**Option Greeks: Options and Risk Parameters
**Published originally at Investopedia.com

By John Summa, CTA, PhD, Founder of OptionsNerd.com

**T**his segment of the options Greeks tutorial will summarize the key Greeks and their roles in the determination of risk and reward in options trading. Whether you trade options on futures or options on equities and ETFs, these concepts are transferable, so this tutorial will help all new and experienced options traders get up to speed.

There are five essential Greeks, and a sixth that is sometimes used by traders.

**Delta**

Delta for individual options, and position Delta for strategies involving combinations of positions, are measures of risk from a move of the underlying price. For example, if you buy an at-the-money call or put, it will have a Delta of approximately 0.5, meaning that if the underlying stock price moves 1 point, the option price will change by 0.5 points (all other things remaining the same). If the price moves up, the call will increase by 0.5 points and the put will decrease by 0.5 points. While a 0.5 Delta is for options at-the-money, the range of Delta values will run from 0 to 1.0 (1.0 being a long stock equivalent position) and from -1.0 to 0 for puts (with -1.0 being an equivalent short stock position).

In the next part of this tutorial, this simple concept will be expanded to include positive and negative position Delta (where individual Deltas of options are merged in a combination strategy) in most of the popular options strategies.

**Vega**

When any position is taken in options, not only is there risk from changes in the underlying but there is risk from changes in implied volatility. Vega is the measure of that risk. When the underlying changes, or even if it does not in some cases, implied volatility levels may change. Whether large or small, any change in the levels of implied volatility will have an impact on unrealized profit/loss in a strategy. Some strategies are long volatility and others are short volatility, while some can be constructed to be neutral volatility. For example, a put that is purchased is long volatility, which means the value increases when volatility increases and falls when volatility drops (assuming the underlying price remains the same). Conversely, a put that is sold (naked) is short volatility (the position loses value if the volatility increases). When a strategy is long volatility, it has a positive position Vega value and when short volatility, its position Vega is negative. When the volatility risk has been neutralized, position Vega will be neither positive nor negative.

**Theta**

Theta is a measure of the rate of time premium decay and it is always negative (leaving position Theta aside for now). Anybody who has purchased an option knows what Theta is, since it is one of the most difficult hurdles to surmount for buyers. As soon as you own an option (a wasting asset), the clock starts ticking, and with each tick the amount of time value remaining on the option decreases, other things remaining the same. Owners of these wasting assets take the position because they believe the underlying stock or futures will make a move quick enough to put a profit on the option position before the clock has ticked too long. In other words, Delta beats Theta and the trade can be closed profitably. When Theta beats Delta, the seller of the option would show gains. This tug of war between Delta and Theta characterizes the experience of many traders, whether long (purchasers) or short (sellers) of options.

**Gamma**

Delta measures the change in price of an option resulting from the change in the underlying price. However, Delta is not a constant. When the underlying moves so does the Delta value on any option. This rate of change of Delta resulting from movement of the underlying is known as Gamma. And Gamma is largest for options that are at-the-money, while smallest for those options that are deepest in- and out-of-the-money. Gammas that get too big are risky for traders, but they also hold potential for large-size gains. Gammas can get very large as expiration nears, particularly on the last trading day for near-the-money options.

**Rho**

Rho is a risk measure related to changes in interest rates. Since the interest rate risk is generally of a trivial nature for most strategists (the risk free interest rate does not make large enough changes in the time frame of most options strategies), it will not be dealt with at length in this tutorial.

When interest rates rise, call prices will rise and put prices will fall. Just the reverse occurs when interest rates fall. Rho is a risk measure that tells strategists by how much call and put prices change as a result of the rise or fall in interest rates. The Rho values for in-the-money options will be largest due to arbitrage activity with such options. Arbitragers are willing to pay more for call options and less for put options when interest rates rise because of the interest earnings potential on short sales made to hedge long calls and opportunity costs of not earning that interest.

Positive for calls and negative for puts, the Rho values will be larger for long-dated options and negligible for short-dated ones. Strategists who use long-term equity anticipation securities (LEAPS) should take into account Rho since over longer time frames the interest rate share of an option’s value is more significant.

**Figure 1: Greeks and what they tell us about potential changes in options valuation.**

**
**One other Greek is known as the Gamma of the Gamma, which measures the rate of change of the rate of change of Delta. Not often used by strategists, it may become an important risk measure of extremely volatile commodities or stocks, which have potential for large changes in Delta.

In terms of position Greeks, a strategy can have a positive or negative value. In subsequent tutorial segments covering each of the Greeks, the positive and negative position values for each strategy will be identified and related to potential risk and reward scenarios. Figure 1 presents a summary of the essential characteristics of the Greeks in terms of what they tell us about potential changes in options valuation. For example, a long (positive) Vega position will experience gains from a rise in volatility, and a short (negative) Delta position will benefit from a decline in the underlying, other things remaining the same.

Last, by altering ratios of options in a complex strategy (among other adjustments), a strategist can neutralize risk from the Greeks. However, there are limitations to such an approach, which will be explored in subsequent parts of this tutorial. (For more, see Getting To Know The Greeks.)

**Conclusion**

A summary of the risk measures known as the Greeks is presented, noting how each expresses the expected changes in an option’s price resulting from changes in the underlying (Delta), volatility (Vega), time value decay (Theta), interest rates (Rho) and the rate of change of Delta (Gamma). It was also shown what it means to have positive or negative position Greeks.