Greeks are factor sensitivities of a derivative price to a unit change in different input variables. They are also referred to as dimensions of risk measures or hedge parameters for any given derivative position. These measures are used by hedgers and speculators in Options trading to gain insight into the perceived risk/reward exposure.
Before we get into the detail of Greeks, it is important to understand the fundamentals of option pricing. An option is a derivative of an underlying asset. Black–Scholes formula (1973) for European call and put option helps us understand the underlying structure of the option. Below are the formulas for a call and put option (Black and Scholes (1973)).
Here, is the cumulative distribution function of the standard normal distribution. Similarly, S = Underlying Price, X = Strike Price, t = Time in years until Maturity, r = continuously compounded risk-free interest rate, and σ= Implied Volatility of underlying stock. We will be using these notations throughout our article.
Looking at the equation (1) and (2), we can infer that the theoretical value of Call price or a Put price changes based on the following input variables
- Price Changes to the Underlying Security (S)
- Time remaining until Expiration (t)
- Alterations in Interest Rates (r)
- Volatility Changes ( – Supply and Demand for the Option)
Option Greek gives us insight into how much change can be expected when there is a unit change in these input variables. It would answer questions such as how much risk are we being exposed to by getting into an option position? Given an option position, we can use Greeks and understand our risk/reward exposure to each input parameter. Out of many Greeks that are out there, some are the first-order derivate of the above input variable, while others are second or third derivate. However, out of many such, five Greeks get used a lot and are considered primary option Greeks (Passarelli, Dan (2012)). Traders use these 5 Greeks to either speculate an opportunity or hedge their position. These 5 primary Greeks are listed below.
- Delta
- Gamma
- Vega
- Theta
- Rho
Now, let us get familiar with some acronyms that we will be using in this paper.
OTM: When an Underlying Asset price is lower than the strike price. It means if we were to execute our options today, it would have no value.
ITM: When an Underlying Asset price is higher than the strike price. It means if we were to execute our options today, it would have a value equal to the difference.
ATM: When an Underlying Asset price is around the strike price.
Also, before we get into each of these Greeks, let us have a look at an Option Greek Snapshot of a random stock. The following page shows an option Greek Snapshot for XPSA.
Option Greeks Snapshot
I have randomly picked an option chain for equity stock XPSA currently being traded on the NASDAQ exchange and will mature on December 18, 2020. We will also randomly pick a strike price of X=2.50. Now looking at the live chart snapshot, we can see that Greek values for both calls and puts option for a strike price of x=2.50.
Figure 1:
Call: Delta = 39.43%, Gamma = 0.39147, Rho = 0.00062, Theta = -0.00453, Vega = 0.00248
Put: Delta = -52.54%, Gamma = 0.30929, Rho = -0.00221, Theta = -0.00602, Vega = 0.00257
DELTA
Delta is the partial derivative of the option price with respect to the underlying security’s price. It is also referred to as sensitivity of the option price with regard to changes in the underlying price. If the options are priced using the BSM model, the call delta is given by the expression (Black and Scholes (1973)).
Since the distribution in Equations 5 and 6 follows the standard normal distribution, the value of Call Delta ranges between 0 and 1while the value for put delta ranges between 0 and -1. It is used during hedging as a hedge ratio. Details on how this is used during hedging are listed in our hedging section.
In simple terms, this delta is the amount by which the price of an option changes when there is a $1 increase in the price of the underlying security. It can be expressed as a decimal or percentage.
Referring to our Option Greek Snapshot, XPSA had a Call Delta of 39.43% or 0.3943 and a Put Delta of -52.54% or 0.5254. This means that if the stock price of XPSA goes from 1.90 to 2.90, then the call option price would be 0.21 (current price) + 0.3943 (call delta) = 0.60 and the put option price would be 0.89 (current price) + [-0.5254] (put delta) = 0.3644.
Below are the Graphs of Call and Put Delta vs stock price. This will help us understand the behavior of Delta when an option is in OTM, ATM, and ITM. The strike price for both the call and put option is 300. In the Graph below, we can see how Call Delta ranges from 0 and grows to 1, and Put delta starts at 0 and shrinks to -1. Also, we can see how the Delta changes rapidly during the moment when the stock price is approaching the strike price and changes very slowly when it is far from the Strike Price i.e., either in the money (ITM) or out of the money (OTM).
Figure 2 & 3:
GAMMA
Gamma is the partial derivate of Delta with respect to the underlying security. If the options are priced using the BSM model, the call Gamma is given by the expression (Black and Scholes (1973)).
And the put gamma is.
In simpler terms, Gamma is the amount that Delta changes as compared to a $1 increase in the price of the stock which may be important where Delta becomes particularly sensitive to changes in the stock price. Gamma values for both call and put are always positive, however rarely negative. We will see in graphs below how it is high for at-the-money options and is roughly normally distributed.
Referring to our Option Greek Snapshot, XPSA had a Call Delta of 39.43% or 0.3943 and a Put Delta of -52.54% or 0.5254. It also had a Gamma of 0.39147 for call and a Gamma of 0.30929 for the put option. These values means that If the stock price of XPSA goes from 1.90 to 2.90, then the call Delta would be 0.3943 (current delta) + 0.39147 (call gamma) = 0.78577 and the put option price would be 0.5254 (current delta) + [-0.30929] (put gamma) = 0.21611.
Below are call and put Gamma behaviors for the same option with a strike price of 300
Figure 4 & 5:
We can see how they are at a maximum when the option is at the money. In addition to this, the value of Gamma keeps on increasing as you approach the expiration date. This web article uses a 3d model to illustrate this more intuitively. [https://medium.com/hypervolatility/options-greeks-delta-gamma-vega-theta-rho-23f0321b64ba]
VEGA
Vega is the partial derivate of the option value with respect to implied volatility. Implied volatility () is the market expectation of deviation from the current price of the underlying asset until the expiration date (Sinclair, Euan (2010)). If the options are priced using the BSM model, the call Vega is given by the expression (Black and Scholes (1973)).
Both Call Vega and Put Vega can be derived using the formula above.
In Simple terms, VEGA is the theoretical change in the amount of the option price with an increase in unit volatility.
Referring to our Option Greek Snapshot in Figure 1, XPSA had a Call Vega of 0.00248 and a Put Vega of -0.00248. This means that if the implied volatility of XPSA goes from 152.53 to 153.53(1 percent increase), then the call option price would be 0.21 (current price) + 0.00248 (call Vega) = 0.21248 and the put option price would be 0.89 (current price) + 0.00248 (put Vega) = 0.89248
Figure 6 & 7:
In the above chart of Vega vs underlying security prices, we can see how Vega is higher around the strike price. Additionally, it also keeps on decreasing as you move towards your expiration date as the market expectation of deviation from the current price lessens.
THETA
Theta is the negative of the partial derivate of the option value with respect to time. It is also referred to as Time decay. We all know that an option has an intrinsic component and a time value component. This time value component decays with the passage of every day and eventually bring down the time value component of the option price to 0 (Farid, Jawwad (2015)). If the options are priced using the BSM model, the call theta is given by the expression (Black and Scholes (1973)).
And the put theta is
Once we calculate the Theta value from the above equation, we will have to divide it by 365 to get the daily decay rate. Also, since this works against the option price, the value of Theta is always negative.
Referring to our Option Greek Snapshot in Figure 1, XPSA had a Call Theta of -0.00453and a Put Theta of –0.00602. This means that if nothing else changed but another day passed, then the call option price would be 0.21 (current price) + 0.00453 (call theta) = 0.20547 and the put option price would be 0.89 (current price) + 0.00602 (put theta) = 0.88398
Figure 8 & 9:
In the above Graphs, we can see how Call Theta behaves differently at three different money stages. It decays maximum when the option is at the money and when closer to the expiration date.
RHO
Rho is the partial derivative of the option value with respect to the interest rate. If the options are priced using the BSM model, then call rho is given by the expression (Black and Scholes (1973)).
And the put rho is given by
Rho is normally expressed as a change in value when one percent point changes. Thus, we will need to divide it by 100 after calculating rho in the above equation.
In Simple terms, Rho is the price of an option changes with respect to a unit increase in the risk-free rate (i.e., short term US Treasury Bill rate). Also, since options are short-termed and the fact that interest costs add up for a smaller component of the overall change in the option price, they are not as popular as others (Turitto, Vito (2018)).
Referring to our Option Greek Snapshot in Figure 1, XPSA had a Call Rho of 0.00062 and Put Rho of -0.00221. This means that if nothing else changed but risk free interest rate increased by 1%(100 points), then the call option price would be 0.21 (current price) + 0.062 (call rho*100) = 0.272 and the put option price would be 0.89 (current price) + [-0.221] (put rho*100) = 0.669
Figure 10 & 11:
Above are the graphs for put and call Rho. In the above graphs, we can see how Rho grows fast around Strike prices.
Now that we have gone through 5 primary Greeks, we will see how these measures are applied during hedging and speculation.
Hedging
Among all the risk exposures expressed while dealing with securities, Delta risk exposure, or directional risk accounts for most of the risk (Turitto, Vito (2014)). Below are some strategies Hedgers use to hedge their options or portfolio.
Delta–Hedging
Delta- hedging is an options strategy where a hedger tries to attain a risk-neutral position. This section follows closely chapter 12 of Dan Passarelli (2012). Let us say that you have 1 Put Options with 100 shares and a delta of -0.25 trading at $100 priced at $4000. You can hedge this risk by going long on underlying assets. For the above scenario, you would need to buy (Delta * S) that is (0.25* 100) = 25 shares of the underlying asset.
Let us say that if the underlying asset goes up to $140. Since the stock went up to $40, your put-call would be now valued at (-0.25*40*100) = -1000. Since you had hedged the stock by buying 25 shares of the underlying asset using delta. You would gain 40*25= $1000. This is how we hedge a put option by inferring to its delta and avoid directional risks.
Delta–Gamma Hedging
This section follows closely the article of Turitto, Vito (2014), “Option Greeks and Hedging Strategies.”. Since the Delta Values are not linear and change from near 0 when out of the money and neat 1 when at the money or deep in the money, we need to constantly re-evaluate our put option’s delta and re-hedge it by short or long positions. We would run into the same problem as with delta hedging where we will have to adjust our position. However, with both Delta-Gamma hedged, we will have to adjust fewer times. For example, let us say that we have the same put option with 100 shares and a delta of -0.25, gamma of -0.0125, trading at $100 price at $4000. We can hedge this negative gamma of – (0.0125) by buying another call option with similar gamma. Thus, we would attain a gamma neutral position. However, buying another call option would incur an additional delta. Let us say that the delta for this call option is 0.22. Now we will need to hedge out aggregate delta. i.e. -0.25 + 0.22 = -0.03. meaning we will be buying (0.03*100) = 3 shares. This hedge would cover both call and put options.
Speculation
These risk exposures expressed by Option price sensitiveness also provides an opportunity for speculators to receive better rewards.
Delta Rewards
Speculators are more likely to seek out options with comparatively higher delta. They will now be able to calculate expected profit based on expected movements and find opportunities in the market (Ursone, Pierino (2015)). For example, let us say that a speculator is bullish on 2 different stock options where one has a delta of 0.5(expected rise $10) and another has a delta of 0.45 (expected rise $10). In this case, the speculator will be able to know that he would be making $5 in one while $4.5 in another for the same price movement. This way, the speculator will be able to pick the better option. Since the market tends to be irrational, a speculator with this information will be able to leverage their speculation.
Vega Rewards
Higher Vega or Lower Vega can be used by speculators in terms of their opinion of future Implied volatility (Ursone, Pierino (2015)). So, if a speculator anticipates that IV would increase, then he is better off selecting stocks with higher Vega. A 1% increase in IV would yield a higher return for the speculator. Similarly, If the speculator anticipates that IV would recede, then the speculator can select options with lower Vega and Write Calls or Puts.
Summary
Greeks help a trader to either acquire a position or manage their position by showing them option’s exposures to changes in Time (Theta), Price (Delta/Gamma), Volatility (Vega), Interest Rates (Rho). These also help trader examine their existing exposure to various option’s centric risks. One key thing to note is that these Greek values are dynamic and are constantly changing. In addition to this, they display different behaviors at various money stages. i.e. ITM, OTM, and ATM. In all our examples, we assumed that change in 1 Greek did not create a cascade and affected other Greeks. But in reality, Changes in one are more likely to affect other Greek and make it a little complex. However, after understanding these values and assessing all the risks and rewards, we will be making more intelligent trade decisions.
Works Cited
Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3), 637-654
Banerji, Gunjan (2020), “The Invisible Forces Exacerbating Market Swings.” The Wall Street Journal, Dow Jones & Company, www.wsj.com/articles/the-invisible-forces-exacerbating-market-swings-11582804802
Davies, Paul J (2019), “Markets Are Calm, Then Suddenly Go Crazy. Some Investors Think They Know Why.” The Wall Street Journal, Dow Jones & Company, www.wsj.com/articles/markets-are-calm-then-suddenly-go-crazy-some-investors-think-they-know-why-11562666400.
Farid, Jawwad (2015), An Option Greeks Primer Building Intuition with Delta Hedging and Monte Carlo Simulation Using Excel. Palgrave Macmillan, 59-78
Leoni, Peter (2014), The Greeks and Hedging Explained. Palgrave Macmillan, 24-39
Passarelli, Dan (2012). Trading Option Greeks: How Time, Volatility, and Other Pricing Factors Drive Profits. Bloomberg Press, 229-264
Ravo, Nick (2020), “These Options Terms Are Greek to You.” The Wall Street Journal, Dow Jones & Company, www.wsj.com/articles/these-options-terms-are-greek-to-you-11601858160.
Rosenberg, Yuval (2009). “Volatility Indexes Offer Hedge for Investors.” CNNMoney, Cable News Network, money.cnn.com/2009/02/27/magazines/fortune/volatility_index_notes.fortune/index.htm?postversion=2009030213.
Sinclair, Euan (2010), Option Trading: Pricing and Volatility Strategies and Techniques. Wiley, 63-88
Turitto, Vito (2014), “Option Greeks and Hedging Strategies.” Medium, HyperVolatility, medium.com/hypervolatility/option-greeks-and-hedging-strategies-14101169604e.
Turitto, Vito (2018), “Options Greeks: Delta, Gamma, Vega, Theta, Rho.” Medium, HyperVolatility, medium.com/hypervolatility/options-greeks-delta-gamma-vega-theta-rho-23f0321b64ba.
Ursone, Pierino (2015), How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega. Wiley. 95-108 Holton, Glyn (2013). “Black-Scholes (1973) Option Pricing Formula.” GlynHolton.com, 3 June 2013, www.glynholton.com/notes/black_scholes_1973/.