In mathematical finance, the Greeks serve as essential measures denoting the sensitivities or derivatives of a derivative instrument’s price—like an option—to alterations in one or more underlying parameters. These parameters are crucial elements upon which the value of the instrument or an entire portfolio of financial instruments relies. The term “Greeks” originates from the convention of denoting these sensitivities using Greek letters, a tradition also followed in representing other financial measures.
These sensitivities, also known as risk sensitivities, risk measures, or hedge parameters, showcase the extent of the impact on the price of an option or derivative due to changes in underlying factors. The most prevalent Greeks are the first-order derivatives: delta, vega, theta, and rho, each representing distinct aspects of an option’s behavior concerning changes in the underlying parameters. Additionally, gamma, a second-order derivative of the value function, measures the rate of change in delta based on alterations in the underlying asset’s price.
A comprehensive understanding of these Greeks is imperative for traders engaging in options trading. Proficiency in interpreting these measures enables traders to make informed decisions, manage risks effectively, and implement strategies aligned with market movements and changes in underlying parameters.
In this article, our focus will be on comprehensively exploring and gaining a deep understanding of the four primary Greeks associated with option trading, which are as follows:
- Delta (Δ): Delta measures the rate of change in an option’s price concerning an INR 1 change in the underlying asset’s price. It shows the sensitivity of the option’s price to movements in the underlying asset. For calls, delta ranges from 0 to 1; for puts, it ranges from -1 to 0.
- Gamma (Γ): Gamma represents the rate of change in an option’s delta concerning an INR 1 change in the underlying asset’s price. It measures the curvature or acceleration of the delta. Gamma is higher for options that are at-the-money (ATM) or near-the-money (NTM) compared to deep-in-the-money (ITM) or out-of-the-money (OTM) options.
- Theta (Θ): Theta measures the rate of decline in an option’s price concerning the passage of time. It indicates how much the option’s value decreases as each day passes, assuming other factors remain constant.
- Vega (ν): Vega measures the rate of change in an option’s price concerning a 1% change in implied volatility. It shows the impact of changes in implied volatility on the option’s price.
- Rho (ρ): Rho measures the rate of change in an option’s price concerning a 1% change in the risk-free interest rate. It indicates how much the option’s price will change with fluctuations in the interest rate.
Delta (Δ)
Delta is a theoretical estimate of how much an option’s premium may change given a ₹1 move in the underlying. For an option with a Delta of .50, an investor can expect about a ₹0.50 move in that option’s premium given a ₹1 move, up or down, in the underlying. For purchased options owned by an investor, Delta is between 0 and 1.00 for calls and 0 and -1.00 for puts. For sold options, as the investor essentially has a negative quantity of contracts, we find that short puts have a positive Delta, while short calls have negative Delta.
For example, the XYZ 100 call has a 0.50 Delta and is trading at ₹2 with XYZ stock at ₹101. If XYZ rises to ₹102, the investor would expect that the 100-strike call would now be worth around ₹2.50 as seen below:
- ₹1 increase in underlying price x 0.50 Delta = ₹0.50 anticipated change in option premium.
- Original Premium: ₹2.00 + ₹0.50 estimated change = ₹2.50 estimated new premium after ₹1 stock price increase.
With a ₹1 move down in XYZ, the investor would expect to see this same 100-strike call option decrease in value to around ₹1.50. As the stock price rises and the call option goes deeper in the money, Delta typically approaches 1.00 because of the increased likelihood the option will be in the money at expiration. As expiration approaches, in-the-money-option Deltas are also more likely to be moving slowly toward 1.00 because at expiration an option either has a Delta of either 0 or 1.00 with no time premium remaining.
The graph below demonstrates the potential relationship between Delta and stock price with a strike price of 100:
Interpretation
- Call Delta (Blue Line): It represents the rate of change in the call option price concerning changes in the stock price. As the stock price increases, the call delta tends to approach 1. This means that call options become more sensitive to changes in the stock price and are more likely to be in the money (positive delta).
- Put Delta (Orange Line): It shows the rate of change in the put option price concerning changes in the stock price. Put deltas tend to move towards -1 as the stock price decreases. This implies that put options become more sensitive to downward stock movements and are more likely to be in the money (negative delta).
Outlined below is a summary:
- Calls display positive Deltas (as indicated by the model). They exhibit a direct correlation with changes in the underlying stock price. When the stock price increases, the call Delta typically rises. Conversely, as the stock price decreases, the call Delta tends to decline. Call Deltas vary within the range of 0 to +1.00.
- Puts demonstrate negative Deltas (as generated by the model). They showcase an inverse correlation with changes in the underlying stock price. An increase in the stock price usually results in a decrease in Put Delta. Conversely, a decrease in the stock price generally leads to an increase in Put Delta. Put Deltas fall within the range of 0 to -1.00.
The Delta of an option is influenced by the stock price, days remaining until expiration, and implied volatility. When implied volatility rises, Delta tends to approach 0.50, reflecting an increased probability of a wider range of strikes potentially ending in the money due to the expected underlying movement.
For instance, consider a scenario where the XYZ 20 strike call, with a stock price of ₹21 and an implied volatility of 30%, has a Delta of 0.60. If the implied volatility climbs to 40%, the Delta might decrease to 0.55. This shift occurs because traders perceive a heightened likelihood of the strike being out-of-the-money at expiration, resulting in the adjustment of Delta to account for this altered perception.
Let’s explore the impact of changes in implied volatility on Delta:
Observing the impact of varying implied volatilities on Delta, we note that higher implied volatilities expand the range of potential strike prices that could potentially become ‘in play,’ leading to more Deltas approximating 0.50. On the other hand, stocks with lower implied volatility levels tend to exhibit higher Deltas for in-the-money options and lower Deltas for out-of-the-money options.
For some traders, Delta serves as an implied probability, indicating the likelihood of an option concluding in the money at expiration. Hence, an at-the-money option typically boasts a Delta of around 0.50, signifying a 50% chance of being in-the-money upon expiration. Conversely, deep-in-the-money options display substantially higher Deltas, indicating a significantly higher probability of expiring in-the-money.
Analyzing the Delta of a far-out-of-the-money option can provide investors with insight into its probability of possessing value at expiration. An option possessing a Delta of less than 0.10 (equivalent to less than a 10% probability of being in-the-money) is generally perceived as having a low likelihood of reaching in-the-money status at any point. Consequently, such options necessitate a substantial movement in the underlying asset to hold value at expiration.
The time remaining until the option’s expiration significantly influences Delta. When considering the same strike, an in-the-money call with an extended time until expiration consistently exhibits a lower Delta compared to the same strike call with a shorter time until expiration. Conversely, for out-of-the-money calls, the option with an extended expiration period tends to display a higher Delta than the option with less time remaining.
Let’s explore the impact of changes in time to expiration on Delta
As the option approaches expiration, certain tendencies in Call Deltas become apparent:
- In-the-money call options tend to increase their Deltas towards 1.00.
- At-the-money call options consistently maintain Deltas around 0.50.
- Out-of-the-money call options tend to decrease their Deltas towards 0, assuming other parameters remain constant.
However, it’s essential to note that Delta is a dynamic metric and undergoes continuous changes during market hours. It often may not precisely predict the exact change in an option’s premium due to its evolving nature.
Gamma (Γ)
Gamma is closely related to delta. Gamma is the rate of change in delta for every ₹1 change in the underlying price. Gamma represents the acceleration at which an option’s price increases or decreases. Think of gamma as the next rupee move.
Let’s understand by example
Consider a call option displaying a Delta of 0.80 and a Gamma of 0.02. If the stock decreases by ₹1, the Delta will reduce by the Gamma amount. The estimated new Delta value would be around 0.78. Conversely, if the stock increases by ₹1, the Delta will rise by the Gamma amount, resulting in an approximate new Delta value of 0.82.
In options trading, long positions, whether they are calls or puts, consistently exhibit positive Gamma. Conversely, short call and short put positions demonstrate negative Gamma. Stock positions, however, do not possess Gamma as their Delta remains constant at either 1.00 (long) or -1.00 (short), and this value does not fluctuate.
Positive Gamma signifies that
- Long call options experience an increase in Delta towards +1.00 when the stock price rises, and a decrease towards 0 when the stock price falls.
- Long put options witness a decrease in Delta towards -1.00 if the stock price decreases, and an increase towards 0 when the stock price rises.
Conversely, for a short call with negative Gamma
- The Delta becomes more negative as the stock price ascends and less negative as it descends.
Gamma tends to be higher for options positioned at the money and those closer to expiration. Specifically, a front-month, at-the-money option typically exhibits higher Gamma compared to a long-term option sharing the same strike. This occurs because the Delta of near-term options tends to approach either 0 or 1.00 more rapidly as expiration approaches.
The elevated Gamma translates to more pronounced changes in Delta as the underlying asset moves, especially when the underlying hovers around the strike price near expiration. Investors can expect more significant fluctuations in Delta owing to the higher Gamma value.
Gamma tends to be comparatively lower in longer-dated options due to the wider range of strikes potentially ending in the money over the extended time horizon. At-the-money options generally exhibit the highest sensitivity in Delta concerning underlying asset movements, contributing to their higher Gamma values.
When the stock price aligns precisely with a strike at expiration, an option’s Gamma peaks as the Delta rapidly shifts from 1.00 towards 0 or vice versa. During these instances, Gamma can be exceptionally high as the Delta undergoes rapid changes with the underlying at the strike while approaching expiration.
In contrast, deep-in-the-money or far-out-of-the-money options showcase lower Gamma than at-the-money options. Deep-in-the-money options already possess high positive or negative Deltas. If these options move further into the money, the Delta approaches 1.00 (or -1.00 for puts), causing a reduction in Gamma as the Delta cannot surpass 1.00. Conversely, as the stock gravitates towards the strike of deep-in-the-money options, the Gamma escalates, resulting in the Delta decreasing by approximately the current Gamma magnitude.
Gamma attains its peak when the Delta lies within the .40-.60 range, often observed when an option is at the money. Conversely, deeper-in-the-money or farther-out-of-the-money options tend to feature lower Gamma values since their Deltas do not adjust as swiftly with movements in the underlying asset.
As the Deltas of options approach 0 or 1.00 (or 0 or -1.00 for puts), Gamma typically reaches its lowest point.
Let’s look at the graph and table below for the above explanation:
Gamma for different Implied Volatility
Implied volatility fluctuations significantly impact Gamma. When implied volatility declines, the Gamma of at-the-money calls and puts tends to rise. Conversely, as implied volatility escalates, the Gamma of both in-the-money and out-of-the-money calls and puts tends to decrease.
This phenomenon occurs due to low implied volatility options exhibiting more pronounced Delta changes in response to underlying movements. On the other hand, high implied volatility in an underlying product foresees greater potential movement, resulting in lesser Delta changes.
Below graph and table below explain the IV effect on gamma
Theta (Θ)
Theta illustrates the impact of time decay on an option’s value. Options, over time, witness a gradual reduction in worth. Daily, options contracts tend to diminish in value. Both long and short-option holders should comprehend Theta’s effects on an option premium. Theta is represented in an actual INR premium amount and can be calculated on a daily or weekly basis. In theory, Theta signifies the potential daily/weekly decay of an option’s premium, assuming all other factors remain constant.
Time decay, or Theta, doesn’t follow a linear pattern. The rate of decay, theoretically, intensifies as the expiration date draws near. Initially, Theta indicates a gradual decay that accelerates as expiration approaches. Upon expiration, options possess no time value and only hold intrinsic value, if any. Pricing models factor in weekends, so options generally decay over five trading days within a seven-day period. However, there’s no universal method for decaying options, leading to different models showing time decay’s impact differently. If a pricing model decays options rapidly, current market values may seem higher in comparison to the model’s theoretical values. Conversely, if the model displays slower decay, current market values may appear cheaper in contrast to the model’s theoretical values.
For instance, if XYZ were trading at ₹50 and a 50 strike call was trading at ₹3 with a Theta of 0.05, an investor might expect that option to decline by approximately ₹0.05 per day, assuming all other factors remain constant. If a day passes without a change in the option price, it implies a change in another variable. Usually, it could indicate an increase in implied volatility. If the option depreciated more than ₹0.05, an investor might infer a possible drop in implied volatility. Moreover, as expiration approaches, Theta is likely to become increasingly negative. On the penultimate trading day, with one day remaining until expiration, the Theta should equate to the entire remaining time value in the option.
Let’s look into an example that demonstrates how Theta behaves as time progresses.
It’s noticeable how the time value diminishes more rapidly as the expiration date approaches.
Theta for different Implied Volatility
The implied volatility of a product plays a significant role in determining the time premium, subsequently impacting Theta amounts. Typically, higher implied volatility levels correspond to higher Theta amounts. However, it’s important to note that selling options in high implied volatility stocks does not guarantee immediate earnings from time decay. Often, options trade at heightened implied volatility levels due to historical volatility or specific events like earnings announcements or product releases. Here’s a chart illustrating the general Theta amounts across different implied volatilities:
Vega (ν)
Vega represents the amount by which option prices change for every 1% alteration in implied volatility within the underlying security. Unlike other components of an option’s price such as spot price, strike price, and time to expiration, Vega remains an unknown element as future volatility cannot be accurately predicted. Notably, Vega does not affect the intrinsic value of an option and solely hinges on volatility alterations, not stock price movements.
Rising volatility leads to an increase in an option’s price, as market participants anticipate potential above-average movements before expiration. Conversely, Vega diminishes as expiration approaches, owing to reduced time for volatility to transpire.
For instance, consider XYZ trading at ₹50, where a call option with 12 months until expiration possesses an implied volatility of 30%, a Vega of ₹0.15, and a current market value of ₹4. If the implied volatility instantaneously escalates by 2% to 32%, the option premium might surge by: ₹0.15 × 2 = ₹0.30, reaching around ₹4.30, assuming all else remains constant. Conversely, a 5% decrease in implied volatility may cause the option to lose approximately: ₹0.15 × 5 = ₹0.75 in value.
This demonstrates that fluctuations in implied volatility can significantly impact option prices, ranking second in importance after underlying price movements.
Longer-term options typically carry higher prices, where a 1% change in implied volatility results in a more significant value shift of the premium compared to an option with a lower premium. Higher implied volatility corresponds to a higher option value. Refer to the graph below for a visual illustration of this concept.
RHO (ρ)
Rho measures an option’s sensitivity to changes in interest rates. Like Vega, interest rate changes affect longer-term options more than near-term ones, as depicted in the chart below. Interest rates are integral to pricing models that determine an option’s price based on its “hedged value,” an approach where an investor uses long or short stock positions to manage risk associated with option positions. Rho is positive for purchased calls because higher interest rates tend to increase call premiums. Conversely, Rho is negative for purchased puts because higher interest rates generally decrease put premiums.
For instance, if interest rates are currently at 3.00% and the Rho on a ₹100 call option is +0.45, a sudden increase in interest rates to 4% would cause the premium to rise by ₹0.45. Conversely, if the Rho for the put was -0.45, the put premium would decline by ₹0.45 per share. This assumption considers that other pricing factors remain constant.
One might wonder, “Why do interest rates affect option prices?” It’s related to the cost of carrying the position over time. Pricing models factor in the cost of capital used to offset risk, accounting for expenses incurred in borrowing money or interest lost on existing funds. For instance, a professional investor looking to buy a deep-in-the-money put with a Delta nearly reaching -1.00 would need to hedge by buying 100 underlying shares. Buying these shares requires borrowing money, and if interest rates rise, the expense of carrying the long stock position increases. Consequently, should interest rates increase, the investor buying this put would pay less to cover the rising expenses of carrying the position over time.
In general, higher stock prices and longer time until expiration result in greater sensitivity to changes in interest rates, denoted by higher absolute Rho values.
In summary, an option position hedged to Delta neutral that requires the investor to buy the underlying shares (such as a short call or long put) would have a negative reaction to increasing interest rates. Conversely, a position involving selling short stock to achieve hedging would be negatively impacted by a decrease in rates.
Refer to the below graph for option values at different interest rates
“The Python code for the above examples is accessible in the Python code category.“
