##### Greek Options are specific risk aspects that capture the sensitivities of an option value to market parameters such as time, interest rate and volatility.

An important aspect to consider in derivatives markets is the concept of Greek options. This article provides an overview of their usefulness, characteristics and functions.

## Greek Options and Their Uses

### Purchasing and Premiums

Greek options, and particularly delta hedging, are useful tools for managing risk and minimizing volatility. When we purchase an option, we can trade the cash instrument and hope to realize more profit from that trade than we pay initially in premium for the option. When we sell an option, we hope that the premium that we are paid up front is more than the losses we will sustain from trading the cash.

When we buy options, we are said to be buying volatility. We make money if the spot rate is volatile enough for us to pay for the option. When we sell options, we are selling volatility. We make money if spot is calm enough that we don’t have to hedge the exposure too frequently.

However, Greek options delta hedging is not the only way to make money with options. The genius of derivatives is that they allow us to take positions in (or to hedge against fluctuations in) other aspects of the cash instrument’s price evolution. Derivatives are dangerous if we do not understand or address each potential dimension of their risk.

With a simple vanilla option, for instance, we can make money if implied volatility moves in our favor. With currency futures, currency forwards and currency options, we can speculate on the spread between interest rates in two different countries for a particular maturity date. With some exotic options, we can buy an option that appreciates in value over time (all other things being constant) and that also appreciates in value with movement lower in implied volatility.

Options dealers and savvy options traders use time-proven techniques to break down the risks in an options position or in a portfolio of options, futures, forwards and cash positions into information that is more easily understandable and therefore more easily positioned or hedged. This method of analysis employs the Greek options, as well as using simulation, scenario analysis and value-at-risk analysis.

The Greeks (Delta, Gamma and Vega) get their name from the fact that the sensitivities of an option to various market parameters are labelled with letters from the Greek alphabet.

## Types of Greek Options

### Delta

The first of the Greek options is the delta. The delta of an option is the sensitivity of the option’s price to very small changes in the price of the underlying instrument.

When we talked about trading spot around the options position in order to realize profit that would pay for the option’s premium, we were talking about trading the delta.

By taking an opposite position equal in size to the option’s delta, we protect the option against profit and loss variability due to small changes in the spot rate.

For example, consider an equity call option with a strike price of $50 when the underlying price is $50. Because it is an at-the-money option, we know that the delta is 50. The delta is expressed in terms of a percentage of the notional amount. An option that is hopelessly out-of-the-money very near to expiration has a delta of 0. Also, near expiration, an option that is completely in-the-money with no danger of being thrown out-of-the-money has a delta of 100. Everything else is placed in-between. At-the-money options have a delta of 50.

Our equity call has a positive delta because it is a long position in a call. If we exercise the call, we will end up being long the stock.

An equity put struck at-the-money would have a negative delta of 50. If we exercise the put, we will end up being short the stock.

Similarly, shorting a call implies a negative delta and shorting a put implies a positive delta.

To delta hedge our long at-the-money equity call struck at $50, we need to know the notional amount. For the sake of argument, let’s set it at $100. Therefore, the delta position implied by our option is $50 (i.e. 50/100 x $100).

If we take a short position in the cash market (assuming that shorting the stock is feasible and liquid enough) at the spot price of $50, we have protected the option’s sensitivity from small changes in the spot price.

If spot goes to $48, the $2 we make on the short stock position will offset the $2 we will lose on the change in price of the option. Similarly, if spot goes to $52, the $2 we make on the option premium will be offset by the $2 we will lose on the short stock position.

Assuming that we own the option, if we plot a graph of the curve of the option premium (on the y-axis) against the price of the underlying instrument (on the x-axis), everything else remaining constant, we obtain a convex curve. The slope of this convex curve is the option’s delta.

### Gamma

The next of the Greek options, the Gamma, begins to get interesting for larger moves in the stock price.

If the spot goes to $70, we might expect to make $1,150 on the option price while only losing $1,000 on the short stock position.

How does this work?

If spot goes to $52, the delta might change to 52. If spot goes to $55, the delta might change to 57. The option position behaves as if it is a miraculous trade that seemingly gets longer as spot goes higher in a non-linear fashion.

Since we have only hedged our exposure to a position that is long $50 at $50, the hedged option position will continue to make money on the incremental position (i.e. the part that appeared to get longer from $50 to $70 at an average rate of, say, $65).

In a volatile environment, the greater the convexity of the option curve, the more bang for our long-option buck and the more pain we will endure if we are short the option.

Mathematically, gamma is the second derivative of the option’s price with respect to the underlying cash price. In essence, it’s the sensitivity of the delta (or rate of change of the delta) with respect to the cash price.

### Vega

We know that options will be expensive when volatility is actually high or when it is thought to be heading higher. We also know that options are cheap when volatility is low or when it is believed to be heading lower.

There are two kinds of volatility between which we must distinguish: actual volatility and implied volatility.

Actual volatility is a measure of how much the spot price actually moves around for a given time period.

Implied volatility is the volatility used in the calculation of the option’s price. Without going into the mathematics, suffice it to say that we can back out (or “imply”) the volatility used to calculate an option’s price if we know the value of each of the other variables used in the formula used to evaluate the option. For the Black-Scholes-Merton model, the list of these remaining variables typically includes the underlying cash price, the maturity date, the delivery date, the strike price, and the risk-free rate of interest.

Some of the more developed derivatives markets, such as the foreign exchange (Forex) options market, actually trade in terms of implied volatility or “vol” instead of specifying a price at which they may buy or sell the option in question.

The sensitivity of an option’s price to changes in its implied volatility, all other things being constant, is our last of the Greek options, called the vega.

### Theta

Theta is the sensitivity of the option’s premium price to the passing of time, otherwise referred to as time value decay. It incorporates the common idea of “time is money” by representing how much the price of the option will fall/decay with the passing of a single day.
For “at the money” and “in the money” options, theta tends to be greater at an increasing rate the closer you get to the option expiry date. For “out of the money” options, the reverse is true since that type of option has very little premium left to decay.

Rho

Rho is the sensitivity of the option’s price to the small changes in interest rates. When interest rates rise, the price of a call option will rise while the price of a put option will fall.

## Greek Options Overview

In essence, Greek options are ways of mitigating risk, monitoring interest rates and managing an options portfolio’s volatility. Properly used and understood, Greeks are incredibly useful in the process of portfolio management.

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