Cryptocurrency options: “Delta” and “Gamma.” Part 1

delta.theta
6 min readApr 10, 2023

In this article, we will examine the impact of the greeks delta and gamma on the option price, potential pitfalls, and ways to use them to reduce risks when trading cryptocurrency options.

An option is a contract between a buyer and a seller. The buyer pays a premium to the seller for the option, and the seller has an obligation to fulfill it. From the seller’s perspective, trading options is akin to trading the underlying asset using a limit order, for which they receive payment (though there are some distinctions).

For instance, if you are a seller of a BTC PUT option with a strike price of 27,000 USD, you are agreeing to buy BTC at that price, even if its value on the expiration date is 25,000 USD. In this case, you would acquire BTC at a price of 27,000 USD.

To understand what influences the price of an option, we must examine the Black-Scholes formula:

For a CALL:

For a PUT:

The remaining parameters are determined as follows:

The variables in the Black-Scholes formula include:

  • F: the price of the underlying asset
  • X: the strike price
  • σ: volatility
  • T: time to expiration
  • R: interest rate

An important aspect to consider when using the Black-Scholes formula to calculate the price of an option is that it assumes volatility will remain constant until the expiration date.

The most critical variable in the formula is F, the price of the underlying asset. As this price continually fluctuates, understanding its impact on the option price is essential.

Originally, when the Black-Scholes formula was developed, options were primarily used for delta-neutral strategies. Later on, speculators became involved. To better comprehend this, let’s revisit the concept of delta.

Delta, a greek symbol, represents the rate at which an option’s price changes when the price of the underlying asset shifts by one unit (e.g., one dollar or one point) while maintaining the same volatility. For instance, if the delta is 0.7, the option price will change by 0.7 when the underlying asset’s price increases by one dollar. For those familiar with mathematics, delta is the first derivative of the option price.

Revisiting delta, the value for a call option at the central strike price (when the strike price is close to the current price of the underlying asset) is 0.5, while for a put option, it is -0.5. As a call option becomes more profitable, its delta approaches 1, and for a put option, it approaches -1. The difference between the deltas of a call and put option is equal to 1.

An “in-the-money” option signifies that the strike price is lower than the current price of the underlying asset for a call option and vice versa for a put option.

When selling an option, the delta’s sign changes to the opposite. For a sold call, delta turns negative, and for a sold put, it becomes positive. The delta of the underlying asset is always equal to 1.

For instance, assume you purchased 1 BTC at $28,000 and want to hedge against a price drop for one week by buying a put option. How many put options must you buy? To answer this question, use the formula: asset delta + option deltas = 0. In this case, our asset delta is equal to 1, and the put delta at a strike price of $28,000 is -0.5. The equation will look like this: 1–0.5x = 0, where x represents the number of put options needed for protection against a price drop. Solving this equation, we find that x=2, meaning we need to buy 2 put options to hedge against a price drop for one week.

However, it is crucial to remember that insurance comes at a cost. In this example, 1 put option is priced around $750, while 2 put options amount to $1500. Thus, the insurance for a week costs 5.4% of the investment.

As the price of the underlying asset and volatility continuously change, the option delta also varies. The constant pursuit of maintaining a delta of 0 can lead to increased transaction costs.

From a practical perspective, this information on delta may be sufficient. To delve deeper, we will explore gamma, its applications in hedging, and the concept of gamma exposure.

Gamma is the second derivative of the option price concerning the underlying asset price, or the acceleration of delta.

This means that the option price’s change not only depends on the alteration in the underlying asset price but also occurs with a specific acceleration. This acceleration is what determines gamma. Gamma is consistent for options with the same strike price at a given time. Purchasing options yields a positive gamma, while selling options results in a negative gamma. The formula for option gamma is provided below. The closer the option is to being at the money and its expiration date, the higher its gamma.

From a practical perspective, there are three key aspects of gamma to understand, the first of which is the gamma squeeze.

When you trade and purchase options, someone sells them to you. Typically, a market maker sells you the underlying asset or options. The market maker’s role is to consistently provide liquidity in the exchange order book, enabling traders to execute orders.

Suppose a trader believes that BTC’s value will increase and decides to buy it on the spot market. From the market maker’s viewpoint, they now have a short position in the spot market that requires hedging. To hedge this, the market maker buys a call option. When another trader observes the call option rising, they begin purchasing it. In response, the market maker sells the call option and buys back the spot. At each iteration, customers compel the market maker to inflate the market, driving the underlying asset’s price higher. How can this situation be avoided?

A gamma-delta neutral strategy can be employed. When selling an option, your gamma position turns negative. To hedge the gamma, you buy a call option, resulting in a total gamma of 0 for the position. Then, balance the delta for this option position through the spot market. This method decreases the probability of a gamma squeeze and simplifies delta hedging, as it does not necessitate continuous delta balancing. However, this approach may reduce potential profits (due to option purchases), which might not be acceptable for market makers.

It is crucial to recognize that strikes with the maximum negative gamma can pose significant challenges for market makers. They are liquidity providers in the options market, and when the underlying asset’s price moves to these levels, it can create serious issues with position balancing.

However, since these strikes often act as “magnets” for the underlying asset price, traders can utilize this information to determine the most probable levels of market movement. In the next chapter, we will examine the tools that can help identify these levels and incorporate this information into trading strategies.

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