unimodally negative. If gamma is trimodal, then kappa/vega is trimodal and of the same sign; and theta is trimodal and opposite in sign to gamma and kappa/vega.
Thus, if the sign and mode of any one of the non-delta risks are known, then the sign and mode of the other risks are known as well. In effect, there is really only one gammaikappa/theta risk cluster (by sign and modality) in single-month positions, with gammaikappa collapsing into one dummy (by sign) risk factor. For the statistically minded, there is only one degree of freedom.
The passage of time affects risk in single-month positions identically with single-option positions. In back-month positions kappa/vega is large while gamma and theta are low and wide. In front-month positions theta and gamma become larger although narrower while kappa/vega grows smaller and narrower in range. All long single-month positions grow more expensive to carry ow ing to time decay but are less affected by implied volatility change as time passes to expiration.
The interrelation of delta, gamma/theta, kappa/vega, and time risk modalities enormously simplifies option risk analysis of single-month option positions. Option positions now become strategic in which risk may be evaluated and not just measured, which is the topic of the next section.
LlMITED- AND U N LIMITED-RISK ANALYSIS
Theoretical consideration of option risks is validated Only if it pro vides some practical guide to the relative risk exposure of the dif ferent option positions for trading purposes. Risk analysis must be both descriptive and evaluative. Previous sections have formally defined and quantitatively measured option risk profiles. In this section we begin to evaluate them.
For trading evaluation option positions must be divided be tween those that have limited- and those that have unlimited-risk exposure. A limited-risk option (LRO) position is one in which po tential dollar loss is always finite and fixed in the worst-case risk scenario. Limited-risk exposure does not mean that dollar losses may not be large or severe, but only that they are fixed and finite no matter how large. The simplest LRO position is a long call or put.
L1MITED- AND U N LIMITED-RISK ANALYSIS 61 An unlimited-risk option (URO) position has no fixed limit on the potential dollar loss in the worst-case risk outcome. Without a fixed limit on the amount of dollar loss that an option position may sustain in the worst-case scenario, risk is catastrophic. The poten tial dollar loss from an option position is unlimited and, therefore, could potentially be greater than the trader's entire capital, no matter how large. Unlimited risk is potentially catastrophic be cause it exposes the trader in the long run to bankruptcy. The simplest URO position is a short call or put.
Option risks such as rho, skew, or risks connected with expi ration are almost always limited and will be considered in other chapters. But what are the limits on delta, gamma, theta, and kappa/vega risk? Which option risks are limited, and which are unlimited?
The delta risk is limited (or neutral) when delta converges asymptotically to zero change, or is positive on the upside and negative on the downside of the asset or futures price movement. The delta risk is unlimited when negative on the upside and pos itive on the downside, that is, the position loses unlimited money on either the upside or the downside.
The gamma and theta risks are limited whether they are pos itive or negative; they are always finite risks, no matter how large the loss. The same is not true for kappa/vega, however. There is no limit on the amount of profitlloss due to kappa/vega, and it is a potentially unlimited or catastrophic risk. In particular, it is usu ally a negative kappa/vega that is associated with unlimited-risk exposure in the case of an implied volatility blowout on the up side. Option positions which are negative kappa/vega, therefore, are unlimited-risk positions and generally poor strategy.
In normal markets, a positive kappa/vega risk exposure often is limited, although possibly large. In the event of a high implied volatility market, being positive with respect to kappa/vega might possibly represent an unlimited-risk situation should implied lev els drop dramatically and fast. However, in normal markets, this risk should be limited. A market situation of high implied levels and positive kappa/vega will be taken up in Chapter 7.
A risk profile with a negative gamma, negative kappa/ vega, and positive theta represents an unlimited-risk position. A positive gamma, positive kappa/vega, and negative theta risk pro file constitutes a limited-risk exposure for single-month positions
62 POSITION RISK PROFILES
in normal markets, resulting primarily from the positive kappa/ vega risk stance. Of course, each of these asset-based risks also has a modality over a range of asset or futures prices. It is pos sible for an option position to be both limited and unlimited in delta or kappa/vega risk, depending upon the direction of asset prices. A short call, for example, has only a limited delta risk on the downside of asset prices but an unlimited risk on the up side. Its kappa/vega risk, however, is unlimited on both the up side and downside, reflecting the unimodal nature of single-option kappa/vega risk.
For most multi-option positions, however, kappa/vega risk is bi or trimodal and may be both positive and negative over a range of futures prices. Kappa/vega risk, therefore, may be potentially both limited and unlimited. For purposes of initial risk evaluation, however, bi- or trimodal risk positions will be considered limited in kappa/vega risk (with normal volatility markets) if the number of options long in the total position is even with or greater than the number of short. As long as there is at least one long option to cover the risk of a short option, even negative kappa/vega risk in bi- or trimodal risk positions may be considered limited. The only single-month position in which this equal balance in long and short options may not limit risk is a fence, which will be the topic of a separate section in Chapter 7.
Table 4.3 summarizes the level of risk for the basic non calendar-option positions. A URO position is subject to unlimited risk (U) in at least one direction of one risk in a risk profile. A LRO position only has limited risk (L) in both directions for all risks, or is a net even option spread.
Unlimited-risk option (URO) positions include the short call or put, short straddle/strangle and short ratio spreads, fences, and cartwheels. Some may be surprised to see a covered call position classified as a URO position, but a short call, long asset or future position is just a synthetic short put. A URO position may experi ence loss either as a result of an unlimited move in asset or futures prices or as a result of sudden increases in the implied volatility of option prices. A short straddle position may experience unlimited loss due to a delta extreme range move on limit days, and option prices may trade at extremely high implied volatility levels. In ei ther case, the potential dollar loss is unlimited and catastrophic for those negative kappa/vega risks.
LlMITED- AND U N LIMITED-RISK ANALYSIS
Table 4.3 Single-month option position risk summary (* = unlimited risk)
63
Option Position Delta GammalI'heta KappaNega
Unlimited risk position
Short call or put Short straddle/strangle Bull and bear fen(;e Short ratio spread, calls,
or puts
Short ratio spread, calls, and puts (short wrangle) Bull or bear cartwheel
Limited risk position
Long call or put Long straddle/strangle Bull or bear sptead Long ratio spread, calls,
or puts
Long ratio spread, calls, and puts (long wrangle) Long or short butterfly/condor Synthetics * * * * * * * * * * * *
Limited-risk option (LRO) positions include all the remaining positions except time spreads. These include the long options, long straddle/strangle, spreads and long ratio spreads, butterflies, long wrangles, and synthetics. All of these risks are limited bidirection ally in both delta and kappa/vega. Likewise, if one is in a spread position, risks are limited in either direction as long as the num ber of long options is equal to or greater than the number of short options in the total position. Some LROs are delta neutral and some are not.
The first prudent principle and strategic goal of every finan cial business is to avoid the risks of bankruptcy, even before
64 POSITION RISK PROFILES
consideration of profitability. It makes no sense to earn high profits if the likelihood of bankruptcy is still higher. This perspective to ward risk will be referred to as the prudent market-maker strategy and represents the safe bet in the long run.
The strategies recommended in this text are of the limited-risk type. A prudently rational trader knows that any option strategy that is exposed to unlimited risk, no matter how small the prob ability, will eventually suffer catastrophe under the law of large numbers. As a Wall Street maxim notes, "There are bold traders and old traders but no bold, old traders."
Limited risk does not necessarily mean limited profits. Limited risk option strategies may be highly profitable in several market situations-that is, those that are unprofitable to unlimited-risk traders! It is possible to have limited option risks yet unlimited profit potential. Prudent and informed market makers will strive to follow strategies that have these limited-risklunlimited-reward characteristics.
A URO position resembles the risk exposure of a martingale gambling strategy, which increases the amount of the bet every
time a loss occurs. In a series of losses, one could bet $1, $2, $4, ...
in a double-up martingale. If a gambler has sufficient capital and is content with modest profit, he or she may consistently earn money for long but limited periods of time with martingale strategies.
However, there eventually comes a series of consecutive-run losses that ultimately bankrupts the gambler. At that point, the gambler will no longer be able to play. For example, betting only a single dollar to start in a double martingale will bankrupt a gambler with a stake of less than $255 within 1000 plays or so. Starting with only a dollar bet, a stake of $1000 may expect to be wiped out in the course of over 4000 plays. There is no way in the long run to beat the odds consistently if they are against you. The improbable should never be assumed to be the impossible.
A naive option trader employing martingale strategies may make money for long periods before losing the entire stake in a rare or freak futures price run. This is the risk of the unlimited risk option trader.
One could argue, of course, that the dollar loss on the short put is limited to a complete collapse of futures or asset prices, which is unlikely, and that it would still be a fixed loss. Indeed, the loss on a short put, at least in terms of intrinsic (delta) value, is limited. However, the value of the potential loss is catastrophically huge
RISK DETERMINATION 65
with most futures margins. Premium expansion for at-the-money options can easily reach $5000 per option in high implied volatility situations, and the margin required may increase exponentially.
Of course, at some level of capital (for example, $1 million or more) it may be possible to sell some small number of options net short and be exposed to negligible catastrophic risk. However, using $ 1 million of capital to short two or three options is probably not the optimum trading strategy, or use of margin or capital.
If the option speculator's profits are akin to gambling returns, then profits of limited-risk market makers may be considered sim ilar to the returns to the house, that is, returns that take the op posite side of the bet from the gambler but that ultimately offset or hedge this bet with the bet of another gambler. The returns of the house (or market maker) come as a percentage of totaJ wagers, whatever the side of the bet or the event outcome. (See Reichen stein and Davidson, 1987, for a gambling interpretation of option trading from the perspective of horse racing.)
RISK DETERM I NATION
This section introduces a quick way to determine approximately the general limit of delta and kappa/vega risk of any single-month option position. (Time spread risk determination is taken up in Chapter 6). An option trader will always want to know what the exact risk profile is for each single-month carryover position at all times. Market-maker positions, however, have a tendency to grow larger during a complete cycle and include spreads covering every strike, with total carryover positions rising into the thousands. As we have already observed, any carryover position, no matter how large or complex, can be represented in one of the basic position types (Table 4. 1); but identifying which one may require the use of option software (see Appendix).
Usually, a trader will have such option software available to perform this risk analysis on each cycle position. Nevertheless, a trader should be able to determine by hand calculation, with out computer assistance, an approximate catastrophic risk profile for his or her carryover position. A trader may never have to do such manual calculations, but he or she should know how they are done.
66 POSITION RISK PROFILES
Approximate single-month position, catastrophic risk exposure may be quickly estimated with some simple position numbers and calculations. There are three primary risk calculations: upside delta risk, downside delta risk, and kappa/vega risk. Consider fu tures options for illustration.
To calculate the potential upside delta risk, add the net total delta of the futures position and net call position, where each call carries the equivalent of one whole delta point.
a upside risk = Net total futures + Net calls
If the result is positive, then the trader appears to have some limited upside delta risk, which is good. A negative delta would indicate potential unlimited risk.
To calculate the potential downside delta risk, add the net total of the futures position and the net put position, where each put carries the equivalent of one whole delta point.
a downside risk = Net total futures + Net puts
If the result is negative, then the trader has a negative poten tial delta on the downside and is exposed only to limited risk. A positive downside delta exposure is an unlimited risk.
These risk calculations estimate upside and downside delta risk exposure but do not indicate a position's exposure to volatility risk. To approximate the position kappa/vega risk, add the net to tals of calls and puts (long-positive), which gives net total options.
Kappa/vega risk = Total net puts + Net calls
= Net total options
A positive total indicates that a position potentially has only limited kappa/vega risk in the event of a volatility explosion. A negative total indicates a position with unlimited-risk expo sure. Consider the example of a single-month position shown in Table 4.4.
In this position, the delta upside risk is short 5, (short 10 fu tures and net long 5 calls). This result represents unlimited delta risk on the upside. Downside delta risk is short 30 (short 10 futures and net long 20 puts). The kappa/vega risk is long 25 options (net
RISK DETERMINATION
Table 4.4 Hypothetical single-month position Position Long 10 130 calls Short 5 100 calls Short 20 100 puts Long 40 90 puts Short 10 futures Subtotal (Net long 5 calls)
(Net long 20 puts)
(Net short 10 futures)
67
long 5 calls and net long 20 puts), which is a positive kappa/vega exposure.
Summarizing this example, the upside delta risk is negative (unlimited), the downside delta risk is negative (limited), and the kappa/vega risk is positive (limited). This position risk profile re sembles a bear cartwheel (see also the figure on page 81 in the Appendix to this chapter). Note that what is of importance is the number of options, in full delta points, rather than the dollar value of the net option position or its delta neutrality.
A trader must always know what the effect would be on the profitlloss of a position if futures prices were to experience a se ries of limit price moves on the upside or downside, or if implied volatility levels went from normal to high ranges (or from high to normal in high-volatility periods). If any of these trading events, no matter how unlikely, were to happen, would the position ex perience an unlimited loss? If the answer is yes, the position is catastrophically risk exposed.
These risk determinations are only approximate and may sometimes be deceptive, and of course they do not include time spread risks. What is often critical to the risk of a carryover po sition is the futUres standard deviation between the constituent strike spreads. Obviously, a long 110 call may be a good delta hedge against a short 100 call if the standard deviation is 15 points, but not if it is only 2 points.
68 POSITION RISK PROFILES
The risk determinations above also do not indicate the absolute dollar risk exposure by delta or kappa/vega, or the risk modalities, which are very important for position adjustment. For these rea sons, option software analysis of position risk is almost indispens able. Nevertheless, a prudent trader should know how to do quick catastrophic risk determinations without such assistance.
APPENDIX: Position Risk Profiles-Selected Single-Month Positions
Positions
1. Long Call Buy 1 100 call 10. Long Call Short 1 100 call
2. Short Call Sell 1 100 call Ratio Spread Long 2 105 calls
Buy 1 100 put 1 1. Short Call Long 1 100 call
3. Long Put Ratio Spread Short 2 105 calls
4. Short Put Sell 1 100 put 12. Bear Cartwheel Long 1 100 call
5. Long Straddle Buy 1 100 call Short 2 105 calls
Buy 1 100 put Short 1 100 put
6. Short Straddle Sell 1 100 call Long 2 95 puts
Sell 1 100 put 13. Long Butterfly Short 1 100 call 7. Bull Spread Buy 1 95 call Short 1 100 put Long 1 105 call
Sell 1 105 call Long 1 95 put
8. Bear Spread Buy 1 105 put 14. Long Wrangle Short 1 100 call
Sell 1 95 put Long 2 105 calls
9. Bull Fence Buy 1 105 call Short 1 100 put
Sell 1 95 put Long 2 95 puts
Where Futures price 100
Day to Expiration 30 Implied Volatility 15 Interest Rate 10 1 Standard Deviation 5.7 2 SD 1 1 .4 ScaJe Key:
A. $ = Dollar payoff at expiration (solid line) and at 30 days (light line).
B. A = Delta
C. r = Gamma
C. e = Theta
C. K = KappaJVega*
*Vega, of course, is not a Greek letter and therefore has no direct Greek alpha betical equivalent but in popular usage has become the substitute for kappa.
70 $ 5 .00 A 0 -$5.00 1 .00 B 0 -1 .00 +$ . 1 0 C 0 -$ . 1 0 Long Cal1 80 90 $ e o 1 2 SO 1 00 Futures Price ($) 1 1 0
POSITION RISK PROFILES
RISK DETERMINATION 71 Short Call $ 5.00 $ A 0 -$5 00 1 00 B 0 -' 00 +$ 1 0 e C 0 -$ 1 0 r 0 1 SO 80 90 1 00 1 1 0 1 20 Futures Price ($)
72 $ 5 00 A 0 -$ 5 00 1 00 B 0 -1 00 +$ 1 0 C 0 -$ l O Long Put BO <)0 $ o 1 2 SD 1 00 1 1 0 Futures Price ($)
POSITION RISK PROFILES
RISK DETERMINATION $5 00 A 0 -$5 00 1 00 B 0 - 1 00 +$ 1 0 c 0 -$ 1 0 Short Put 80 90 $ o 2 SD 1 00 1 1 0 Futures Price ($) 73 1 20
74 $5 00 A 0 -$5 00 1 00 B 0 -1 00 +$ 1 0 C 0 -$ 1 0 Long Straddle 80 90 o 2 5D 1 00 1 1 0 Futures Price ($)
POSITION R ISK PROFILES
RISK DETERMINATION 75 Short Straddle $5.00 A 0 -$5 00 1 00 B 0 -1 00 +$ 1 0 C 0 -$ 1 0 2 SD 80 90 1 00 1 1 0 1 20 Futures Price ($)
76 POSITION RISK PROFILES