Greek letters are very important observation indicators in option theory and trading. As you get deeper into the actual practice, you will feel their magical effects more and more. Different Greek letters represent different factors that affect option prices. The main function of Greek letters in options is to measure and control risks. For practical investors and traders, mathematical operations themselves are not the most important, but a deeper understanding and application and extension to various option investment strategies are the most important.
1. Delta
For easy memorization, we nicknamed her "Direction Fairy". Below I will analyze Delta from the perspective of peeling an onion:
①Definition
Delta measures the impact of changes in the underlying price (taking crypto as an example: a 1USD change in BTC or ETH) on the option price, that is, the option price changes accordingly when the underlying price changes by one unit.
② Position
How many BTC positions are equivalent to the current option contracts held? For example, holding 1 option with a delta of 0.2 is equivalent to 1 positive BTC.
③Direction
Represents direction; the Delta of a single-leg option call option is positive, and the Delta of a put option is negative; a combination of + represents a bullish position, and a combination of negative numbers represents a bearish position (as long as it is the same underlying, it does not matter if they expire at different times)
④Value
The Delta value of an option is between -1 and 1, which means that the underlying price changes faster than the option value. The absolute value of Delta for an at-the-money option is about 0.5, and the absolute value of Delta for an in-the-money option is usually greater than 0.5 and less than 1. The absolute value of Delta for a deep in-the-money option is close to 1. The absolute value of Delta for an out-of-the-money option is smaller, and the absolute value of Delta for an in-the-money option is larger. Therefore, the Delta of an option is often used to indicate the degree of in-the-moneyness of an option. The absolute value of Delta can also be expressed as the possibility of an option becoming in-the-money when it expires. The larger its absolute value, the greater the possibility of closing at in-the-money.
⑤Probability
Delta represents the probability, the percentage of whether this option will be valuable in the end [the most core concept]
2. Vega
For easy memorization, we nicknamed him "Lang Lang Ge". Vega represents the change in option price caused by each point change in implied volatility. You never know when Lang Lang will make a big move.
① Sensitive areas
Like Theta, Vega is most sensitive at ATM. The further the price of the underlying asset deviates from the option strike price, the smaller the Vega is, that is, the less sensitive the option price is to changes in volatility; the closer the price of the underlying asset is to the option strike price, the larger the Vega is, that is, the more sensitive the option price is to changes in volatility.
② Time impact
As time goes by, the changes in Vega near the ATM have little impact on option prices, because as it approaches expiration, it is already destined to be in the money or out of the money, and the subsequent changes in Vega have little impact.
③ The Vega of an in-the-money or out-of-the-money option will gradually approach that of an at-the-money option as IV increases.
3. Theta
For easy memorization, we nicknamed him "Time Management Master". Theta measures the impact of changes in expiration time on option prices, that is, the change in option prices should occur when the expiration time passes by one unit.
① Positive and negative signs
Buyers are negative and sellers are positive.
② Acceleration
Theta decay will gradually accelerate as the expiration date approaches. [Qualitative understanding] The absolute value of Theta is the largest near the strike price. In other words, near the strike price, the change in expiration time has the greatest impact on the option value. For in-the-money options and out-of-the-money options, the closer to the expiration date, the closer Theta is to 0; while the absolute value of Theta for at-the-money options is larger.
③Relationship with IV
Theta will increase the price as IV increases, which is the same reason why sellers sell when IV is high.
4. Rho
For ease of memorization, we nicknamed it "Little Brother" because its impact on the entire option price is relatively small and can be ignored, and it is all priced in the option price. Rho measures the impact of interest rate changes on option prices, that is, the corresponding change in option prices for a one-unit change in interest rates. Rho is the partial derivative of option value with respect to the risk-free interest rate.
The impact of risk-free interest rates on option prices. This is a medium- to long-term strategy. Generally speaking, it is not recommended for crypto novices to start with long-term options, as market volatility itself is much greater than that of U.S. stocks.
① Impact of rise and fall on call and put
When the interest rate rises: the call price rises; the put price falls; when the interest rate falls: the call price falls; the put price rises.
The more in-the-money an option is, the greater the impact of interest rate changes on the option value; the more out-of-the-money an option is, the smaller the impact of interest rate changes on the option value.
②Convergence value
Rho increases monotonically with the price of the underlying security. For call options, the higher the underlying price, the greater the impact of interest rates on the option value; for put options, the lower the underlying price, the greater the impact of interest rates on the option price. 3. Rho converges monotonically to 0 as the option expires. In other words, the closer the option is to expiration, the smaller the impact of interest rate changes on the option value.
Gamma
For easy memorization, we nicknamed him "Running God". Gamma is placed at the end because it is different from the other letters and is a second-order derivative. It is also the most difficult to understand, and the core of risk control for option sellers is to control negative Gamma. Gamma measures the impact of changes in the underlying security price on Delta, that is, the corresponding change in the option Delta when the underlying security price changes by one unit. The Gamma indicator indirectly measures the second-order impact of changes in the underlying security price on the option price, and measures the sensitivity of Delta to the underlying security price.
The Gamma of the option Buy side is always positive, while the Gamma of the Sell side is always negative.
① It is the first-order derivative of Delta, and also the underlying (the second-order derivative of BTC and ETH price changes on option prices). It can be roughly understood as the concept of acceleration. Delta will get faster and faster near the actual value due to the Gamma factor (example: stove). Buying calls or puts are positive Gamma; selling calls or puts are negative Gamma
② Option sellers make money slowly but lose money quickly. Gamma is the enemy of option sellers. Why do sellers not hold their options until expiration? Gamma Risk. When Gamma is relatively small, Delta changes slowly, and trading adjustments to ensure Delta neutrality do not need to be too frequent. However, when the absolute value of Gamma is large, Delta is very sensitive to changes in the price of the underlying security, and frequent adjustments are required to ensure Delta neutrality.
③Gamma is largest near ATM over time; the seller's P&L profit and loss will fluctuate greatly at the end; (the Gamma of deep in-the-money and deep out-of-the-money contracts gradually decreases, and the PL fluctuates less, but this is not the most important thing)
④ As IV decreases, gamma gradually increases near ATM, driving the overall P&L vibration.
VI. Conclusion
In order to facilitate the memory of many novices, a limerick suitable for crypto option trading is specially summarized:
In teaching practice, many novices are obsessed with various neutral hedging strategies, which is actually influenced by the quantitative hedge funds in recent years. There is no strategy that can earn all the money from the Greek letters, and there will always be risk exposure.
The most interesting thing about options is that they allow us to have a deeper understanding of some common sense from the fields of financial investment and mathematics, or in other words, some common sense words are reflected through options.
Finally, I would like to end with a famous quote: "All gifts given by fate have already been secretly marked with a price."
(Note: Zweig once used this sentence to comment on Marie Antoinette)
