The Greeks—delta, gamma, theta, vega and rho—are five variables that help identify the risks of an option position. The risks investors face in options are not one-dimensional. In order to deal with changing market conditions, an investor should be aware of the magnitude of these changes. To see if the changes are large or small, whether they create a major or minor risk, option theory and option pricing models provide investors with variables identifying the risk characteristics of their option position. These variables are referred to as the Greeks. There are five Greeks that we monitor: delta, gamma, theta, vega and rho.

Because the Greeks are derivatives of the Black & Scholes formula, we will begin by explaining some more about that.

The Black and Scholes formula, sometimes known as the Black, Scholes and Merton formula, is the market standard tool for pricing options. This formula prices option as a function of the current stock price S_{0}, the time to maturity of the option T, its strike X, volatility σ and interest rate r:

call = S^{0}N(d_{1}) - Xe^{-rT} N(d_{2})
put = Xe^{-rT} N(-d_{2}) - S_{0}N(-d_{1}) with

where N(x) is the cumulative normal distribution function for the standard normal distribution, i.e. the probability that a random variable~N(0,1) (with a standard normal distribution) is less than x. Before discussing the formula, let’s state the underlying assumptions. The Black and Scholes formula assumes:

- Returns are IID (independent and identically distributed) with a normal distribution.
- Future volatility is known and constant.
- Future interest rate is known, constant and the same for borrowing and lending.
- Stock path is continuous, and continuous trading is possible.
- Transaction costs are null.

To develop the theory we assume all these assumptions hold. This formula is the market standard because it is extremely robust with respect to violations of its assumptions.

The first Greek that will be discussed is the delta. Basically the delta is the sensitivity of an option’s theoretical value to a change in the price of the underlying contract. More straightforward, the delta is the change of the value of an option when the underlying value rises by 1 dollar. For example:
Δ_{call} =∂c/∂S = N(d_{1}) and Δ_{put} = ∂p/∂S = N(d_{1}) – 1,
with N(d_{1}) as in the BS formula.
The value of a call option increases when the stock price goes up, so the delta of a call option is positive. Conversely the value of a put option decreases when the stock price goes up, so the delta of the put option is negative.

One can note that N(x) is a probability density function, so it takes value in [0,1]. The delta of one call is then always in [0,1] and delta of one put in [-1,0]. Because the underlying level is usually 100 stocks the delta of the option is multiplied by 100. For example an option with a delta of 0.25 is regarded as a delta 25. The higher the delta the more similar the change of the value of the option will be to the underlying stock. The value of an option with delta 100 will move exactly at the same rate as the underlying stock. Note also that the derivative operation is linear so we can compute delta of each option and sum them to get delta of the whole portfolio (it then can be outside of [-1,1] of course).

When an option gets closer to expiration, its delta will change, since the probability of expiring in or out of the money changes and the normal distribution narrows and centres around the mean. As an option gets closer to expiration, in-the-money options will move towards delta 100 and out-of-the-money options will move towards delta 0. At-the-money options, on the other hand, will stay around delta 50.

As the underlying stock changes in price, delta changes as well. This is to be expected as d_{1} is a function of the stock price.

A practical interpretation of the delta is the hedge ratio: the amount of shares that should be bought or sold to neutralize the directional risk of an option. From the BS formula we can see another interpretation. Roughly speaking, we can say that the delta of an option is its probability of expiring in the money. (For a put we will take absolute value). This approximation only works for European options though.

Summarizing: there are three interpretations of delta:

- The change in value of an option if the underlying increases by 1 dollar.
- The hedge ratio: the number of shares to be bought or sold to neutralize the directional risk of the position.
- The chance that the option will be in-the-money on expiration

→ OTM calls: delta tends to 0 as we approach expiration.

→ ITM calls: delta tends to 100 as time passes.

As the volatility increases (decreases) the delta of a call goes towards (away from) 0.50 and the delta of a put towards (away from) -0.50. So if the volatility rises (decreases) the delta of an in the money option decreases (increases). In case of an out of the money option this is exactly the opposite.

As time decays the delta of a call moves away from 0.50 and the delta of a put away from -0.50. As the time goes by the delta of an in the money call moves towards 1 and the delta of an out the money towards 0.

Gamma is the derivative of delta as a function of the stock price. Since delta is the derivative of the option value as a function of the underlying stock, gamma is the change of delta when the stock price increases by 1 dollar. It is written as follows:

Γ = δ_{2}c/δS^{2} = N'(d_{1}) / S_{0} σ √T

with d_{1} as in the BS formula and N’ the first derivative of the Gaussian cumulative density function, that is the usual Gaussian density:

One often says that gamma reaches its maximum value when an option is ATM. This is correct as a first approximation however the real maximum is reached when the stock price is just below the strike price. This effect is shown in the left part of the figure above for a stock trading at 100 dollar. Given a strike X, volatility σ , a rate r and a time to expiration T, the stock value giving maximum gamma is S _{max Γ}= Xe ^{-(r+3σ^2/2)T}.

The gamma curve of a call and a put are identical. This is consistent with what we said about calls and puts in general as well as gamma in particular so far.

As the time to expiration decreases, the gamma and theta of at-the-money options increase. Just before expiration these variables can become dramatically large.

As the above figure shows; the graph narrows but the total surface underneath the graph remains unchanged. As a consequence the graph gets a much higher top. The higher top symbolizes the increase in gamma and theta as the time to expiration decreases.

Because of the behavior of ITM, ATM and OTM calls we see that delta curve will steepen around the strike as expiration approaches. Therefore gamma will increase for the ATM option as time passes. This is however not true for OTM and ITM options.

Gamma is an important risk parameter, because it determines how much money we can gain or lose on our delta-neutral portfolio as the stock price changes. In the following example we will evaluate the P/L of an option position as a consequence of movement of the underlying. We will assume a constant gamma of 2.7, so the delta changes by 2.7 per dollar movement of the underlying.

Assume we buy the 80 call 1000 times at 5.52 with a stock price of 79 dollars. To be delta neutral, we should sell 51,100 shares. The stock price develops as follows: at T=0 the price is 79, at t=1: 84, at t=2:76 and at t=3:79.

At t = 1 and t = 2, I readjust my hedge in order to be delta neutral, and at t = 3 I close my position.

Here are three ways to calculate the change in value of our position the first using cash flow, the second using delta, and the third using gamma.

So eventually we make a profit of 132,300. If we are long options and thus have a long gamma position we need to buy stock if the stock price decreases and sell stock if the stock price increases (buy low, sell high), so we always make a profit if the stock moves. Check for yourself that this is valid for both calls and puts.

In this case we use the average delta method. That is, we:

- Compute the average delta position during the stock move.
- Multiply this by the interval to calculate the profit.

At time t, we hedge so we buy/sell stock so delta is neutral again. Let’s look at this more carefully:

- At t=0, stock trades 79, we start a delta neutral position, that is we have 51,100 stocks short
- At t=1, stock trades 84. Delta of the option position is 64.6*1000 (from options) -51100 (from stocks). Between t=0 and t=1, my delta position went from 0 to 13,500. My average delta for the move was then (13,500+0)/2=6750 (6.75 per call). To compute the PnL of my position I multiply these deltas by the amount of stock move: 6570*5=33,750 dollar. To realize this profit I need sell stocks to be delta neutral again.
- At t=2, stock trades 76. Delta of my option position is 43.0*1000 and delta of my stock position is -64600 ...

In the previous example we calculated the average delta position by taking the average of the starting delta position and the final delta position. This can also be achieved using the gamma, as the gamma defines the change of the delta per dollar.

Let’s clarify how:

- At t=0, stock trades 79, delta neutral, gamma is 2,700.
- At t=1, stock trades 84. Stock moved by 5 so my new delta position is 5*2,700. At the beginning of the move my delta was 0, so my average delta is 5*2,700/2. Stock moved by 5 so the portfolio gained 5* average delta=5* 5 *2,700/2. Portfolio is hedged so that the delta is 0 again. We call this scalping the gamma. A long gamma position enables you to buy low and sell high.
- At t=2, stock trades 76. This is an 8 dollar moves, my new delta position is the 8 * 2700 ...

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