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3

Option Risks

INTRODUCTION TO OPTION RISKS

Fair-value option models link statistical theory to investment risk, which modern capital theory defines as the probability that the actual return from an investment will differ from the expected return (Hagin, 1979; Gehm, 1983). Defined this way, investment risk is useful in evaluating portfolios that have equal profit but differ in risk taken. The chances of doubling one’s money on a single toss of the dice in a casino promises a better rate of return than the interest earned in a bank account, but runs radically different risks. Capital may seek the largest profit, but at the “best” risk/reward ratio.

The link between probability theory and investment risk makes it possible to quantify option investment risk in very precise ways. Except for strike, any change in the other variables in the fair- value option model (interest rate, futures price change, days to expiration, and volatility) may bring about changes in option prices over the duration of the option cycle until expiration. Thus, these variables represent risks to an option portfolio.

This chapter briefly characterizes each fundamental risk to an option’s price. These risks are defined as delta ( , futures price change), gamma ( , change in delta), theta ( , time decay change), kappa (K)/vega (implied volatility change), and rho (P, interest rate change). Vertical spreads across single calendar strikes also nro subject to a skew risk in the implied volatility strike spread. Holding calendar or time spreads exposes option positions to additional risks: time delta (spread futures price change) and time kappa/vega (spread option implied volatility change). In addition to these market-based risks, there are also extramarket financial risks.

For the purpose of illustration risk to an option will generally be quoted in dollar-and-cent amounts for a single hypothetical option in this text. However, the real dollar risk must be multiplied by the value of the actual option contract, which will vary depending upon the type of option. For example, a 10-cent gamma risk may not seem large when one is considering only one option, but this 10 cents must be multiplied by the dollar value of the actual option on the asset or futures contract. For example, if the futures option or asset moves $500 for every whole futures point, a 10- cent risk is really a $50 risk for one option ($0.10 x $500), and a 10-option position would have a $500 gamma risk. Likewise, a single stock option represents an option on 100 shares of stock and a 10-cent risk on a stock option is really a risk of $10.00.

In trading options it is easy to lose your shirt if you do not understand the variety of risks involved. These risks will be illustrated in the following sections using simple option positions.

DELTA ( ) RISK

The delta risk of an option is a ratio reflecting the dollar amount of change in an option price for every dollar change in the underlying futures price. The delta risk is also known as the hedge ratio.

 

Dollar change in option price

Positive dollar change in asset price

When an option’s value increases with a positive change in asset price, delta is positive. If an option’s delta is +.30, for example, the option will increase in value 30 cents for every positive dollar change in asset price. If an option’s delta is -.30, then it will decrease in value 30 cents for every positive dollar change in asset price.

Figure 3.1 shows the pnyoff of a long 100 strike call and its delta over a range of futures prices from 90 to 110 at 60 days, 30 days, and one dny to (or at) expiration (with implied volatility levels at 15 and interest nt 10 percent).

 

 

Futures price

Figure 3.1 Delta and gamma risks of a long 100 strike call and its payoff.

 

When futures or the asset itself are at the strike of the call, the delta of the long call always is a +.50. It can range upward to + 1.00 when futures are above the strike and to zero when futures are below the strike. A short call would have a negative delta, since it would be the inverse or opposite side of the long call. The long put has a negative delta and the short put a positive delta. The delta of these and other option positions are discussed in more detail in Chapter 4.

A very deep in-the-money option will usually have a delta close to 1.00, that is, the option price will move almost exactly as the asset price itself. This correspondence makes intuitive sense, since a deep in-the-money option is likely to expire with the value of the underlying asset and, therefore, will move in value almost equally. Likewise, a very far out-of-the-money option usually has a very small delta, which is asymptotically approaching zero. This pattern also makes intuitive sense, since a far out-of-the-money option is likely to be worthless at expiration and to be relatively unresponsive to underlying asset price changes.

An at-the-money option will always move in price about half that of an asset contract, for a delta of ±.50, reflecting the equal probability of an option expiring in or out of the money. Generally, no single option can have a delta greater than +1.00 or less than -1.00 since no single option can change, dollar for dollar, more than a single asset contract at expiration. The underlying asset itself will always have a delta of +1.00 or -1.00, depending upon whether a long or short position is held.

The passage of time increases or decreases the delta of an op¬tion. An in-the-money option will show an increasing delta and an out-of-the-money option a decreasing delta. Option delta, there¬fore, drifts in time. This delta drift becomes more pronounced as the option expiration approaches reflecting the time shrinkage of the standard deviation, all else being equal.

In statistical terms, an option’s delta is also the probability that at expiration the futures will settle above the option strike, making the option in-the-money. A delta risk is a futures or asset price-based risk. A high positive delta indicates that in-the-money options are likely to remain in-the-money, and a low positive delta indicates the likelihood that the option will expire worthless. At- the-money options with a delta of +.50, therefore, have a 50:50 chance of expiring in- or out-of-the-money.

A positive delta indicates a bullish option position and a neg¬ative delta reflects a bearish one. A long call or short put is a positive delta position, and a short call or long put is a negative delta, for example. The delta of a total option position is simply the net sum of the delta’s of all separate positions.

Some option traders and market makers, however, often prefer to remain neutral about market direction, that is, delta neutral. To create a delta neutral option position, a trader must offset the deltas of the separate options. For example, if the delta of an at- the-money option is always +.50, then an option position that is long $2 at-the-money calls, and short $1 future, will be delta neu¬tral, that is, 2 X .50 - 1.00 = 0. A long or short straddle can also be a delta neutral position if the strike of the straddle is at the current futures price, for example, (+.50) + (—.50) = 0. The Δ’s of many single-month option strategies will be found in Chapter 4.

 

GAMMA   AND LAMBDA   RISK

An option’s delta is not a constant, but changes as the asset price changes and makes the option more or less in-, at-, or out-of-the- money. The change in an option’s delta as the asset price moves up or down is referred to as gamma risk.

 

Net change in Δ

Dollar change in asset price

While delta is a measure of risk to price from asset price change, gamma is a measure of the risk due to the stability of the delta. As the asset price increases $1.00, the delta of an out-of-the-money long call will increase, for example, from .30 to .35. This .05-cent increase in the delta is the gamma of the call position at that futures price range, and represents the added risk to an option when delta changes.

Gamma can be negative as well as positive. A long call or long put always has a positive gamma, and a short call or short put always has a negative gamma throughout the entire futures price range, with a Hingle modal peak centering around at-the-money options. The deep in- or out-of-the-money wing options have the least change in delta and, therefore, the lowest gamma (Figure 3.1).

For example, at 60 days to expiration, the long 100 call still has some positive probability of expiring in-the-money and, therefore, will have a positive delta and a positive gamma over the range of futures prices from 90 to 110. When futures are at strike (100), gamma is 0.075 cents at 60 days to expiration; gamma rises to 0.10 cents at 30 days, but with a smaller range.

By one day to expiration, however, the same option that was once very deep out-of-the money has virtually no chance of expiring in-the-money, and its delta and gamma have shrunk to zero. By one day to expiration, the gamma of the long 100 call will have exploded to over 0.25 cents when futures are at strike, but very deep in-the-money options will have a high positive delta but gamma at zero since delta has reached its upward limit of one. Generally, at-the-money options show an increase in gamma as expiration approaches, while the deep in-the-money or out-of-the-money option gamma decreases over time. The gamma risk of many singlemonth option positions is found in Chapter 4.

Another measure closely related to delta is the elasticity, or lambda, of an option position; lambda is defined as the percent change in the value of an option position relative to the percent change in the value of the underlying asset.

 

Percent change in option value

Percent change in asset price

The lambda of an option will always be greater than one, since an option is a leveraged instrument relative to the underlying asset. A deep in-the-money option with little time remaining to expiration will have a lambda approaching one, reflecting equal proportional changes between option and asset. As an option becomes out-of-the-money, its lambda will become much greater than one, indicating its higher leveraged power for gain or loss relative to the underlying security.

 

THETA   RISK

The change in option price due to the days remaining to expiration is known as time decay, or theta risk.

 

 

Change in option’s value

One-day change in time remaining to expiration

The risk should be obvious since, all else being equal, an option contract with fewer days remaining is worth less than an equivalent one with more days to expiration, for the extra days add value. Thus, option prices tend to decline as expiration approaches, and indeed, decline more rapidly the closer the expiration is.

For example, the theta risk of a short 100 call at 60 days to expiration is only +2.5 cents per day when futures remain at the strike price, which means that the trader will profit by this amount for one day by time decay. At 30 days, the short call will have a theta just under +4 cents, which will rise to over +10 cents on the last day before expiration so long as futures remain at a strike of 100. If there is a move in either direction, however, the in- or out-of-the-money option will have a smaller theta risk (Figure 3.2).

If an option trader is long an option, each day brings some decrease in the option’s value, all else being equal, and the trader is exposed to negative theta. Short options will have a positive theta, for the short option seller will profit from the time decay value of an option. To be short premium is to have a positive theta, because the trader expects to make a profit from time value decay. Generally, at-the-money options with the least amount of time remaining to expiration will experience the largest positive or negative theta values. The time value decay is slowest with long trading life remaining in the option.

Note that the theta of a long call is strikingly similar in form to its gamma risk over this range of asset prices, except that theta is about half the magnitude of gamma and of opposite sign. The identity of risk between gamma and theta in single-month options will be important in the evaluation of risk in the more com¬plex option positions and strategies that are discussed in later chapters.

KAPPA (K)/VEGA RISK

Even if there is no change in futures or asset price risk (delta or gamma) or in time risk (theta), an option price may be affected by changes in the market’s valuation of implied volatility. This valuation is formally referred to as kappa, although popular usage also refers to this risk as vega. Kappa /vega risk is one of the most important risks to an option price and is defined in this manner:

 

Futures price

Figure 3.2 Gamma, theta, and kappa/vega risks of a long 100 call.

 

 

Dollar change in option price

Positive one-point implied volatility change

For example, an increase in the implied volatility from 15 to 16 (one point) would increase the value of a long 100 strike call by a positive 16 cents at 60 days to expiration when futures are trading at the strike price (Figure 3.2). The kappa/vega of the long call would be positive over all futures price ranges but would move either upward or downward as the call became in or out of the money. Of course, there is no limit on the increase in implied volatility. Were a 10-point increase in implied levels to happen, the long (short) call would have a profit (loss) of $1.60.

This amount must be multiplied by the actual value of the option contract to determine the precise dollar risk. For a single stock option on 100 shares of stock, the amounts above must be multiplied by 100; and for a futures option that has a value of $500 per contract point, the above kappa/vega amounts would be multiplied by that full point value. Thus, a $1.60 kappa/vega risk in the above example would represent an $800 risk to the holder of one futures option and $8000 to the holder of 10!

An option position will have a positive, negative, or neutral kappa/vega risk at any single asset or futures price. Positive kappa/vega risk will result in increased profits from increased implied volatility, and negative kappa/vega will result in a loss from increased implied volatility change. A long call is positive kappa/vega and a short call, negative. A long put is also positive kappa/vega and a short put, negative. To be long volatility means that a trader has a positive kappa/vega option position.

Remember, kappa/vega risk does not vary directly with delta risk; they must be treated separately. Implied volatilities of op¬tions may move independent of futures price changes in either price direction. For purposes of illustration, let us suppose an important announcement (such as weekly USDA export sales) is scheduled for the following day; the price directional implications of this announcement are unknown. Option prices might rise on an increased market estimate of the possible future price change after the announcement, and might fall just as rapidly after the announcement if futures price change did not occur, all else being equal. The increase and decrease in option prices before and after this announcement reflect changes in the market's implied volatility price.

Paradoxically, sudden sharp futures price changes (up or down) may occur in conjunction with falling implied volatilities. This divergence might seem counterintuitive, since if real volatility is up, one might suppose that implied volatility must be up also. However, one would be ignoring that implied volatility is the market’s estimate of future volatility, not historical volatility. An increase in current volatility accompanied by a fall in implied volatility merely suggests that the market expects future volatility to fall, however high it currently is.

It is not uncommon for option traders to be long a call and find, when futures prices do move sharply higher, that their call has increased only marginally in price even with a moderate positive delta. While such traders are likely to be upset and perhaps even blame the market maker for somehow cheating them out of their profit, what the option traders do not realize is that the price paid for the call had already been based on a high-volatility esti-mate. Once the move had occurred, the market believed no further moves were likely and lowered its estimate of volatility. Although the option traders were correct about price movement, they were whipsawed by the market’s changing estimate of volatility. These traders did not understand kappa /vega risk (see letters to Barron’s, April 18, and July 18,1988). One exaggerated example of the possibility for a trader to be correct about the direction of the market but still lose money in options because of an implied volatility change occurred in the stock market crash of October 1987, when short out-of-the-money call options rose in price after the 500-point drop in the market, reflecting a market-implied volatility level of well over 100!

In practice, the implied volatilities of options are one of the major risks to any option trader. Much analysis, evaluation, and strategy are devoted to kappa/vega risk.

 

RHO (P) RISK

One of the variables in the BSM futures option-pricing model is the risk-free rate of interest. This interest rate represents the cost of carry of an option position, or the opportunity cost to capital of trading in options, that is, what unoccupied capital may safely earn. A positive cost of curry earns interest, and a negative cost of carry incurs interest pityments.The appropriate risk-free rate of interest is taken from the yield curve of the U.S. Treasury bill market that matches the option cycle. If interest rates change, the cost of carry and the value of an option will also change, all else being equal. Change in the cost of carry that leads to change in the value of an option is referred to as rho risk.

Consider a futures call which has a hypothetical fair value of $100 and a zero cost of carry, that is, the cost of capital is free. The purchase or sale of the fair-value call at $100 results in neither profit nor loss over the long run. But if capital is not free, the buyer of the $100 call must deposit cash to purchase the option (a negative carry). Thus, he or she forfeits the earned interest on the $100, and the seller gains the use of these funds, which may be deposited to earn interest after having satisfied margin requirements. If the call is to have a fair value that includes the cost of carry, the call will sell at some discount to $100 that equals the cost of interest rate carry.

Therefore, as interest rates increase, the fair value of futures options falls, all else being equal. For example, the fair value of a 100 call with 200 days to expiration, a volatility of 15, and a riskfree rate of interest of 8 percent, is $4.24. If interest rates were to rise from 8 to 15 percent, however, the value of the option would drop to $4.08. The 16-cent difference represents the discount on $4.24 of an additional 8 percent annual interest prorated for 200 days.

For nonfutures options, such as stock options, an increase in interest may either increase or decrease the value of the option. An increase in interest will increase the carrying cost of long stock positions, making a call option more desirable than holding stock. Likewise, an increase in interest will make shorting the stock more attractive than holding a long put, in which case the value of the put will decline.

The impact on option prices of a change in the risk-free rate of interest is not so large as that of some other option price risks. Interest rates rarely move so dramatically over six-month periods as the previous example might imply, although the early 1980s did witness extraordinarily high historical interest rate changes. In some foreign option markets these interest rate changes may also not be unusual.

Interest rates, nevertheless, remain an important market risk and necessitate sophisticated strategies for market makers. These will be discussed in Chapter 5.

 

SKEW RISK

The central implied volatility of an option strike series is that found in the level of the at-the-money straddle. But if the 100 call is trading at an implied level of 15, what about the 105 call, or any other options of this cycle, whatever the strike?

All strikes of one calendar series need not trade at the at-the- money implied level. A 100 call may be trading at a 15 volatility, but a 105 call at a 20 implied volatility. This difference in the implied volatility levels of single-cycle options across strikes is termed the implied volatility slope, or skew.

Positive skew refers to both out-of-the-money puts and out-of- the-money calls (wings) that are trading at higher volatility levels than at-the-money options (center). For a flat skew, all strikes trade at approximately the same volatility levels. In a negative skew the option wings trade at lower volatilities than the center strike options. Since puts and calls each have different out-of-the- money/in-the-money volatility spreads, there is a put skew and a call skew in every single-cycle strike series. These hypothetical skew relations are shown in Figure 3.3.

Skew risk exists in any spread option position because the degree of implied skew may change. If a skew goes from positive to flat to negative, vertical spread traders are affected, as changes in skew affect option spread prices. If an implied volatility skew exists, then it generally is a linear or curvilinear function. Sometimes saw-toothed and choppy skews exist temporarily as a result of transient imbalances or trades on close. These are relatively unimportant and may safely be disregarded.

 

 

 

 

Figure 3.3 Implied volatility skew

On many if not most futures and financial option markets some form of positive put and call skew exists. In practice, option traders tend to value out-of-the-money options higher than the BSM model suggests. In other words, option market participants trade as if a more frequent or extreme price volatility than log-normal probability suggests is likely. From Chapter 2 we know that actual long-term returns of financial assets have greater extremes than would be expected by the normal distribution, and that these are consistent with implied volatility skew.

Positive skew patterns are also consistent with assumptions of popular option trading strategies. Option speculators may prefer to be long out-of-the-money options, owing to their relatively low absolute cost and low absolute risk of loss as well as their infrequent potential for big payoffs (for example, the lottery ticket play). Conversely, option traders may prefer selling at-the-money options because of the greater absolute time decay (positive theta) profits. These market demand situations may tend to lower the implied volatilities of the center and raise those of the wings.

The only common exceptions to the general pervasiveness of positive skews in futures and financial option markets are the sometimes negative call skew in stock index futures options, and the sometimes negative or flat skew in commodity futures puts. Stock option participants devalue out-of-the-money calls and some commodity option markets devalue out-of-the-money puts with negative skews. The topic of skew risk will be taken up again in Chapter 8.

 

TIME SPREAD RISK

The discussion of option risks so far has been limited to single¬month positions only, either as single options or as some type of vortical spread. Time spreading means holding a simultaneous position in more than one calendar month; it is also known as a horizontal or calendar spread. The option leg that is the closest to expiration is termed the front month, and the option leg furthest from expiration is known as the back month. An option leg botween the front and back months would be the middle month. Time spreading neutralizes some of the risks of holding single- month positions. At the same time, calendar spreads nro subject to other time-based risks that may be as severe or more severe than those risks neutralized. The two time-based risks for calendar spreads are termed time delta and time kappa/vega risks.

Time delta risk is defined as the delta risk to any futures option time spread caused by a change in the underlying futures time basis. Time delta risk is only applicable to futures option positions, and not to cash or stock option positions, since the underlying instrument never expires in stock calendar option spreads (that is, the shares of common stock), while futures price time spreads reflect differential prices of monthly contracts. Time delta risk is obvious to any futures option trader, of course, but less obvious sometimes to traders who only trade stock or cash option spreads. Stocks and the cash market do not have a time basis. Futures and futures options do, and this is the source of time delta risk.

All option calendar spreads, whether futures or nonfutures, are subject to time kappa/vega risk as well. A position composed of two different contract months risks having the implied volatility spread of the different months’ change. The implied volatility of a time spread is by no means constant, and exposes a time option spread to price risk, sometimes more than the risk neutralized in a single-month strategy.

Both time delta time kappa/vega are extremely important and will be discussed further in Chapter 6.

 

EXTRAMARKET RISKS

The risks to the value of an option discussed so far are generally known and may be quantitatively measured. There is another class of option risks that are less well known or measurable. They are the extramarket risks that frequently result from financial irregularities or calamities of one sort or another.

A financial risk to all options and futures traders is the possibility that a trader may lose his or her entire capital in a clearinghouse suspension or bankruptcy. Market makers and exchange option traders are required to keep trading funds in an exchange- regulated clearinghouse that guarantees the trades of its members. Although the clearinghouse system protects a trader from the default of any individual trader with whom he or she trades, it does not protect an exchange trader from the default of the clearinghouse itself. Any floor trader takes the risk that another trader at his or her clearinghouse may suffer losses so large that the clearinghouse itself becomes temporarily or permanently insolvent. This risk is real, and over the years there have been several stock or futures clearinghouse failures, such as Volume Investors in 1985, Fossett in 1989, and Stotler in 1990. Others are bound to happen in the future. Fortunately, in these clearinghouse suspensions virtually no floor traders lost account capital, but of course, it is not impossible that this might happen in the future.

An attendant risk in a suspension of a trader’s clearinghouse is that the trader will not have the means to trade the carryover position, exposing the trader to market risk. If a trader is inadvertently long or short in the market and unable to adjust his or her position because a clearinghouse suspension has frozen all accounts, the trader may incur steep losses whether or not the clearinghouse eventually remains insolvent.

Some means do exist to lessen extramarket risks. For instance, the general public may be somewhat safer trading through large brokerages; and stock and currency options are guaranteed settlement through the Options Clearing Corporation. Nevertheless, the large over-the-counter market in options, especially bonds and currencies, has no clearinghouse guarantee system. Therefore, traders are exposed to what is known as counterparty risk, which exists when the opposite party to a contract defaults without clearinghouse guarantees. In this case, hypothetical profits turn into losses.

Extramarket risks should be considered carefully by prudent option traders, however improbable. Although they are rare and freak, these risks may be devastating to a market maker’s or dealer’s capital.