by Dave Caplan

A trader may see if current market volatility is high and implied and the historical volatilities are at the upper band of their normal range, and volatility has been rising.  He then initiates a “Neutral Option Position.”  If the trader has incorrectly forecasted that volatility was going to decline, he can suffer a loss due to a greater actual volatility than anticipated.  The effects of these types of volatility increases on the position may be quantified by using the option risk variables.  The risk of implied volatility fluctuation is measured by the position’s “vega.”  “Delta” is the rate of change of an options price with respect to a unit change in the underlying price.  The rate of change of an option’s delta with respect to unit price change in the underlying is called “gamma.”

If a trader purchases options during periods of declining or down-trending volatility, the market does not punish the long volatility player quickly, but lets his position die a slow death.  If the trader was taking a long volatility position based on his forecast volatility as compared to the current implied and historical volatilities, he might, for example initiate a long at-the-money straddle position, again with 120 days until expiration.  If the trader has guessed wrong on his forecast volatility, the position does not change much day after day, giving a trader a hope that volatility will increase and “bail him out”.  Since the trader over-estimated volatility, the lack of implied volatility increases and market movement causes the position to erode from time decay.

Each option has its own weighting with respect to the price movement of the underlying.  This weighting is expressed by the number of equivalent underlying (futures), or the amount an option moves for a minimum price fluctuation of the underlying.  For example, at-the-money options move by half the move in the futures.  The term for this measurement is called delta.  The at-the-money option has a delta of 0.5.  Futures always have a delta of 1.  In mathematics, this term is commonly termed the rate of change.  This measurement is also called the hedge ratio, because it represents the ratio of options needed to equal one future.  This figure does not take other factors into account, such as volatility and time erosion.  Additionally, this number changes with market movement.  The more in-the-money an option is, the higher the value; the further out-of-the-money the smaller the value of this measure is. 

The sum of the absolute values of this ratio (delta), for a call and put of the same month and strike price equals one.  If December gold is trading at $350, the December 350 calls and puts each have a delta of .5, equal one.  If the 360 calls have a ratio of .35, .65 is the hedge ratio for the 360 put with the future at $350.  If we examine the 350 strike, we would find that no matter what volatility is trading at, this ratio will still be .5 for both the calls and puts.  This also means that there is 50/50 chance of these options being in-the-money at the expiration, or a 50% probability.

In-the-money options are affected by volatility changes.  For example, the Dec 360 put, with Dec gold at $350 has a hedge ratio of .65.  An increase in volatility would decrease this measurement, while a decrease in volatility would increase this figure.  This is because volatility is a risk measurement tool of the perceived range by the marketplace of the underlying, in this case December gold futures.  A higher volatility figure implies a larger range of movement for the underlying future.  Therefore, in-the-money options have less weight with higher volatilities because the market is implying that the market is more likely to move away from where it currently is.  A volatility decrease would increase the probability of the market staying closer to its current price level until expiration.

You can predict what an implied volatility change would do to out-of-the-money options.  If implied volatility increases, the hedge ratio of the in-the-money option decreases.  If implied volatility increases, the hedge ratio of the in-the-money option decreases.  The opposite is true for decreases in volatility.  Mathematically, using the same gold option example.

IF DEC 360 PUT PRICE RATIO = .65
THAN 360 CALL = APPROX .35
VOLATILITY INCREASE:
360 PUT HEDGE RATIO <.65
THAN 360 CALL HEDGE RATIO >.35
IMPLIED VOLATILITY DECREASE:
360 PUT HEDGE RATIO >.65
THAN 360 CALL HEDGE RATIO <.35

The length of time of an options life acts identical to a change in volatility; more time means higher delta for out-of-the-monies and lower deltas for in-the-monies, a shorter options life means an increase in in-the-money delta, and decrease in out-of-the-money deltas, assuming all other variables (price and volatility) remain constant.

The hedge ratio (delta) or price ratio, changes for each option with a change in the underlying price.  This measurement, or rate of change of the hedge ratio given a unit change in underlying price, is represented in models by the Greek letter “gamma.”

This measure may be illustrated by using our gold option example again.  We know that the at-the-money option will always have an approximate price ratio of .5.  We assumed that with Dec gold at $350, the Dec 360 put had a price ratio of .65, with the 360 call .35.  If our unit change for this “gamma” measure is $10 in the underlying, we can estimate the measure if the futures were to move up to $360.  If the Dec gold futures are trading at $360, the 360 strike is now the at-the-money strike.

Therefore, the new price ratios are .5 and .5 for the calls and puts respectively.  The change from .65 to .5, and from .35 to .5 is equal to .15.  This number, .15 is the gamma for the 360 strike, or change in price ratio.

This is an important measure for risk analysis of an options position.  This risk of the options position is directly related to price risk in the underlying.  The moment a trader initiates a position three possibilities exist as to the directional market bias the position has; long the market, short the market, or neutral the market.  Gamma, or the change in price ratio, allows the trader to measure how market movement affects the directional bias.  This measure is one of the tools necessary in making adjustments to the “core” position, as the underlying market moves.

A position that is net long premium, long options, is long this measure.  Keeping time and volatility constant, this type of position, an outright call or put purchase, or a straddle or strangle purchase, is “always the right way,” with respect to market moves.  The position becomes “more long” as the futures move up, and “more short” as the futures price declines.

Short premium positions are short or negative this measure (gamma).  Short outright calls or puts, straddles and strangles, are always the “wrong way”, regarding market moves, keeping time and implied volatility constant.  These positions become more short, market-direction wise, as the futures move up, and more “long” with declining futures prices.

The limits of a hedge ratio are from -1 to 1, negative ratios are for short positions, positive for long positions.  This means that the futures equivalents have a limit from 0 to the number of contracts, for the option cannot be more valuable than the underlying.  The change in price ratio (gamma) therefore must have its own limits.

The at-the-money options are the most sensitive to change in price ratio.  As the strikes go further from the at-the-money strike, this measure decreases.  Additionally, this measure is the same for calls and puts with the same strike and expiration date.  With the passage of time, options are more sensitive to this change.  In other words, the risk of change in price ratio increases as expiration nears.

The risk is a benefit for long premium positions, and a disadvantage for short premium positions.  The change in price ratio for at-the-money options is more sensitive to volatility changes.  Volatility increases, decrease the change in price ratio; volatility decreases, increase this measure.  This effect is based on the volatility effect respective to hedge ratio.  Higher volatility results in a narrowing margin between in-the-money and out-of-the-money hedge ratios, therefore the change is smaller.  Lower volatility causes a widening margin between in-the-money and out-of-the-money hedge ratio, therefore the rate of change is larger. 

The volatility long trader is betting that the actual market behavior will be volatile enough for the position to prosper from the positive change in price ratio.  The volatility short player is betting that the price behavior of the market will not punish the position, with a negative change in price ratio, thereby allowing time to pass, and the premium to erode.

In summation, it is important to note that the change in price measure only allows for approximation of hedge ratio estimate.  As the hedge ratio changes with changes in futures prices, so does the change in price ratio (gamma).