Chapter 10 Bull and Bear Spreads

Chapter 10 Bull and Bear Spreads
第十章 牛市价差和熊市价差

While delta neutral strategies are perhaps the most popular among active option traders, many traders prefer to trade with a bullish or bearish perspective in the underlying instrument. The trader who wishes to take a directional position in the underlying instrument has the choice of doing so in either the instrument itself, buying or selling a futures contract or stock, or in the option market. If he chooses the option market, the trader has the opportunity to integrate option pricing theory into a bull or bear strategy in order to take advantage of theoretically mispriced options.

虽然 delta 中性策略在活跃的期权交易者中很受欢迎，但许多交易者还是更倾向于对标的资产采取看涨或看跌的立场。希望对标的资产进行方向性操作的交易者可以选择直接交易该资产——买入或卖出期货合约或股票，或在期权市场中操作。如果选择期权市场，交易者便有机会将期权定价理论融入看涨或看跌策略中，以利用理论上被错误定价的期权。

NAKED POSITIONS
裸头寸

Since the purchase of calls or the sale of puts will create a positive delta position, and the sale of calls or purchase of puts will create a negative delta position, we can always take a directional position in a market by taking an appropriate naked position in either calls or puts. If all options are overpriced (high implied volatility), we might sell puts to create a bullish position, or sell calls to create a bearish position. If all options are underpriced (low implied volatility), we might buy calls to create a bullish position, or buy puts to create a bearish position.

由于买入看涨期权或卖出看跌期权会形成正 delta 头寸，而卖出看涨期权或买入看跌期权会形成负 delta 头寸，因此我们可以通过适当的裸头寸来在市场中采取方向性操作。如果所有期权被高估（隐含波动率较高），我们可以卖出看跌期权构建看涨头寸，或卖出看涨期权构建看跌头寸。如果所有期权被低估（隐含波动率较低），则可以买入看涨期权构建看涨头寸，或买入看跌期权构建看跌头寸。

The problem with this approach is that, as with all non-hedged positions, there is very little margin for error. If we purchase options, we will lose money not only if the market moves in the wrong direction, but also if the market fails to move enough in our favor to offset the time premium in the option. If we sell options, we face the prospect of unlimited risk if the market moves violently against us. An experienced trader will always look for a way to improve the risk/reward characteristics of his position by looking for positions with the greatest possible margin for error. This philosophy applies no less to directional strategies than to volatility strategies. In directional strategies, as in volatility strategies, this can often be done by finding an appropriate spread.

这种方式的问题在于，与所有非对冲头寸一样，容错空间非常小。如果买入期权，不仅市场走势不利会造成亏损，即使市场未能足够有利于我们以抵消期权的时间价值也会亏损。而如果卖出期权，则市场剧烈波动可能带来无限风险。有经验的交易者总会寻找改善头寸风险回报的方式，确保头寸有尽可能大的容错空间。这一理念同样适用于方向性策略和波动率策略。在方向性策略中，也可以通过寻找合适的价差来实现。

BULL AND BEAR RATIO SPREADS
牛市和熊市比例价差

If a trader believes that implied volatility is too high, one sensible strategy is a ratio vertical spread. For example, with the underlying market at 100 suppose a June 100 call has a delta of 50 and a June 110 call has a delta of 25. A delta neutral spreader can:

buy 1 June 100 call (50)
sell 2 June 110 calls (25)

当交易者认为隐含波动率过高时，可以选择使用比例垂直价差策略。例如，在标的市场价格为 100 的情况下，假设 6 月 100 看涨期权的 delta 为 50，而 6 月 110 看涨期权的 delta 为 25。此时，采用 delta 中性策略的交易者可以：

买入 1 份 6 月 100 看涨期权（50）
卖出 2 份 6 月 110 看涨期权（25）

Since the spread is delta neutral, it has no particular preference for upward or downward movement in the underlying market.

由于该价差为 delta 中性，对标的市场的上涨或下跌没有特别倾向。

Now suppose the same trader believes that this ratio vertical spread is a sensible strategy, but at the same time he is also bullish on the market. There is no law that requires him to do this spread in a delta neutral ratio. If he wants this spread to reflect his bullish sentiment, he might change the ratio slightly:

buy 2 June 100 calls (50)
sell 3 June 110 calls (25)

现在假设同一交易者认为这个比例垂直价差是合理的策略，同时他对市场持看涨观点。没有规定必须按照 delta 中性的比例来进行价差操作。如果他希望反映自己的看涨情绪，可以稍微调整比例：

买入 2 份 6 月 100 看涨期权（50）
卖出 3 份 6 月 110 看涨期权（25）

The trader has essentially the same ratio vertical spread, but with a bullish bias. This is reflected in the total delta for the position of +25.

该交易者的比例垂直价差现在带有看涨倾向，整体 delta 为 +25。

There is, however, an important limitation using this type of ratio strategy to create a bullish or bearish position. In our example the trader is initially bullish, but the position is still a ratio vertical spread. As such, it has a negative gamma. If the trader has underestimated volatility, and the underlying market moves up too quickly, the spread can invert from a positive to a negative delta. If the market rises far enough, to 130 or 140, eventually all options will go deeply into-the-money and the deltas of both the June 100 and June 110 calls will approach 100. Eventually, the trader will be left with a delta position of -100. Even though the trader was correct in his bullish sentiment, the position was primarily a volatility spread, so that the volatility characteristics of the position eventually outweighed any considerations of market direction.

然而，用此类比例策略构建看涨或看跌头寸有一个重要限制。在我们的例子中，交易者最初持看涨立场，但该头寸仍为比例垂直价差，因此具有负 gamma。如果交易者低估了波动性，而标的市场上涨速度过快，该价差的 delta 可能会从正变为负。若市场上涨到 130 或 140，所有期权将变为深度实值状态，6 月 100 和 6 月 110 的看涨期权的 delta 将接近 100，最终交易者将持有-100 的 delta 头寸。尽管交易者的看涨预期是正确的，但由于该头寸主要是波动率价差，波动率的影响最终超过了市场方向的考虑。

The delta can also invert with a backspread. Unlike a negative gamma position, where the inversion is caused by swift price movement in the underlying contract, a backspread can invert when the market is less volatile than expected. For example, suppose that with the underlying market at 100 a trader believes that implied volatility is too low. He might decide on a delta neutral backspread:

buy 2 June 110 calls (25)
sell 1 June 100 call (50)

delta 也可能在反向价差中发生倒置。与负 gamma 头寸因标的合约价格快速变动而导致倒置不同，反向价差在市场波动性低于预期时可能发生倒置。例如，在标的市场价格为 100 的情况下，交易者认为隐含波动率过低，可能选择 delta 中性反向价差：

买入 2 份 6 月 110 看涨期权（25）
卖出 1 份 6 月 100 看涨期权（50）

If, however, he is bullish on the market, he can, as in the last example, change the ratio to reflect this sentiment:

buy 3 June 110 calls (25)
sell 1 June 100 call (50)

如果他对市场持看涨立场，可以如前例调整比例以反映该情绪：

买入 3 份 6 月 110 看涨期权（25）
卖出 1 份 6 月 100 看涨期权（50）

His delta position of +25 reflects this bullish bias.

此时的 delta 头寸为 +25，带有看涨倾向。

We know that as time passes, or as volatility declines, all deltas move away from 50. If time begins to pass with no movement in the underlying contract, the delta of the June 100 call will remain at 50, but the delta of the June 110 call will start to decline. If, after a period of time, the delta of the June 100 call declines to 10, the delta of the position will no longer be +25, but will instead be -20. Since this spread is a volatility spread, the primary consideration, as before, is the volatility of the market. Only secondarily are we concerned with the direction of movement. If the trader overestimates volatility, and the market moves more slowly than expected, the spread which was initially delta positive can instead become delta negative.

随着时间推移或波动率下降，所有 delta 值都会偏离 50。如果标的合约没有波动，6 月 100 看涨期权的 delta 会保持在 50，而 6 月 110 看涨期权的 delta 则会开始下降。假设一段时间后，6 月 100 看涨期权的 delta 降至 10，此时头寸 delta 将不再是 +25，而是-20。由于该价差是波动率价差，波动率仍然是主要考量因素，市场方向则其次。如果交易者高估了波动性，而市场移动比预期更缓慢，最初为正 delta 的价差可能转为负 delta。

BULL AND BEAR BUTTERFLIES AND TIME SPREADS
牛市和熊市蝶式价差及时间价差

Butterflies and one-to-one (non-ratioed) time spreads can also be chosen in such a way as to reflect a trader's bullsh or barsh blad like ratio spreads, however, their delta characteristics can also invert as market conditions change.

与比例价差类似，蝶式价差和一对一（非比例）时间价差也可以反映交易者的牛市或熊市观点。不过，这类策略的 delta 也会随市场条件变化而反转。

With the underlying market at 100, a delta neutral trader might buy the June 95/100/105 call butterfly (buy a 95 call, sell two 100 calls, buy a 105 call). The trader hopes that the market will sit still at 100 so that at expiration the butterfly will widen to its maximum value of five points. If, however, a trader wants to buy a butterfly, but is also bullish on the market, he can choose a butterfly where the inside exercise price is above the current price of the underlying contract. If the underlying market is currently at 100, he might choose to buy the June 105/110/115 call butterfly. Since this position wants the underlying market at 110 at expiration, and it is currently at 100, the position is a bull butterfly. This will be reflected in the position having a positive delta.

在标的市场价格为 100 时，一位 delta 中性交易者可能买入 6 月 95/100/105 看涨期权蝶式价差（买入 95 看涨期权，卖出两个 100 看涨期权，买入 105 看涨期权）。该交易者希望市场停留在 100 附近，以便在到期时蝶式价差扩大到最大值五点。然而，如果交易者看涨市场，也可以选择行权价高于当前标的价格的蝶式价差。如果当前标的市场为 100，交易者可能选择买入 6 月 105/110/115 看涨蝶式价差。由于该头寸希望标的市场在到期时达到 110，而目前价格为 100，因此该头寸属于牛市蝶式，表现为正 delta。

Unfortunately, if the underlying market moves too swiftly, say to 120, the butterfly can invert from a positive to a negative delta position. Since at expiration the butterfly always has its maximum value with the underlying contract right at the inside exercise price, the trader will now want the market to fall back from 120 to 110. Whenever the underlying market is below 110, the position will be bullish; whenever the underlying market is above 110, the position will be bearish.

如果标的市场快速上涨至 120，蝶式价差可能会从正 delta 反转为负 delta。由于在到期时蝶式价差的最大值总是出现在标的合约正好位于中间行权价时，此时交易者会希望市场从 120 回落至 110。当标的市场低于 110 时，该头寸呈现多头；当标的市场高于 110 时，该头寸呈现空头。

Conversely, if the trader is bearish, he can always choose to buy a butterfly where the inside exercise price is below the current price of the underlying market. But again, if the market moves down too quickly and goes through the inside exercise price, the position will invert from a negative to a positive delta.

相反，如果交易者看空，可以选择行权价低于当前标的价格的蝶式价差。不过，如果市场快速下跌并穿过中间行权价，则该头寸会从负 delta 反转为正 delta。

In a similar manner, a trader can choose time spreads that are either bullish or bearish. A long time spread always wants the near-term contract to expire exactly at-the-money. A long time spread will be initially bullish if the exercise price of the time spread is above the current price of the underlying market. (footnote 1: In the futures market the situation may be complicated by the fact that different futures months may be trading at different prices. Instead of choosing a traditional time spread where both options have the Same exercise price, the trader may have to choose a diagonal spread to ensure that the position is either bullish (delta positive) or bearish (delta negative).) With the underlying market at 100, the June/March 110 call time spread (buy the June 110 call/sell the March 110 call) will be bullish since the trader will want the underlying market to rise to 110 by March expiration. The June/March 90 call time spread (buy the June 90 call/sell the March 90 call) will be bearish, since the trader will want the underlying market to fall to 90 by March expiration. However, like a long butterfly, a long time spread has a negative gamma, so that if the market moves through the exercise price, the delta can invert. If the market moves from 100 to 120, the June/March 110 call time spread, which was initially bullish, will become bearish. If the market moves from 100 to 80, the June/March 90 call time spread, which was initially bearish, will become bullish.

同样，交易者可以选择偏向牛市或熊市的时间价差。长时间价差总是希望近期合约在到期时恰好位于平值。如果时间价差的行权价高于当前标的价格，长时间价差会呈现初始看涨立场。（脚注 1：在期货市场中，由于不同月份的期货价格可能不同，交易者可能需选择对角价差，以确保头寸偏多（正 delta）或偏空（负 delta）。）在标的市场价格为 100 时，6 月 /3 月 110 看涨时间价差（买入 6 月 110 看涨期权 / 卖出 3 月 110 看涨期权）为看涨头寸，因为交易者希望标的市场在 3 月到期时上涨至 110。6 月 /3 月 90 看涨时间价差（买入 6 月 90 看涨期权 / 卖出 3 月 90 看涨期权）则为看跌头寸，因为交易者希望标的市场在 3 月到期时下跌至 90。然而，像长蝶式价差一样，长时间价差具有负 gamma，因此如果市场穿越行权价，delta 可能会反转。如果市场从 100 上升至 120，最初看涨的 6 月 /3 月 110 看涨时间价差将转为空头。如果市场从 100 跌至 80，最初看跌的 6 月 /3 月 90 看涨时间价差将转为多头。

VERTICAL SPREADS
垂直价差

It is common for traders to take bull or bear market positions by choosing appropriate ratio spreads, butterflies, or time spreads, but in each of these positions volatility is still the primary concern. A trader can be right about market direction, but if he is wrong about volatility, the spread may not retain the directional characteristics that the trader intended.

交易者通常会通过选择适当的比例价差、蝶式价差或时间价差来采取牛市或熊市头寸，但这些策略中波动性仍然是主要关注点。即使交易者对市场方向判断正确，但如果对波动性判断错误，该价差可能无法维持预期的方向性特征。

If a trader focuses initially on the direction of the underlying market, he might look for a spread where the directional characteristics of the spread are the primary concern, and volatility is of secondary importance. He would like to be certain that if the spread is initially bullish (delta positive) it will remain bullish under all possible market conditions. If it is initially bearish (delta negative) it will remain bearish under all possible market conditions.

如果交易者更关注标的市场的方向性特征，可能会选择一类策略，以方向性为主要考量，波动性为次要考量。他希望确保价差初始为看涨（正 delta）时，在任何市场条件下仍保持看涨；若初始为看跌（负 delta），在任何市场条件下也仍为看跌。

The class of spreads which meet the above requirements are known as vertical spreads. A vertical spread always consists of one long (purchased) option and one short (sold) option, where both options are of the same type (either both calls or both puts) and expire at the same time. The options are distinguished only by their different exercise prices. Typical vertical spreads might be:

buy 1 June 100 call
sell 1 June 105 call

or

buy 1 March 105 put
sell 1 March 95 put

符合上述条件的策略被称为垂直价差。垂直价差始终由一个买入期权和一个卖出期权构成，两个期权类型相同（均为看涨或看跌期权），且具有相同到期日，仅行权价不同。典型的垂直价差包括：

买入 1 份 6 月 100 看涨期权
卖出 1 份 6 月 105 看涨期权

或

买入 1 份 3 月 105 看跌期权
卖出 1 份 3 月 95 看跌期权

Vertical spreads are not only initially bullish or bearish, but they remain bullish or bearish no matter how market conditions change. Two options which have different exercise prices but which are identical in every other respect cannot have identical deltas. In the first example, where the trader is long a June 100 call and short a June 105 call, the June 100 call will always have a delta greater than the June 105 call. If both options go deeply into-the-money or very far out-of-the-money, the deltas may be almost identical. Even then, the June 100 call will have a delta fractionally greater than the June 105 call. In the second example, no matter how market conditions change, the March 105 put will always have a delta greater than the March 95 put.

垂直价差不仅在初始时为看涨或看跌，并且无论市场条件如何变化，方向性特征都不变。具有不同行权价但其他条件相同的两个期权，其 delta 不可能相同。在第一个例子中，交易者持有 6 月 100 看涨期权并卖出 6 月 105 看涨期权时，6 月 100 看涨期权的 delta 始终大于 6 月 105 看涨期权。如果两个期权都深度价内或深度价外，delta 可能几乎相同，但 6 月 100 看涨期权的 delta 始终会比 6 月 105 看涨期权略大。在第二个例子中，无论市场如何变化，3 月 105 看跌期权的 delta 始终大于 3 月 95 看跌期权。

At expiration, a vertical spread will have a minimum value of zero, if both options are out-of-the-money, and a maximum value of the amount between exercise prices, if both options are in-the-money. If the underlying market is below 100 at expiration, the June 100/105 call spread will be worthless because both options will be worthless. If the market is above 105, the same spread will be worth five points because the June 100 call will be worth exactly five points more than the June 105 call. Similarly, the March 95/105 put spread will be worthless if the underlying market is above 105 at expiration, and it will be worth 10 points if the market is below 95. The expiration values of typical bull and bear vertical spreads are shown in Figures 10-1 and 10-2.

到期时，若两个期权均处于价外，垂直价差的最低价值为零；若两个期权均处于价内，则最大价值为行权价之差。若到期时标的市场低于 100，6 月 100/105 看涨价差将一文不值，因为两个期权均无价值；若市场高于 105，该价差将价值五点，因为 6 月 100 看涨期权的价值比 6 月 105 看涨期权多五点。同样，若到期时标的市场高于 105，3 月 95/105 看跌价差将无价值；若市场低于 95，该价差将价值 10 点。典型的牛市和熊市垂直价差的到期价值如图 10-1 和图 10-2 所示。

Since a vertical spread at expiration will always have a value between zero and the amount between exercise prices, a trader can expect the price of such a spread to be somewhere within this range. A 100/105 call vertical spread will trade for some amount between zero and five points; a 95/105 put vertical spread will trade for some amount between zero and 10 points. The exact value will depend on the likelihood of the underlying market finishing below the lower exercise price, above the higher exercise price, or somewhere in between. If the market is currently at 80 and gives little indication of rising, the price of the 100/105 call vertical will be close to zero, while the price of the 95/105 puts vertical will be close to 10. If the market is currently at 120 with little likelihood that it will fall, the price of the 100/105 call vertical will be close to five points, while the price of the 95/105 put vertical will be close to zero.

由于垂直价差到期时的价值始终介于零和行权价差之间，交易者可以预期其价格在这个范围内。例如，100/105 看涨垂直价差的交易价格将在零到五点之间；95/105 看跌垂直价差的交易价格将在零到十点之间。具体的价差值取决于标的市场最终收于下限行权价之下、上限行权价之上，或两者之间的概率。如果当前市场在 80，且上涨可能性较低，则 100/105 看涨垂直价差价格接近于零，而 95/105 看跌垂直价差则接近 10 点。如果当前市场在 120，且下跌可能性较低，则 100/105 看涨垂直价差价格接近五点，而 95/105 看跌垂直价差则接近于零。

If a trader wants to do a vertical spread, he has essentially four choices. If he is bullish he can choose a bull vertical call spread or a bull vertical put spread; if he is bearish he can choose a bear vertical call spread or a bear vertical put spread. For example:

bull call spread:

buy a June 100 call
sell a June 105 call

bull put spread:

buy a June 100 put
sell a June 105 put

bear call spread:

sell a June 100 call
buy a June 105 call

bear put spread:

sell a June 100 put
buy a June 105 put

交易者在做垂直价差时基本有四种选择。若看涨，可选择看涨垂直看涨价差或看涨垂直看跌价差；若看跌，可选择看跌垂直看涨价差或看跌垂直看跌价差。例如：

看涨看涨价差：

买入 6 月 100 看涨期权
卖出 6 月 105 看涨期权

看涨看跌价差：

买入 6 月 100 看跌期权
卖出 6 月 105 看跌期权

看跌看涨价差：

卖出 6 月 100 看涨期权
买入 6 月 105 看涨期权

看跌看跌价差：

卖出 6 月 100 看跌期权
买入 6 月 105 看跌期权

Note that a trader who is bullish can buy a 100 call and sell a 105 call, or buy a 100 put and sell a 105 put. And a trader who is bearish can buy a 105 call and sell a 100 call, or buy a 105 put and sell 100 put. This may seem counter intuitive, since one expects spreads which consist of puts to have characteristics which are the opposite of those which consist of calls. Regardless of whether a vertical spread consists of calls or puts, whenever a trader buys the lower exercise price and sells the higher exercise price, the position is bullish; whenever a trader buys the higher exercise price and sells the lower exercise price, the position is bearish. Call and put vertical spreads which expire at the same time and which consist of the same exercise prices have approximately the same delta, so they have approximately the same bullish or bearish characteristics. (footnote 2: We are assuming for the moment that all options are European, with no possibility of early exercise.)

需要注意的是，看涨的交易者可以买入 100 看涨期权并卖出 105 看涨期权，或者买入 100 看跌期权并卖出 105 看跌期权；而看跌的交易者可以买入 105 看涨期权并卖出 100 看涨期权，或者买入 105 看跌期权并卖出 100 看跌期权。看似反直觉，因为通常认为由看跌期权组成的价差特性应与看涨期权相反。无论垂直价差是由看涨期权还是看跌期权构成，只要买入低行权价、卖出高行权价，仓位就是看涨的；反之，买入高行权价、卖出低行权价，则仓位为看跌。到期时间相同且行权价格相同的看涨和看跌垂直价差的 delta 基本一致，因此它们的看涨或看跌特性也基本相同。（脚注 2：这里假设所有期权为欧式期权，不能提前行权。）

Given the many different exercise prices and expiration months available, how can a trader choose the vertical spread which best reflects his directional expectations and which gives him the best chance to profit from those expectations?

在多种行权价格和到期月份的期权选择中，交易者如何选择最符合其市场预期、并有望从中获利的垂直价差？

Since options have a limited life with a fixed expiration date, a trader who wants to use options to take advantage of an expected market move must first determine his time horizon. Is the movement likely to occur in the next month, in the next three months, in the next nine months? If it is currently May and the trader foresees upward movement, but believes the movement is unlikely to occur within the next two months, it does not make much sense to take a position in June options. If his expectations are long-term, he may have to take his position in September, or even December, options. Of course, as he moves further and further out in time, market liquidity may become more, rie wile co dete us are to aken in conside is is he very confident, and therefore willing to take a very large directional position? Or is he less certain and willing to take only a limited position? Two factors determine the total directional characteristics of a vertical spread:

1. The delta of the specific vertical spread
2. The size in which the spread is executed

由于期权有固定的到期日，想利用期权捕捉市场走势的交易者必须先确定时间预期。是预期价格在下个月、未来三个月，还是九个月内发生变化？假设当前为 5 月，交易者预期上涨，但认为在未来两个月内不太可能发生，那么选择 6 月期权意义不大。如果是长期预期，可能需要考虑 9 月甚至 12 月期权。当然，时间越长，市场流动性可能会有所下降。在确定仓位时，还需考虑两个因素：交易者是否非常自信，因此愿意承担较大的方向性仓位？还是不确定，只愿意承担有限的仓位？垂直价差的方向性特性取决于两个因素：

1. 特定垂直价差的 delta 值
2. 执行价差的规模

For example, a trader who wants to take a position which is 500 deltas long (equivalent to purchasing five underlying contracts) can either choose a vertical spread which is 50 deltas long and execute it 10 times, or choose a vertical spread which is 25 deltas long and execute it 20 times. Both positions will leave him long 500 deltas.

例如，若交易者希望建立 500 delta 的多头仓位（相当于买入五份标的合约），可以选择一个 50 delta 的垂直价差并执行 10 次，或选择一个 25 delta 的垂直价差并执行 20 次，最终都能使仓位达到 500 delta。

The delta value of a vertical spread is determined by various factors: time to expiration, volatility, and distance between exercise prices. Since a trader will be required to choose an expiration date which covers the period of expected directional movement, and since a trader will always make his best estimate of volatility over a given period, in practice the delta will be a function of the exercise prices which he chooses. The greater the distance between exercise prices, the greater the delta value associated with the spread. A 95/110 bull spread will be more bullish than a 100/110 bull spread, which will, in turn, be more bullish than a 100/105 bull spread. This is shown in Figure 10-3.

垂直价差的 delta 值受多个因素影响：到期时间、波动率和行权价之间的距离。交易者需要选择覆盖预期走势期间的到期日，并对该期间的波动率进行最佳估计，因此实际 delta 值主要取决于选择的行权价差。行权价差越大，价差的 delta 值越高。例如，95/110 的看涨价差比 100/110 的看涨价差更为看涨，而 100/110 看涨价差又比 100/105 更为看涨，如图 10-3 所示。

Once a trader decides on an expiration month in which to take his directional position, he must decide which specific spread is best. That is, he must decide which exercise prices to use. A common approach is to focus on the at-the-money options. If a trader does this, he will have the following choices:

Bull call spread:	buy an in-the-money call sell an at-the-money call	or	buy an at-the-money call sell an out-of-the-money call
Bear call spread:	buy an at-the-money call sell an in-the-money call	or	buy an out-of-the-money call sell an at-the-money call
Bull put spread:	buy an at-the-money put sell an in-the-money put	or	buy an out-of-the-money put sell an at-the-money put
Bear put spread:	buy an in-the-money put sell an at-the-money put	or	buy an at-the-money put sell an out-of-the-money put

当交易者确定了方向性仓位的到期月份后，需要选择具体的价差组合，即决定使用哪些行权价。通常的方法是关注平值期权。若采用这种方法，交易者将有以下选择：

牛市看涨价差：	买入一个价内看涨期权 卖出一个平价看涨期权	或	买入一个平价看涨期权 卖出一个价外看涨期权
熊市看涨价差：	买入一个平价看涨期权 卖出一个价内看涨期权	或	买入一个价外看涨期权 卖出一个平价看涨期权
牛市看跌价差：	买入一个平价看跌期权 卖出一个价内看跌期权	或	买入一个价外看跌期权 卖出一个平价看跌期权
熊市看跌价差：	买入一个价内看跌期权 卖出一个平价看跌期权	或	买入一个平价看跌期权 卖出一个价外看跌期权

With four different bull spreads and four different bear spreads available, how can the trader choose the spread which represents the best value? One way to do this is to use a theoretical pricing model to evaluate several different vertical spreads. This has been done in Figure 10-4 for six different options on an underlying futures contract: an in- themoney, at-the-money, and out-of-the-money call, and an in-the money, at-the-money, and out-of-the-money put. The assumptions include an underlying price of 100, 12 weeks to expiration, a volatility estimate of 20%, and an interest rate of 8%.

在四种不同的看涨价差和四种不同的看跌价差中，交易者如何选择最佳价差？一种方法是使用理论定价模型评估多种垂直价差组合。图 10-4 对六种不同的期权进行了此类评估：实值、平值和虚值的看涨期权，以及实值、平值和虚值的看跌期权。假设条件为标的价格 100、到期时间 12 周、波动率估值 20%、利率 8%。

Suppose we are interested in doing a bull call spread. Two possibilities are to buy a 95 call and sell a 100 call, or buy a 100 call and sell a 105 call. The spreads and their theoretical values are:

Spread	Theoretical Value	Delta
95/100 call spread	2.87	+20
100/105 call spread	1.88	+20

假设我们想做一个看涨认购价差，有两种选择：买入 95 看涨期权并卖出 100 看涨期权，或买入 100 看涨期权并卖出 105 看涨期权。两种价差及其理论价值如下：

价差组合	理论价值	Delta 值
95/100 看涨价差	2.87	+20
100/105 看涨价差	1.88	+20

With both spreads having the same delta of +20, it may appear that the 100/105 spread is a better value since it is likely to be less expensive. But should this be the sole determinant in choosing a strategy? As in all spreads, the option trader's goal is to create a position with positive theoretical edge, by either purchasing high value at a low price, or selling low value at a high price. In order to achieve this goal we need to know not only the theoretical values of the spreads, but also the prices of the spreads in the marketplace.

由于两种价差的 Delta 均为 +20，100/105 价差可能显得更划算，因为其成本较低。但是否应仅此作为选择策略的依据？在所有价差中，期权交易者的目标是通过低价买入高价值，或高价卖出低价值，以获得正向理论收益。为实现此目标，不仅需要了解价差的理论价值，还需要了解市场上的实际价格。

From an option trader's point of view, the relative prices of options in the marketplace is usually represented by the implied volatility. We know the value of the spreads based on our volatility input of 20%. What might be the prices of the spreads if implied volatility in the option market is something other than 20%? We can answer the question by using a theoretical pricing model with the same underlying price, time to expiration, and interest rate, but with volatilities lower and higher than 20%. This has been done in Figure 10-5 using volatilities of 16% and 24%, along with our estimated volatility of 20%.

从期权交易者的角度看，市场中期权的相对价格通常通过隐含波动率表示。我们知道在 20% 的波动率输入下的价差理论价值。如果市场隐含波动率不是 20%，价差价格可能会是多少？可以通过使用相同的标的价格、到期时间和利率，但设置低于和高于 20% 的波动率来计算。图 10-5 展示了在 16%、20% 和 24% 波动率下的理论价格。

Going back to our bull call spread, suppose implied volatility is lower than our estimate of 20%, say 16%. We can see that the price of the 95/100 spread will be approximately 3.01, while the price of the 100/105 spread will be approximately 1.78. We have two choices: we can pay 3.01 for a spread that is worth 2.87 (the 95/100 spread), or we can pay 1.78 for a spread that is worth 1.88 (the 100/105 spread). Clearly, we will prefer the 100/105 spread since it will result in a positive theoretical edge of .10. If we were to buy the 95/100 spread we would end up with a negative theoretical edge of .14.

回到我们的看涨价差，假设隐含波动率低于预估的 20%，为 16%。此时，95/100 价差约为 3.01，而 100/105 价差约为 1.78。我们有两个选择：要么花费 3.01 买入理论价值为 2.87 的 95/100 价差，要么花费 1.78 买入理论价值为 1.88 的 100/105 价差。显然，我们会选择 100/105 价差，因为这将带来 0.10 的正向理论收益。而若选择 95/100 价差，则会得到-0.14 的负向理论收益。

Now suppose implied volatility is higher than our estimate of 20%, say 24%. We can see that the price of the 95/100 spread will be approximately 2.77, while the price of the 100/105 spread will be approximately 1.94. Again, we have two choices: we can pay 2.77 for a spread that is worth 2.87 (the 95/100 spread), or we can pay 1.94 for a spread that is worth 1.88 (the 100/105 spread). Now we will prefer the 95/100 spread since it will result in a positive theoretical edge of .10. The 100/105 spread would result in a negative theoretical edge of .06.

假设隐含波动率高于预估的 20%，为 24%。此时，95/100 价差约为 2.77，而 100/105 价差约为 1.94。我们依然有两个选择：要么花费 2.77 买入理论价值为 2.87 的 95/100 价差，要么花费 1.94 买入理论价值为 1.88 的 100/105 价差。此时，我们会选择 95/100 价差，因为它将带来 0.10 的正向理论收益，而 100/105 价差则会产生-0.06 的负向理论收益。

What's happening here? Even though both spreads have the same delta values, under one set of circumstances we seem to prefer the 95/100 spread, while under different circumstances we seem to prefer the 100/105 spread. The reason becomes clear if we recall one of the characteristics of option evaluation introduced in Chapter 6:

这是什么原因？尽管两种价差的 Delta 值相同，但在不同情况下，我们的偏好却不同。这一现象的原因在于第六章介绍的一个期权定价特性：

If we consider three options, an in-the-money, at-the-money, and out-of-the-money option which are identical except for their exercise prices, the at-the-money option is always the most sensitive in total points to a change in volatility.

如果我们考虑三个相同到期日的期权，仅行权价不同，其中一个价内、一个平值、一个价外，那么平值期权对波动率变化最为敏感。

This means that when all options appear overpriced because we believe implied volatility is too high, in total points the at-the-money option will be the most overpriced. When all options appear underpriced because we believe implied volatility is too low, in total points the at-the-money option will be the most underpriced. This characteristic leads to a very simple rule for choosing bull and bear vertical spreads:

这意味着当我们认为隐含波动率偏高时，平值期权的总点数最为高估；而当我们认为隐含波动率偏低时，平值期权的总点数最为低估。这一特性引出了选择看涨和看跌垂直价差的简单规则：

If implied volatility is too low, vertical spreads should focus on purchasing the at-the-money option. If implied volatility is too high, vertical spreads should focus on selling the at-the-money options.

如果隐含波动率偏低，垂直价差应重点购买平值期权；如果隐含波动率偏高，垂直价差应侧重卖出平值期权。

Now we can see why the 100/105 call spread is a better value if implied volatility is 16%, while the 95/100 spread is a better value if implied volatility is 24%. If implied volatility is too low (10%), we want to buy the at-the-money (100) call. Having done this, we have only one choice in order to create a bull spread— sell the out-of-the-money (105) call. If implied volatility is too high (24%), we want to sell the at-the-money (100) call. Having done this, we again have only one choice in order to create a bull spread-buy the in-the-money (95) call.

因此，当隐含波动率为 16% 时，100/105 看涨价差性价比更高；而隐含波动率为 24% 时，95/100 价差更具优势。若隐含波动率过低（10%），我们应买入平值（100）看涨期权，然后卖出价外（105）看涨期权以形成看涨价差。若隐含波动率过高（24%），我们则应卖出平值（100）看涨期权，并买入价内（95）看涨期权。

The same principle is equally true for put vertical spreads. We always want to focus on the at-the-money option, buying it when implied volatility is too low, and selling it when implied volatility is too high. For example, suppose we want to do a bear put spread with implied volatility too low. In this case we want to buy the at-the-money (100) put. Having done this, we are forced to sell the out-of-the-money (95) put to create our bear spread. We can see from Figure 10-5 that we will pay approximately 1.90 for this spread, but the spread is worth 2.04. We will end up with a delta position of -20, and a positive theoretical edge of .14.

这一原则同样适用于看跌垂直价差。我们应优先关注平值期权，在隐含波动率低时买入，隐含波动率高时卖出。例如，当隐含波动率偏低时，我们可进行看跌价差，先买入平值（100）看跌期权，再卖出价外（95）看跌期权。根据图 10-5，我们将为此价差支付约 1.90，但其理论价值为 2.04，最终获得-20 的 Delta 头寸和 0.14 的正理论收益。

Of course, a trader is not required to execute any vertical spread by first buying or selling the at-the-money option. Such spreads always involve two options, and a trader can choose to either execute the complete spread in one transaction, or leg into the spread by trading one option at a time. In the latter case, he may decide to first trade the in-the-money or out-of-the-money option, and trade the at-the-money option at a later time. This is a practical trading decision he will have to make based on market conditions and the amount of risk he is willing to accept. Regardless of how the spread Is executed, the trader should focus on the at-the-money option, either buying it when implied volatility is too low, or selling it when implied volatility is too high.

当然，交易者在执行垂直价差时，不一定需要先买入或卖出平值期权。这种价差总是涉及两个期权，交易者可以选择一次性完成价差，或分步建仓，先交易价内或价外期权，再交易平值期权。这取决于市场状况和风险承受能力。但无论价差如何执行，交易者应聚焦平值期权，在隐含波动率低时买入，隐含波动率高时卖出。

In practice it is unlikely that one option will be exactly at-the-money, so that it may be difficult to decide which option to buy and which to sell. In such a case, it is usually best to focus on the option which is closest to at-the-money. If the underlying market is at 103, with 95, 100, 105, and 110 calls available, it is logical to focus on the 105 call since it is closest to at-the-money. If implied volatility is too low, a trader will want to buy the 105 call; if implied volatility is too high, a trader will want to sell the 105 call. He can then trade a different option in order to create a bull or bear vertical spread.

实际操作中，可能没有一个期权完全平值，此时应选择最接近平值的期权。例如，若标的市场在 103 点，有 95、100、105 和 110 的期权可选，则 105 期权最接近平值。若隐含波动率偏低，交易者应买入 105 期权；若隐含波动率偏高，则应卖出 105 期权，再交易另一个期权以形成看涨或看跌垂直价差。

Nor does a trader have to include the option which is closest-to-the-money as part of his spread. A trader who has a strong directional opinion can choose a vertical spread where both options are very far out-of-the-money or very deeply in-the-money. The delta values of such spreads will be very low, but a trader can create a highly leveraged position by executing each spread many times. For example, with the underlying market at 100, a trader who is strongly bullish might buy the 115/120 call spread (assuming such exercise prices are available). The cost of this spread will be very low since there is a high probability that the spread will expire worthless. Hence, the trader will be able to execute the spread many times at a relatively low cost. If he is right and the market does rise above 120, the spread will widen to its maximum value of five points, resulting in very large profits. But regardless of the exercise prices chosen, if implied volatility is low, the trader will attempt to buy the closer-to-the-money option, and if implied volatility is high, the trader will attempt to sell the closer-to-the-money option.

交易者并不一定非要在价差中包含最接近平值的期权。若交易者有较强的方向性观点，可以选择两个都处于深度价外或价内的期权来构建垂直价差。这类价差的 Delta 值会非常低，但交易者可以通过多次执行此类价差来获得高杠杆。例如，当标的市场价格为 100 时，若交易者非常看涨，可以购买 115/120 看涨期权价差（假设该执行价格可用）。由于价差很可能最终失效，该价差成本会很低，因此交易者可以以较低成本多次执行该策略。如果市场真的上涨至 120 以上，该价差会扩展至最大值 5 点，带来可观的利润。但无论选择何种执行价格，若隐含波动率较低，交易者会倾向于买入更接近平值的期权；若隐含波动率较高，则会倾向于卖出更接近平值的期权。

The choice of the at-the-money option is slightly different when we move to stock options. If we define the at-the-money option as the one whose delta is closest to 50, then we may find that the at-the-money option is not always the one whose exercise price is closest to the current price of the underlying contract. This is because the option with a delta closest to 50 will be the one whose exercise price is closest to the forward price of the underlying contract. In stock options, the forward price is the current price of the stock, plus carrying costs on the stock, less expected dividends. Assuming an underlying stock price of 99, six months to expiration, a volatility of 28%, interest rates of 10%, and no dividend, we can see from Figure 10-6 that the 105 call has a delta of 50 and therefore acts like the at-the-money option, even though its exercise price is six points higher than the current underlying price. Therefore, any vertical spread should focus on the 105 call or 105 put.

在股票期权中，平值期权的选择有所不同。如果将 Delta 最接近 50 的期权定义为平值期权，可能会发现该期权的执行价格并不总是最接近标的合约当前价格的那一个。这是因为 Delta 最接近 50 的期权执行价格通常更接近标的合约的远期价格。对于股票期权，远期价格为当前股价加上持有成本减去预期股息。例如，假设标的股票价格为 99，距到期时间为六个月，波动率为 28%，利率为 10%，且无股息。从图 10-6 可以看出，105 看涨期权的 Delta 为 50，因此可视为平值期权，尽管其执行价格比当前标的价格高出 6 点。因此，任何垂直价差都应关注 105 看涨或 105 看跌期权。

The approximate prices of vertical spreads with these stock options at implied volatilities of 23% and 33% are shown in Figure 10-7.

图 10-7 显示了这些股票期权在 23% 和 33% 隐含波动率下的垂直价差的近似价格。

As always, we can never be certain of the delta values of a spread because we can never be certain that we have the correct inputs into the model, in particular, volatility. If volatility turns out to be lower than our estimate, the deltas will move away from 50. If volatility turns out to be higher than our estimate, the deltas will move towards 50 This will change the delta values of the spreads. However, if we work around the at-the-money option, spreads with approximately the same amount between exercise prices will have approximately the same delta values. Our primary problem will be deciding whether implied volatility is too high or too low.

我们永远无法确定价差的确切 Delta 值，因为输入模型的各项数据（尤其是波动率）并不完全准确。如果波动率低于预期，Delta 值会偏离 50；如果波动率高于预期，Delta 值会趋向 50，从而改变价差的 Delta 值。然而，当我们围绕平值期权操作时，执行价格差不多的价差大致会拥有相同的 Delta 值。我们主要的问题是判断隐含波动率是过高还是过低。

Notice that in every case, whether with futures options or with stock options, whether using a low volatility or a high volatility, if one leg of our spread always involves an at-the-money option, the vertical spread which includes the in-the-money option always has a higher price than the spread which includes the out-of-the-money option. A new trader might take the view that since both spreads have approximately the same delta values, it must make sense to always purchase the cheaper spread or sell the more expensive. This ignores the purpose of option evaluation: to consider not only the initial cost of a strategy, but to compare that to the strategy's expected return. In our futures option example, a trader will always have to pay more for the 95/100 call spread than the 100/105 call spread. But if implied volatility is too high, the 95/100 spread will also return more to the trader. To see why, suppose a trader is choosing between the 95/100 bull call spread and the 100/105 bull call spread. Consider three possible cases. In case one, the trader is right and the market moves from 100 to 110. If that happens both spreads will be winners since they will both widen to their maximum value of five points. In case two, the trader is wrong and the market drops to 90. In that case both spreads will be losers since they will both collapse to zero. Now consider case three, where the trader is wrong because he expected the underlying market to rise and it didn't. But he wasn't terribly wrong, because it also didn't fall. It simply remained at 100. If that happens the 100/105 spread will collapse to zero, while the 95/100 spread will still widen to its maximum value of five points. The 95/100 spread is always more valuable than the 100/105 spread because it has time on its side. The 100/105 spread needs the market to rise to show a profit. The 95/100 spread doesn't really need the market to rise; it just needs for the market not to fall. The reader can confirm this by calculating the gamma and theta positions of the spreads in Figure 10-5. The 100/105 spread has a positive gamma and a negative theta, whereas the 95/100 spread has negative gamma and a positive theta.

无论是期货期权还是股票期权，无论波动率高低，如果价差的一条腿总是包含平值期权，那么包含价内期权的垂直价差价格始终高于包含价外期权的价差。新手可能认为，既然这两种价差的 Delta 值大致相同，选择购买更便宜的价差或卖出更贵的价差一定更合理。这种想法忽略了期权评估的真正目的：不仅要考虑策略的初始成本，还需对比策略的预期收益。在我们的期货期权示例中，交易者购买 95/100 看涨期权价差的花费总会高于购买 100/105 看涨期权价差的花费，但如果隐含波动率过高，95/100 价差的回报也会更高。假设交易者在 95/100 和 100/105 看涨价差之间选择，考虑以下三种情况：情况一，市场上升至 110，两种价差都将扩大至最大值 5 点，获利；情况二，市场下跌至 90，两种价差都会归零，亏损；情况三，市场停留在 100，100/105 价差会归零，而 95/100 价差则仍可扩大至 5 点。95/100 价差的优势在于它受时间影响较小，100/105 价差需要市场上涨才能获利，而 95/100 价差只需市场不下跌。图 10-5 显示了这种情况：100/105 价差的 Gamma 为正，Theta 为负，而 95/100 价差的 Gamma 为负，Theta 为正。

If a trader is considering a bull vertical spread, how likely is it that the market will rise? That depends on his ability to predict market direction. At the same time, he must ask himself how likely it is that the market will move. That depends on his ability to predict volatility. If a trader believes there is a good chance the market will move (high volatility), and at the same time believes the movement will be to the upside (bullish on direction), it makes sense to buy the 100/105 spread. On the other hand, if the trader believes the market is unlikely to move significantly (low volatility), but at the same time believes that whatever movement does occur is likely to come on the upside (also bullish), it makes sense to buy the 95/100 spread. In both cases the trader is trying to maximize the return on his investment when he is right and minimize the loss when he is wrong.

如果交易者考虑做多垂直价差，他需要判断市场是否会上涨，这取决于他对市场走势的预测能力。同时，他还需评估市场的波动性，这取决于他对波动率的判断能力。如果他认为市场可能大幅波动（高波动）并且预期方向向上（看涨），选择 100/105 价差较为合理；相反，如果他认为市场波动不大（低波动），但预期波动向上，则选择 95/100 价差更为合适。这样可以在预测正确时最大化收益，预测错误时将损失最小化。

Why might a trader with a directional opinion prefer a vertical spread to an outright long or short position in the underlying instrument? For one thing, a vertical spread is much less risky than an outright position. A trader who wants to take a position which is 500 deltas long can either buy five underlying contracts or buy 25 vertical call spreads with a delta of 20 each. The 25 vertical spreads may sound riskier than five underlying contracts, until we remember that a vertical spread has limited risk while the position in the underlying has open-ended risk. Of course, greater risk also means greater reward. A trader with a long or short position in the underlying market can reap huge rewards if the market makes a large move in his favor. By contrast, the vertical spreader's profits are limited, but he will also be much less bloodied if the market makes an unexpected move in the wrong direction.

为什么有方向观点的交易者会选择垂直价差而非标的合约的直接多头或空头头寸？首先，垂直价差比直接持仓风险更小。一个想要持有 500 Delta 多头头寸的交易者，可以选择买入五份标的合约，也可以买入 25 份 Delta 值为 20 的看涨垂直价差。尽管 25 份垂直价差看似风险更大，但垂直价差的风险是有限的，而标的合约的头寸则风险敞口无限。当然，风险大也意味着潜在收益更高。标的合约多头或空头在市场大幅变动时可能获得巨大回报，而垂直价差的收益是有限的，但当市场意外地向不利方向变动时，其损失也会更少。

Experienced traders accept the fact that they are human, and that their directional forecasts will occasionally be wrong. When that happens the vertical spreader has a distinct advantage over the trader with an outright position in the underlying market. By intelligently estimating volatility, an option trader can decide whether he wants time working for or against him. When he chooses to have time on his side (positive theta), he can sometimes show a profit when the outright trader either breaks even or snows a loss. When he chooses to have time working against him (positive gamma), the option trader's losses when he is wrong about market direction will often be less than the losses from an outright underlying position.

有经验的交易者会接受自己的预测有时可能出错的事实。在出错时，垂直价差策略与直接持仓相比更具优势。通过合理地估算波动率，期权交易者可以决定是否希望时间为己所用。当他选择让时间效应发挥正面作用（正 Theta）时，有时即便直接持仓者没有盈利或出现亏损，期权交易者仍可能盈利；当选择让时间效应对自己不利（正 Gamma）时，期权交易者在判断市场方向错误时的损失通常会低于直接持仓者的损失。