

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