In order to have a deep understanding of how option prices are determined, we need to focus on the response of option prices to the factors that determine its price. As we noted above, the price of an European Option is determined by the price of the underlying stock, the exercise or strike price of the option, the interest rate, the volatility of the underlying stock, and the time to expiration. 

The factors that measure the impact on options prices given changes on the determinants of the options prices are called greeks. The greeks are a group of parameters that account for the sensitivity in changes of options prices given changes in the parameters that determine the price in the Black Scholes model. In this post we will learn how to calculate various options greeks in R programming language.

There are five important Greeks namely Delta, Theta, Vega, Gamma and Rho. Each of these is related to one of the variables that determine the option price. We will now analyze each of these greeks.

Delta

Delta is the first greek parameter and measures the sensitivity in option’s prices to changes in stock prices. The delta parameter is always positive for call options because if the stock price increases, the option prices will increase. It is always negative for put options because the option value would decrease when stock prices increase.

The logic behind this can be explained. If the stock price increases, the payoff of the call option would increase as the difference between the stock price and strike price is higher. On the other hand, if the stock price increases, the difference between the put strike and the stock price would increase, and the value of the put option will lower.

If a Delta call is 0.55 and a Delta put is -0.42, and the stock price increases by USD 1, the price of the call will rise approximately by USD 0.55 and the price of the put will fall by about USD 0.42. Delta is the most important parameter of the greeks and many times it is used to build Delta Neutral Portfolios that are portfolios which are slightly affected by changes in the underlying stock price.   

It Is interesting to observe that the option sensitivity with respect to stock price, i.e., Delta, is always between -1 and 1 (positive for call options and negative for put options), and it changes its values according to whether the option is out-of-the-money, in-the-money or at-the-money. The Delta for a call option tends to 1 when the call option is deep in-the-money. Similarly, the Delta approach to zero when the call option is deep out-of-the-money. Finally, Delta is higher when the stock price is near the exercise price or the option is at-the-money.

For a put option the same behavior applies. The Delta for a put option tends to -1 when the option is deep in-the-money, while the Delta is 0 when the option stands deeper out-of-the money.

Theta

Theta describes the sensitivity of option prices with respect to the passage of time. Theta parameter is the negative of the first derivative of the option price with respect to the time to expiration. Time is an important variable in options prices because when the option is near expiration date the option value tends to decrease as there are less chances of price movements in the stock that might cause a profitable position at expiration. 

This impact that the passage of time has on the option prices, is also known as time decay. Generally, Theta is negative for calls and put options but in some circumstances such as deep in-the-money put options, the Theta can be positive. Also the time decay is different between at-the-money and out-of-the money options.

Theta value is generally less than zero for both put and call options. The value of Theta is also related to the moneyness position of the option. For options that are deep out-of-the-money, Theta will increase more in dates far from expiration than at-the-money options whose Theta would decay near expiration. 

Also the Theta value is not linear along the contract option life. When expiration time is far away, time decay causes small changes in option prices but when the time to expiration is near, the time decay has a great impact on the option price and Theta tends to increase. 

When Theta is -12.5432 and a call option is priced at USD 5.5, the Theta can be interpreted as the loss that the option would have in each day until expiration. The Theta parameter should be interpreted in terms of year, so 1 day is equivalent to 1/365 = 0.00273973. Therefore, with a Theta of -12.5432, in each day until expiration the option would loss -12.5432 * 0.00273973 = USD 0.03436 per day if the stock price remains at the same level.

The volatility tends to neutralize the effect of Theta. This is the case when an earnings announcement is coming, the volatility increases and the option value would increase too, even though the time to expiration of the option is near.

Vega

This parameter measures the sensitivity of option prices to changes in the volatility of the underlying stock. The value of Vega is the change in the option price given a variation of 1% in the Implied Volatility. Vega has a positive impact on both put and call options prices.

As with the other Greeks, the Vega parameter is obtained by Option Pricing models such as Black Scholes model.  If we have a call option with price of USD 10.8, when the volatility of the underlying stock increases from 0.2 to 0.4 and the Vega is 15, we can calculate the change in option prices with the following formula:

10.8 (Call Price) + 0.2 (Volatility Change) * 15 (Vega) = USD 13.8 (New option price)

The above formula explains how the increase in volatility of a stock price impacts the option price by the Vega amount. 

Just like other Greeks, the Vega is different for options with different strike prices. Vega tends to be high for options that are at-the-money, while Vega is lower for calls and puts options that are both deeper in-the-money and out-of-the money.

Time to expiration also affects Vega. If the time to expiration is far away from the current date, the Vega tends to be higher.  This is because the more days we have until expiration, more time exists for a big move and therefore more uncertainty in the path of the underlying. Finally if the Implied Volatility increases, this also affects Vega, as there is a higher chance that the option expires near the money. 

Gamma

The Gamma parameter measures how much the Delta parameter changes due to changes in stock prices. In terms of calculus, Gamma is the second derivative of changes in option prices related to changes in stock prices.

Gamma value is the same for a put or a call option and can be either positive or negative. When stock prices change, we can obtain the new value of Delta using the Gamma parameter and changes in prices. 

Suppose Gamma for a call and a put is 0.077 and Delta Call is 0.612 while Delta put is -0.453. If stock price move from USD 150 to USD 152, the new values of Deltas would be:

Delta Call = 0.612 + 2 * 0.077 = 0.764 

Delta Put = -0.453 +2 * 0.077 =  -0.296

A large Gamma for a given option price means that Delta sensitivity is large with respect to changes in stock prices. Gamma tends to be high for options near the money, as on the other hand Gamma is small for options deep in-the-money and options deep out-of-the money. 

When an option is deep in-the-money, it will move similar like a stock and the Delta for this option is closer to 1. In this case, the Gamma value would be small. The options that are near the money would have Deltas around 0.5 and the Gamma for these options would be high. To conclude, we can claim that as an option moves further in-the-money or out-of-the-money, the Gamma value tends to decrease.

Gamma also varies with the time remaining until expiration. For options that are near the money, Gamma increases as time to expiration approaches. On the other hand Gamma will fall when expiration time is near for deep in-the-money and deep out-of-the-money contracts.

Rho

Rho is the change in options prices due to changes in interest rate. For call options, this parameter is always positive and for put options this parameter is always negative. Generally this parameter is the less significant among the Greeks. If we have a call option with a RHO value of 15.65 and the interest rate increases by 1%, then the call price should increase by 0.01 * 15.65 = USD 0.1565.

Just like the other Greeks, the Rho changes as a function of the stock prices and the time to expiration. For call options that are deep in-the-money, the Rho value tends to be high, and for call options that are deep out-of-the-money the Rho value is small. Rho is more sensitive when the option is near the money.

In case of put options, when the put is deeper in-the-money the Rho value tends to decrease, and when the put is deeper in-the-money, the Rho value increases. As in the case for calls options, the Rho for put options is more sensitive for put options that are near the money.

The passage of time also affects this parameter, and the Rho for calls and puts tends to zero when the options approach expiration.

Greeks Combination to Produce Neutral Portfolios

Greeks parameters can be combined by having multiple positions and be used to create portfolios with different behavior. In the above analysis, we study the Greeks by the impact that they have in one position such as a long call or put option. However, many kinds of portfolios can be built that combine short and long options positions as well as options with different expiration times.

If someone wishes to construct a portfolio to hedge against stock price movement, they can buy a call and a put position in order to get zero Delta. In this case, changes in stock price would not affect the portfolio value. Another example is to build a portfolio that is neutral in Delta but bet high to the volatility and so has a high value in the Vega parameter.

The time varying parameter, Theta, is negative when someone holds a put or a call option, as the passage of time impacts the option prices adversely. Someone can combine a long and a short position with puts or calls at different expirations and obtain a positive Theta for that portfolio.  This is called the calendar spread, which can be performed separately for call and for put options.