Python & the Option Greeks

What Are Primary & Secondary Option Greeks?

Have you ever watched options prices bounce around and wondered, What's the deal? Why don't they follow the stock exactly?

Sometimes it's like they're doing their own chaotic dance! Ever seen a call option's price go up when the stock it's tied to goes down? It can feel like a financial puzzle. But here's the secret, there's a clever formula called the Black-Scholes Model that explain the code of option pricing. It reveals five key players influencing those prices: the underlying stock's price, interest rates, volatility (how wild the market is), time until the option expires, and the option's strike price (the price it lets you buy or sell at).

These five factors are like the option world's, each with a unique personality (and a Greek nickname!). They're called the "Greeks," and they're the key to understanding how options should behave when the market throws a curveball.

For traders, the Greeks are like superpowers. They let you peek into the future and calculate how your options portfolio might change if any of those five factors shift. With this knowledge, we can become a financial ninja, dodging risks and building a portfolio that's perfectly balanced for your goals. So, the next time you see options doing their wild thing, remember, it's not magic, it's the Greeks at work.

Primary Greeks

• Delta (Δ)
• Gamma (Γ)
• Theta (Θ)
• Vega (ν)
• Rho (ρ)

Secondary Greeks

• Charm
• Vanna
• Vomma
• Speed
• Zomma
• Color
• Ultima
• Lambda
• Vera
• Epsilon

PRIMARY OPTION GREEKS

Delta (Δ)

• Measures the sensitivity of the option's price to changes in the price of the underlying asset.
• Measures the change in an option's price for every $1 change in the underlying asset's price. It ranges from -1 to 1 for puts and 0 to 1 for calls. A delta of 0.5 means the option price will move $0.50 for every $1 move in the underlying asset.
• Call options have a positive Delta, while put options have a negative Delta.

• Range: Delta values range from -1 to 1 for put options and 0 to 1 for call options.
• Interpretation: Think of delta as the probability that an option will expire, in the money (ITM), meaning it will have intrinsic value at expiration.

Delta for Call Options:

• Positive Values: Call options have positive delta values (0 to 1). A higher delta means the call option is more likely to expire ITM. A delta of 0.50 suggests a 50% probability of the call option expiring ITM.
• Relationship with Underlying Asset: As the price of the underlying asset increases, the delta of a call option increases. Conversely, as the underlying asset's price decreases, the delta of a call option decreases.
• A call option with a delta of 0.75 means that for every $1 increase in the underlying asset's price, the call option's price is expected to increase by $0.75.

Delta for Put Options:

• Negative Values: Put options have negative delta values (-1 to 0). A more negative delta means the put option is more likely to expire ITM. A delta of -0.50 suggests a 50% probability of the put option expiring ITM.
• Relationship with Underlying Asset: As the price of the underlying asset decreases, the delta of a put option becomes more negative. Conversely, as the underlying asset's price increases, the delta of a put option becomes less negative.
• A put option with a delta of -0.30 means that for every $1 decrease in the underlying asset's price, the put option's price is expected to increase by $0.30.

Lets, take an example using python programming:

Black-Scholes formulas for calculating the price of both call and put options

C = S N(d1) - K e^(-rT) * N(d2)

P = K e^(-rT) N(-d2) - S * N(-d1)

• S = Current price of the underlying asset
• K = Strike price of the option
• T = Time to expiration (in years)
• r = Risk-free interest rate (as a decimal)
• sigma = Implied volatility (as a decimal)
• N(x) = Cumulative standard normal distribution function of x
• e = Euler's number (approximately 2.71828)

d1 = (ln(S / K) + (r + (sigma^2) / 2) T) / (sigma sqrt(T))

d2 = d1 - sigma * sqrt(T)

• Cumulative Standard Normal Distribution (N(x)): This function gives the probability that a standard normal random variable will be less than or equal to x. we can find its values in statistical tables or calculate them using software libraries like SciPy.
• Exponential Function (e^(-rT)): This is the discount factor used to bring future values back to the present. In this context, it discounts the strike price (K) to its present value.
• Natural Logarithm (ln(S/K)): This represents the logarithmic return of the stock price over the strike price.

S = 100                     # Stock price
K = 105                     # Strike price
T = 0.5                      # 6 months to expiration
r = 0.05                     # 5% risk-free rate
sigma = 0.20            # 20% volatility        

import numpy as np
from scipy.stats import norm

def delta(S, K, T, r, sigma, option_type='call'):
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    return norm.cdf(d1) if option_type == 'call' else norm.cdf(d1) - 1

S = 100                     # Stock price
K = 105                     # Strike price
T = 0.5                      # 6 months to expiration
r = 0.05                     # 5% risk-free rate
sigma = 0.20            # 20% volatility

call_delta = delta(S, K, T, r, sigma, 'call')
put_delta = delta(S, K, T, r, sigma, 'put')

print("Call Delta:", call_delta)
print("Put Delta:", put_delta)        

Gamma (Γ)

• Measures the rate of change of Delta with respect to changes in the price of the underlying asset.
• It helps in understanding how Delta will change if the underlying asset's price changes.
• High Gamma indicates that Delta could change significantly with small movements in the underlying price.

Call Options:

• Positive Gamma: Call options always have a positive Gamma. A higher Gamma means the Delta of the call option will change more rapidly as the underlying asset's price increases. This is generally desirable for call option holders as it amplifies potential gains in a rising market. However, it also means potential losses can be magnified if the underlying asset's price falls.
• At-the-money (ATM) Calls: Gamma is highest for at-the-money call options, where the strike price is equal to the current market price of the underlying asset. This is because ATM options have the highest potential for large Delta changes as the underlying asset's price fluctuates.

Put Options:

• Positive Gamma: Similar to call options, put options also have a positive Gamma. A higher Gamma means the Delta of the put option will change more rapidly as the underlying asset's price decreases. This is generally desirable for put option holders as it amplifies potential gains in a falling market. However, it also means potential losses can be magnified if the underlying asset's price rises.
• At-the-money (ATM) Puts: Gamma is also highest for at-the-money put options, where the strike price is equal to the current market price of the underlying asset. This is because ATM options have the highest potential for large Delta changes as the underlying asset's price fluctuates.

Now we using python for calculating Gamma:

Γ = N'(d1) / (S σ √T)

• N'(d1) = Standard normal probability density function of d1 (the first derivative of the standard normal cumulative distribution function)
• S = Current price of the underlying asset
• σ = Implied volatility (as a decimal)
• T = Time to expiration (in years)

d1 = (ln(S / K) + (r + (σ^2) / 2) T) / (σ √T)

• Gamma for Call and Put Options: The Gamma formula is the same for both call and put options. This means that the rate of change of Delta with respect to the underlying asset's price is identical for both types of options.
• Standard Normal Probability Density Function (N'(d1)): This function represents the probability of a standard normal random variable taking on a specific value of d1. You can find its values in statistical tables or calculate them using software libraries like SciPy.
• Relationship with Delta: Gamma is closely related to Delta. A higher Gamma indicates that Delta will change more rapidly as the underlying asset's price changes.
• At-the-Money (ATM) Options: Gamma is highest for at-the-money options and decreases as the option moves further into or out of the money. This means that ATM options are the most sensitive to changes in the underlying asset's price, as reflected by their higher Gamma.

def black_scholes_gamma(S, K, T, r, sigma):
    d1 = (np.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T))
    return norm.pdf(d1) / (S * sigma * np.sqrt(T)) / 365

S = 100                     # Stock price
K = 105                     # Strike price
T = 0.5                      # 6 months to expiration
r = 0.05                     # 5% risk-free rate
sigma = 0.20            # 20% volatility

gamma_value = black_scholes_gamma(S, K, T, r, sigma)  

print("Gamma:", gamma_value)

        

Theta (Θ)

• Measures the sensitivity of the option's price to the passage of time, also known as time decay.
• Theta represents the amount by which the option's price decreases as time to expiration decreases, assuming all other factors remain constant.
• Both call and put options usually have a negative Theta.

Theta for Call Options:

• Negative Theta: Call options typically have negative Theta. This means that as time passes, and the option gets closer to expiration, its value decreases. This is primarily due to the diminishing time value of the option. As the time remaining to exercise the option shrinks, the probability of the underlying asset's price moving significantly enough to make the option profitable decreases, hence the decay in value.
• Impact on OTM Call Options: Out-of-the-money (OTM) call options, where the strike price is higher than the current market price of the underlying asset, are particularly susceptible to Theta decay. This is because they have a lower probability of expiring in-the-money (ITM) and their value is primarily derived from time value. As time passes, this time value diminishes rapidly.

Theta for Put Options:

• Negative Theta: Similar to call options, put options also typically have negative Theta. Their value decreases as time passes due to the erosion of time value.
• Impact on OTM Put Options: Out-of-the-money (OTM) put options, where the strike price is lower than the current market price of the underlying asset, are also highly sensitive to Theta decay. Their value diminishes rapidly as time passes due to the decreasing probability of expiring ITM.

Now, we using python programming to calculate theta:

Theta for Call Options (Θc):

Θc = -(S N(d1) σ) / (2 √T) - r K e^(-rT) N(d2)

Theta for Put Options (Θp):

Θp = -(S N(d1) σ) / (2 √T) + r K e^(-rT) N(-d2)

• S = Current price of the underlying asset
• K = Strike price of the option
• T = Time to expiration (in years)
• r = Risk-free interest rate (as a decimal)
• σ = Implied volatility (as a decimal)
• N(x) = Cumulative standard normal distribution function of x
• N'(d1) = Standard normal probability density function of d1
• e = Euler's number (approximately 2.71828)

d1 Calculation (same as for option pricing):

d1 = (ln(S / K) + (r + (σ^2) / 2) T) / (σ √T)

d2 Calculation (same as for option pricing):

d2 = d1 - σ * √T

• Negative Theta: Both call and put options typically have a negative Theta. This means that their value decreases as time passes, all else being equal.
• Time Decay Acceleration: Theta increases (becomes more negative) as the option approaches expiration. This means the rate of time decay accelerates as the option gets closer to its expiry date.
• Impact of Volatility: Higher implied volatility can reduce the magnitude of Theta, as the increased uncertainty about future price movements can offset some of the time decay.
• Impact of Interest Rates: Higher interest rates generally lead to a slightly higher (less negative) Theta for call options and a slightly lower (more negative) Theta for put options.

def black_scholes_theta(S, K, T, r, sigma, option_type='call'):
    d1 = (np.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    common_term = -(S * norm.pdf(d1) * sigma) / (2 * np.sqrt(T))
    second_term = -r * K * np.exp(-r * T) * norm.cdf(d2) if option_type == 'call' else r * K * np.exp(-r * T) * norm.cdf(-d2)

    return (common_term + second_term) / 365         

S = 100                     # Stock price
K = 105                     # Strike price
T = 0.5                      # 6 months to expiration
r = 0.05                     # 5% risk-free rate
sigma = 0.20            # 20% volatility

call_theta = black_scholes_theta(S, K, T, r, sigma, 'call')
put_theta = black_scholes_theta(S, K, T, r, sigma, 'put')

print("Call Theta:", call_theta)
print("Put Theta:", put_theta)         

Vega (V)

• Measures the sensitivity of the option's price to changes in the volatility of the underlying asset.
• Higher Vega means the option's price is more sensitive to changes in volatility.
• Both call and put options have positive Vega.

Vega for Call Options:

• Increased Value with Higher Volatility: When implied volatility increases, the value of a call option also increases. This is because higher volatility increases the probability of the underlying asset's price moving significantly upwards, which is favorable for call option holders.
• Impact on OTM Options: Out-of-the-money (OTM) call options are particularly sensitive to changes in volatility. An increase in volatility can significantly increase the value of an OTM call option due to the increased probability of the option becoming ITM.

Vega for Put Options:

• Increased Value with Higher Volatility: Similar to call options, put options also increase in value when implied volatility increases. This is because higher volatility increases the probability of the underlying asset's price moving significantly downwards, which is beneficial for put option holders.
• Impact on OTM Options: Out-of-the-money (OTM) put options are also highly sensitive to changes in volatility. An increase in volatility can significantly increase the value of an OTM put option due to the increased probability of the option becoming ITM.

The Black-Scholes formula for Vega (ν) is the same for both call and put options:

ν = S N'(d1) √T

• S = Current price of the underlying asset
• N'(d1) = Standard normal probability density function of d1 (the first derivative of the standard normal cumulative distribution function)
• T = Time to expiration (in years)

d1 Calculation (same as for option pricing):

d1 = (ln(S / K) + (r + (σ^2) / 2) T) / (σ √T)

• Positive Vega: Vega is always positive for both call and put options. This means that an increase in implied volatility will always lead to an increase in the option's price, regardless of whether it's a call or a put.
• Magnitude of Vega: The magnitude of Vega is highest for at-the-money (ATM) options and decreases as the option moves further into or out of the money. This means that ATM options are the most sensitive to changes in volatility.
• Time to Expiration: Vega generally increases with time to expiration, as there's more time for volatility to affect the option's price.

def black_scholes_vega(S, K, T, r, sigma):
    d1 = (np.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T))
    # multiply by 0.01 for per 1% change in volatility
    return S * norm.pdf(d1) * np.sqrt(T) * 0.01  


S = 100                     # Stock price
K = 105                     # Strike price
T = 0.5                     # 6 months to expiration
r = 0.05                    # 5% risk-free rate
sigma = 0.20                # 20% volatility

vega_value = black_scholes_vega(S, K, T, r, sigma)  

print("Vega:", vega_value)        

Rho (ρ)

• Measures the sensitivity of the option's price to changes in interest rates.
• For example, a Rho of 0.05 means that if interest rates increase by 1%, the option's price is expected to increase by $0.05.
• Call options have positive Rho, while put options have negative Rho.

Call Options:

• Positive Rho: Call options have a positive Rho. This means that as interest rates increase, the price of the call option is also expected to increase. This is because higher interest rates increase the cost of carrying the underlying asset, making the call option (which gives the holder the right to buy the asset) more valuable.
• Magnitude of Rho: The magnitude of Rho for call options tends to increase with longer time to expiration and higher strike prices (deeper in-the-money). This is because the impact of interest rates is compounded over time, and higher strike prices imply a larger potential loan amount if the option is exercised.

Put Options:

• Negative Rho: Put options have a negative Rho. This means that as interest rates increase, the price of the put option is expected to decrease. This is because higher interest rates increase the cost of carrying the underlying asset, making the put option (which gives the holder the right to sell the asset) less valuable.
• Magnitude of Rho: The magnitude of Rho for put options tends to decrease with longer time to expiration and higher strike prices (deeper in-the-money). This is because the negative impact of interest rates is offset by the increased probability of the option being exercised.

Now, let use python to calculate rho:

Rho for Call Options (ρc):

ρc = K T e^(-rT) N(d2)

Rho for Put Options (ρp):

ρp = -K T e^(-rT) N(-d2)

• K = Strike price of the option
• T = Time to expiration (in years)
• r = Risk-free interest rate (as a decimal)
• N(x) = Cumulative standard normal distribution function of x
• e = Euler's number (approximately 2.71828)

d1 and d2 Calculation (same as for option pricing):

d1 = (ln(S / K) + (r + (σ^2) / 2) T) / (σ √T)

d2 = d1 - σ * √T

• Rho for Call Options (ρc): It is positive, meaning that as interest rates increase, the price of the call option is also expected to increase. This is due to the increased cost of carrying the underlying asset, making the call option more valuable.
• Rho for Put Options (ρp): It is negative, meaning that as interest rates increase, the price of the put option is expected to decrease. This is because higher interest rates make it less attractive to hold a short position in the underlying asset (which is what a put option essentially does).
• Magnitude of Rho: The magnitude of Rho increases with the time to expiration and decreases as the option moves further into the money.

def black_scholes_rho(S, K, T, r, sigma, option_type='call'):
    d1 = (np.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    if option_type == 'call':
        rho = K * T * np.exp(-r * T) * norm.cdf(d2)
    else:  # Put option
        rho = -K * T * np.exp(-r * T) * norm.cdf(-d2)

    return rho * 0.01  # Rho per 1% change in interest rate

S = 100
K = 105
T = 0.5
r = 0.05
sigma = 0.20

call_rho = black_scholes_rho(S, K, T, r, sigma, 'call')
put_rho = black_scholes_rho(S, K, T, r, sigma, 'put')

print("Call Rho:", call_rho)
print("Put Rho:", put_rho)        

SECONDARY OPTION GREEKS

Vanna

• Measures the change in Delta with respect to changes in volatility.
• Useful for assessing the combined effect of volatility and underlying price changes on Delta.
• It helps to understand how vega will change as the underlying asset price fluctuates.

Call Options:

• Positive Vanna: For call options, Vanna is typically positive. This means that as implied volatility increases, the Delta of the call option also increases. In other words, the call option becomes more sensitive to changes in the underlying asset's price. This is generally beneficial for call option buyers in a rising market, as it can amplify potential gains. However, it also means larger potential losses if the underlying asset's price falls.
• Negative Vanna (rare): In some rare cases, usually with deep in-the-money (ITM) call options, Vanna can be negative. This means that as implied volatility increases, the Delta of the call option decreases. This is because the option is already deep ITM, and further increases in volatility are less likely to push it significantly further ITM.

Put Options:

• Negative Vanna: For put options, Vanna is typically negative. This means that as implied volatility increases, the Delta of the put option becomes more negative. In other words, the put option becomes more sensitive to decreases in the underlying asset's price. This is generally beneficial for put option buyers in a falling market, as it can amplify potential gains. However, it also means larger potential losses if the underlying asset's price rises.
• Positive Vanna (rare): Similar to call options, in some rare cases (usually with deep ITM put options), Vanna can be positive. This means that as implied volatility increases, the Delta of the put option becomes less negative.

Charm (or Delta decay)

• Measures the rate of change of Delta over time, with constant price and volatility.
• Useful for understanding how Delta will change as the option approaches expiration

Call Options:

• Positive Charm: A positive Charm for a call option means that Delta will increase over time. This is because as the option approaches expiration, the probability of the option expiring in-the-money (ITM) increases, leading to a higher Delta. This can be beneficial for call option holders, as it means their option's value will become more sensitive to changes in the underlying asset's price.
• Negative Charm: A negative Charm for a call option indicates that Delta will decrease over time. This is more common for out-of-the-money (OTM) call options, as the probability of them expiring ITM decreases as time passes. This decrease in Delta can be detrimental to call option holders, as it means their option's value will become less sensitive to changes in the underlying asset's price.

Put Options:

• Positive Charm: A positive Charm for a put option means that Delta will become more negative over time. This is because as the option approaches expiration, the probability of the option expiring ITM increases, leading to a more negative Delta. This can be beneficial for put option holders, as it means their option's value will become more sensitive to decreases in the underlying asset's price.
• Negative Charm: A negative Charm for a put option indicates that Delta will become less negative over time. This is more common for OTM put options, as the probability of them expiring ITM decreases as time passes. This increase in Delta can be detrimental to put option holders, as it means their option's value will become less sensitive to changes in the underlying asset's price.

Vomma (or Volga)

• Measures the rate of change of vega with respect to changes in implied volatility.
• Indicates how sensitive Vega is to changes in implied volatility.
• It helps to understand how vega will change as implied volatility fluctuates.

Call Options:

• Positive Vomma: A positive Vomma for a call option means that as implied volatility increases, the Vega of the option also increases. This is generally beneficial for call option buyers. It means their option's value will become increasingly sensitive to further rises in volatility, potentially leading to larger gains if the underlying asset's price rises. However, it also means the option's value will be more negatively affected if volatility decreases.
• Negative Vomma: A negative Vomma for a call option indicates that Vega decreases as implied volatility increases. This is generally unfavorable for call option buyers. It means their option's value will become less sensitive to volatility changes, potentially limiting potential gains in a rising volatility environment.

Put Options:

• Positive Vomma: Similar to call options, a positive Vomma for a put option means that Vega increases as implied volatility rises. This is beneficial for put option buyers, as their option's value will become increasingly sensitive to further rises in volatility, potentially leading to larger gains if the underlying asset's price falls. However, it also means the option's value will be more negatively affected if volatility decreases.
• Negative Vomma: A negative Vomma for a put option indicates that Vega decreases as implied volatility increases. This is generally unfavorable for put option buyers. It means their option's value will become less sensitive to volatility changes, potentially limiting potential gains in a rising volatility environment.

Speed

• Measures the rate of change of gamma with respect to changes in the underlying asset's price.
• It helps to predict how gamma will change as the underlying asset price fluctuates.

Call Options:

• Positive Speed: A positive Speed for a call option means that Gamma will increase as the underlying asset's price increases. This implies that the option's Delta will become even more sensitive to price changes as the underlying asset's price rises. In a rising market, this can lead to larger gains for call options due to the accelerating Delta. However, it also means that losses can be magnified if the underlying asset's price unexpectedly falls.
• Negative Speed: A negative Speed for a call option indicates that Gamma will decrease as the underlying asset's price increases. This means the option's Delta will become less sensitive to price changes as the underlying asset's price rises. This can limit potential gains in a rising market but also provide some protection against significant losses if the underlying asset's price falls.

Put Options:

• Positive Speed: A positive Speed for a put option means that Gamma will increase as the underlying asset's price decreases. This implies that the option's Delta will become more sensitive to price changes as the underlying asset's price falls. In a falling market, this can lead to larger gains for put options. However, it also means that losses can be magnified if the underlying asset's price unexpectedly rises.
• Negative Speed: A negative Speed for a put option indicates that Gamma will decrease as the underlying asset's price decreases. This means the option's Delta will become less sensitive to price changes as the underlying asset's price falls. This can limit potential gains in a falling market but also provides some protection against significant losses if the underlying asset's price rises.

Zomma

• Zomma, sometimes called DgammaDvol, measures the rate of change of Gamma with respect to changes in implied volatility. It indicates how sensitive Gamma is to changes in implied volatility, providing insights into how the option's price behavior (Delta) will change when volatility fluctuates.

Call Options:

• Positive Zomma: A positive Zomma for a call option means that Gamma will increase as implied volatility increases. This implies that the option's Delta will become even more sensitive to changes in the underlying asset's price as volatility rises. In a volatile market with an upward trend, this can lead to larger gains for call options due to accelerated Delta. Conversely, if the underlying asset's price falls, losses can also be magnified.
• Negative Zomma: A negative Zomma for a call option suggests that Gamma will decrease as implied volatility increases. This means the option's Delta will become less sensitive to changes in the underlying asset's price as volatility rises. This can limit potential gains in a rising market but also provide some protection against significant losses in a falling market.

Put Options:

• Positive Zomma: A positive Zomma for a put option also indicates that Gamma will increase as implied volatility increases. This implies that the option's Delta will become more sensitive to changes in the underlying asset's price as volatility rises. In a volatile market with a downward trend, this can lead to larger gains for put options. However, if the underlying asset's price rises, losses can also be magnified.
• Negative Zomma: A negative Zomma for a put option suggests that Gamma will decrease as implied volatility increases. This means the option's Delta will become less sensitive to changes in the underlying asset's price as volatility rises. This can limit potential gains in a falling market but also provide some protection against significant losses in a rising market.

Color

• Color measures the rate of change of Gamma over time. Gamma, as you know, measures the rate of change of Delta (an option's price sensitivity to the underlying asset's price). In essence, Color tells us how quickly Gamma will change as time passes, especially as the option approaches its expiration date.

Call Options:

• Positive Color: A positive color for a call option indicates that Gamma will increase as time passes. This means the option's delta will become more sensitive to changes in the underlying asset's price. In a rising market, this can be beneficial as it amplifies potential gains. However, it also increases the risk of larger losses if the underlying asset's price falls.
• Negative Color: A negative color for a call option suggests that Gamma will decrease over time. This means the option's delta will become less sensitive to changes in the underlying asset's price. While this may limit potential gains, it also provides some protection against substantial losses if the underlying asset's price declines.

Put Options:

• Positive Color: A positive color for a put option also indicates that Gamma will increase as time passes. This means the option's delta will become more sensitive to changes in the underlying asset's price. In a falling market, this can be beneficial as it amplifies potential gains. However, it also increases the risk of larger losses if the underlying asset's price rises.
• Negative Color: A negative color for a put option suggests that Gamma will decrease over time. This means the option's delta will become less sensitive to changes in the underlying asset's price. While this may limit potential gains, it also offers some protection against substantial losses if the underlying asset's price increases.

Ultima

• Measures the rate of change of vomma with respect to changes in implied volatility.
• It helps to understand how vomma will change as implied volatility fluctuates.

Call Options:

• Positive Ultima: A positive ultima for a call option indicates that the vomma will increase as implied volatility increases. This means the option's vega (sensitivity to volatility) will become even more sensitive to changes in volatility as volatility rises. In a volatile market, this can lead to larger gains for call options if the underlying asset's price increases. However, it also means larger losses if the underlying asset's price decreases.
• Negative Ultima: A negative ultima for a call option suggests that vomma will decrease as implied volatility increases. This means the option's vega will become less sensitive to changes in volatility as volatility rises. In this case, the impact of increasing volatility on the option's price will be less pronounced.

Put Options:

• Positive Ultima: Similar to call options, a positive ultima for a put option indicates that vomma will increase as implied volatility increases. This means the option's vega will become even more sensitive to changes in volatility, leading to potentially larger gains if the underlying asset's price falls in a volatile market. However, it also means larger losses if the underlying asset's price increases.
• Negative Ultima: A negative ultima for a put option suggests that vomma will decrease as implied volatility increases. This means the option's vega will become less sensitive to changes in volatility as volatility rises, limiting the impact of increasing volatility on the option's price.

Lambda

• It is also known as leverage, lambda measures the percentage change in an option's price for a 1% change in the underlying asset's price. It essentially quantifies the leverage inherent in an option position.
• A high lambda indicates that the option is highly leveraged. This means that even small changes in the underlying asset's price can lead to significant changes in the option's value. While this offers the potential for high gains, it also comes with increased risk.
• A low lambda suggests that the option is less leveraged. This means the option's price will be less sensitive to changes in the underlying asset, offering lower potential gains but also lower risk.
• Higher lambda means higher potential gains (and losses).
• Lower lambda means lower potential gains (and losses).
• Both call and put options can exhibit high or low lambda, depending on their strike price, time to expiration, and the current price and volatility of the underlying asset.

Lambda for Call Options:

• High Lambda: A high lambda for a call option means it is highly leveraged. This translates to a larger percentage change in the option's price for a relatively small percentage change in the underlying asset's price. For example, if a call option has a lambda of 5, a 1% increase in the underlying stock price could lead to a 5% increase in the call option's value. While this magnifies potential gains, it also significantly amplifies the potential for losses if the stock price moves against you.
• Low Lambda: A low lambda for a call option signifies lower leverage. This means the option's price will be less responsive to changes in the underlying stock price. While this limits the potential for outsized gains, it also provides a degree of protection against substantial losses if the stock price declines.

Lambda for Put Options:

• High Lambda: A high lambda for a put option similarly indicates high leverage. This implies that a small percentage decrease in the underlying stock price could lead to a significant percentage increase in the put option's value. This is because put options become more valuable as the underlying stock price falls. However, the high leverage also increases the potential for losses if the stock price unexpectedly rises.
• Low Lambda: A low lambda for a put option signifies lower leverage. This means the put option's price will be less sensitive to downward movements in the underlying stock price. While this limits the potential for significant gains if the stock price drops, it also offers more protection against losses if the stock price unexpectedly increases.

Vera

• Vera (also known as Rhova), measures the rate of change of rho with respect to changes in implied volatility.
• It helps to understand how rho (the sensitivity to interest rates) will change as implied volatility fluctuates.
• A high vera suggests that the option's sensitivity to interest rates (rho) will increase significantly as implied volatility rises. This is important because it can help traders anticipate how their option positions might react to changes in both interest rates and volatility.
• A low vera indicates that the option's sensitivity to interest rates (rho) will be less affected by changes in implied volatility. This can be valuable information for traders who are primarily focused on interest rate risk rather than volatility risk.

Call Options:

• High Vera: For call options, a high vera suggests that as implied volatility increases, the option's sensitivity to interest rates (rho) will also increase significantly. This means that if interest rates rise, the price of the call option will increase even more than it would if volatility was low. This is because higher volatility increases the potential upside of the underlying asset, making the call option more valuable in a rising interest rate environment.
• Low Vera: A low vera for call options indicates that the option's sensitivity to interest rates (rho) won't be heavily influenced by changes in implied volatility. So even if volatility increases, the impact of interest rate changes on the call option's price will remain relatively stable.

Put Options:

• High Vera: For put options, a high vera means that as implied volatility increases, the option's sensitivity to interest rates (rho) will become more negative. This implies that if interest rates rise, the price of the put option will decrease even more than it would if volatility was low. This is because higher volatility increases the potential downside of the underlying asset, making the put option less valuable in a rising interest rate environment.
• Low Vera: A low vera for put options suggests that the option's sensitivity to interest rates (rho) won't be significantly affected by changes in implied volatility. Therefore, even if volatility increases, the impact of interest rate changes on the put option's price will remain relatively consistent.

Epsilon

• While not a widely used Greek, epsilon measures the percentage change in an option's price for a 1% change in the dividend yield of the underlying asset. It's primarily relevant for options on dividend-paying stocks.

Call Options:

• A positive epsilon indicates that the call option's price will increase if the dividend yield increases. This is because higher dividends can make the stock more attractive to investors, potentially driving up its price. Since call options give the holder the right to buy the stock at a predetermined price (the strike price), a higher stock price increases the potential profit for the call option holder.
• A negative epsilon means that the call option's price will decrease if the dividend yield increases. This is because higher dividends can reduce the stock's potential for price appreciation. If investors are receiving a larger portion of their return in the form of dividends, there may be less upward pressure on the stock price, which can negatively impact the value of call options.

Put Options:

• A positive epsilon indicates that the put option's price will decrease if the dividend yield increases. This is because higher dividends make the stock more attractive and can potentially reduce the likelihood of the stock price falling below the put option's strike price.
• A negative epsilon means that the put option's price will increase if the dividend yield increases. This is because higher dividends can potentially hinder the stock's price growth, increasing the likelihood of the stock price falling below the put option's strike price, which is beneficial for put option holders.

Thank you for reading!!

References

• What Are Option Greeks? by TrueData (https://www.truedata.in/blog/what-are-option-greeks/)
• Option Trading School: What the Heck is Vega and Do I Need to Care? by BigTrends (https://thetradinganalyst.com/vega-options/)
• Chapter 12: Option Greeks Part 2 by ICICI Direct (https://www.icicidirect.com/ilearn/stocks/courses/chapter-13-option-greeks-part-3)
• Learn & Understand Vega in Options by Merrill Lynch (https://www.merrilledge.com/investment-products/options)
• Vega by OptionTradingTips (https://m.youtube.com/watch?v=zqKllafB5ig)
• Rho by OptionTradingTips (https://www.optionstrading.org/improving-skills/greeks/rho/)
• Free Option Greeks by OptionTradingpedia (https://www.optiontradingpedia.com/free_option_greeks.htm)