Chapter 4 Volatility
第四章 波动率
What is volatility and why is it so important to an option trader? The option trader, like a trader in the underlying instrument, is interested in the direction of the market. But unlike the trader in the underlying, an option trader is also extremely sensitive to the speed of the market. If the market for an underlying contract falls to move at a sufficient speed, options on that contract will have less value because of the reduced likelihood of the market going through an option's exercise price. In a sense, volatility is a measure of the speed of the market. Markets which move slowly are low-volatility markets; markets which move quickly are high-volatility markets.
什么是波动率，为什么它对期权交易者如此重要？期权交易者和标的资产的交易者一样，关注市场的方向。但与标的资产交易者不同，期权交易者对市场的波动幅度非常敏感。如果标的资产市场波动幅度不足，相关期权的价值就会降低，因为市场突破期权行权价的概率降低，从而导致期权的价值下降。从某种意义上说，波动率是衡量市场波动幅度的指标。市场波动缓慢的是低波动市场；快速波动的是高波动市场。
One might guess intuitively that some markets are more volatile than others. Between 1980 and 1982, the price of gold moved from $300 per ounce to $800 per ounce, more than doubling its price. Yet few traders would predict that the S&P 500 Index might more than double in a similar period. A commodity trader knows that precious metals are generally more volatile than interest rate instruments. In the same way, a stock trader knows that high-technology stocks tend to be more volatile than utility stocks.
我们可以直观地猜测，有些市场比其他市场更波动。1980 年至 1982 年间，黄金价格从每盎司 300 美元上涨到 800 美元，价格上涨超过两倍。然而，很少有交易者会预测标普 500 指数会在类似时间段内翻倍。商品交易者了解，贵金属的波动通常高于利率金融产品。同样，股票交易者也知道，高科技类股票的波动通常高于公用事业类股票。
If we knew whether a market was likely to be relatively volatile, or relatively quiet, and could convey this information to a theoretical pricing model, any evaluation of options on that market would be more accurate than if we simply ignored volatility. Since option models are based on mathematical formulae, we will need some method of quantifying this volatility component so that we can feed it into the model in numerical form.
如果我们能够预测某个市场是相对波动的还是相对平静的，并将这一信息传递给理论定价模型，那么对该市场期权的评估将比忽略波动率时更加准确。由于期权模型基于数学公式，我们需要一种方法量化波动率，并以数值形式输入模型。
RANDOM WALKS AND NORMAL DISTRIBUTIONS
随机漫步与正态分布
Consider for a moment the pinball maze pictured in Figure 4-1. When a ball is dropped into the maze at the top it moves downward, pulled by gravity, through a series of nails. When the ball encounters each nail there is a 50% chance the ball will move to the right, and a 50% chance it will move left. The ball then falls down to a new level where it encounters another nail. Finally, at the bottom of the maze the ball falls into one of the troughs.
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| 图 4-1 随机漫步 |
请先想象图 4-1 中的弹球迷宫。当一个弹球从迷宫顶部掉下时，它在重力作用下穿过一系列钉子向下移动。每当弹球遇到钉子时，它有 50% 的几率向右或向左移动。接着，弹球下落到一个新的位置，遇到另一个钉子。最终，弹球会在迷宫底部掉进某个槽里。
The path the ball follows as it moves downward through the maze of nails is known as a random walk. Once the ball enters the maze nothing can be done to artificially alter its course, nor can one predict ahead of time the path the ball will take through the maze.
弹球在迷宫中通过钉子的下落路径称为随机漫步。一旦弹球进入迷宫，无法人为地改变其路线，也无法预先预测它会经过哪些路径。
If enough balls are dropped into the maze, we might begin to get a distribution of balls similar to that in Figure 4-2. Most of the balls tend to congregate near the center of the maze, with a decreasing number of balls ending up in troughs further away from the center. The distribution which results from dropping many balls into our maze is referred to as a normal, or bell-shaped, distribution.
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| 图 4-2 正态分布 |
如果我们将足够多的弹球投入迷宫，可能会得到类似图 4-2 所示的分布。大多数弹球会集中在迷宫的中央附近，离中心越远，掉入槽中的弹球数量越少。这种由许多弹球掉入迷宫所形成的分布称为正态分布或钟形分布。
If we were to drop an infinite number of balls into the maze we might approximate the distribution with a normal, or bell-shaped, curve such as the one overlaid on the distribution in Figure 4-2. Such a curve is symmetrical (if we flipped it from right to left it would look the same), it has its peak in the center, and its tails always flare down and away from the center.
如果我们投入无限多的弹球，分布将接近图 4-2 上覆盖的正态曲线。这条曲线是对称的（如果从右向左翻转它，看起来仍然一样），它的峰值位于中心，两端的尾部逐渐向下并远离中心扩展。
Normal distribution curves are used to describe the likely outcomes of random events. For example, the curve in Figure 4-2 might also represent the results of flipping a coin 15 times. Each outcome, or trough, would represent the number of heads which occurred after each 15 flips. An outcome in trough zero would represent zero heads and 15 tails; an outcome in trough 15 would represent 15 heads and zero tails. Of course, we would be surprised to flip a coin 15 times and get all heads or all tails. Assuming the coin is perfectly balanced, some outcome in between, perhaps 8 heads and 7 tails, or 9 heads and 0 tails, seems more likely.
正态分布曲线通常用于描述随机事件的可能结果。例如，图 4-2 中的曲线也可以代表掷硬币 15 次的结果。每一个结果，或每个槽位，代表 15 次掷硬币后出现的正面次数。结果在槽位 0 表示没有正面，15 次全是反面；而结果在槽位 15 表示 15 次全是正面。显然，掷硬币 15 次全是正面或全是反面的情况极为罕见。如果硬币是完全平衡的，某个中间的结果，例如 8 次正面 7 次反面，似乎更为常见。
Suppose we change our maze slightly by closing off a row of nails so that each time a ball encounters a nail and goes either left or right, it must drop down two levels before it encounters another nail. If we drop enough balls into the maze we may end up with a distribution represented by the curve in Figure 4-3. Since the sideways movement of the balls is restricted, the curve will have a higher peak and narrower tails than the curve in Figure 4-2. In spite of its altered shape, this curve still represents a normal distribution, although one with slightly different characteristics.
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| 图 4-3 低波动正态分布 |
假设我们稍微改变迷宫的结构，关闭某排钉子，使每次弹球遇到钉子并左右移动时，它必须先下降两层才能遇到另一个钉子。如果投入足够多的弹球，可能会得到图 4-3 所示的分布。由于弹球的横向移动受到限制，曲线的峰值更高，尾部更窄。尽管形状有所变化，这条曲线仍然代表正态分布，只是特征略有不同。
Finally, we might block off some of the spaces between nails so that each time a ball drops down a level it must move two nails left or right before it can drop down to a new level. Again, if we drop enough balls into the maze we may get a distribution which resembles the curve in Figure 4-4. This curve, while still a normal distribution curve, will have a much lower peak and its tails will spread out much more quickly than the curves in either Figure 4-2 or 4-3.
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| 图 4-4 高波动正态分布 |
接着，我们可能阻挡一些钉子之间的空间，使得每次弹球下落时，必须左右移动两钉的距离，才能再下降一层。如果投入足够多的弹球，分布可能会类似于图 4-4。这条曲线的峰值较低，尾部扩展得比图 4-2 和图 4-3 中的曲线更快。尽管如此，这仍然是一条正态分布曲线。
Suppose we now think of the ball's sideways movement as the up and down price movement of an underlying contract, and the ball's downward movement as the passage of time. If we assume that each day the underlying contract can move up or down $1, the price distribution after 15 days might be represented by the curve in Figure 4-2. If we assume the price can move up or down $l every two days, the price distribution after 15 days might be represented by the curve in Figure 4-3. And if we assume that each day the price can move up or down $2, the price distribution might be represented by the curve in Figure 4-4.
现在，假设弹球的横向移动代表标的合约价格的涨跌波动，而弹球的向下移动代表时间的流逝。如果我们假设每一天标的合约可以上下波动 1 美元，那么 15 天后的价格分布可能类似于图 4-2 中的曲线。如果我们假设每两天价格波动 1 美元，15 天后的价格分布可能类似于图 4-3 中的曲线。而如果我们假设每一天价格波动 2 美元，那么价格分布可能类似于图 4-4 中的曲线。
With the underlying contract presently at $100 and 15 days to expiration, how might we evaluate a $105 call? One way is to assume that prices follow a random walk through time and that one of the curves in Figures 4-2, 4-3, or 4-4 represents the likely distribution after 15 days. The comparative value of the $105 call under these three scenarios is shown in Figure 4-5. If we assume a distribution similar to Figure 4-3, we can see that the underlying contract has very little chance of reaching $105. Consequently, the value of the $105 call will be low. If we assume a distribution similar to Figure 4-2, there is an increased probability of the underlying reaching $105, and this will increase the value of the $105 call. Finally, if we assume a distribution similar to Figure 4-4, there is a very real likelihood that the $105 call could finish in-the-money. As a result, the value of the option will increase dramatically.
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| 图 4-5 到底日标的合约价格的正态分布 |
假设当前标的合约价格为 100 美元，距离到期还有 15 天，我们如何评估一份行权价为 105 美元的看涨期权呢？一种方法是假设价格随着时间进行随机漫步，并假设图 4-2、图 4-3 或图 4-4 中的某条曲线代表了 15 天后的可能价格分布。图 4-5 显示了在这三种情景下 105 美元看涨期权的相对价值。如果我们假设分布类似于图 4-3，标的合约达到 105 美元的机会很小，因此该期权的价值会很低。如果假设分布类似于图 4-2，标的合约达到 105 美元的概率增加，该期权的价值也会随之上升。最后，如果假设分布类似于图 4-4，105 美元看涨期权很有可能到期时具有内在价值，期权的价值将大幅上升。
If we assume only that the price movement of an underlying contract follows a random walk, and nothing about the likely direction of movement, the curves in Figures 4-2, 4-3, and 4-4 might represent possible price distributions in a moderate-volatility, low-volatility, and high-volatility market, respectively. In a low-volatility market, price movement is severely restricted, and consequently options will command relatively low premiums. In a high-volatility market the chances for extreme price movement is greatly increased, and options will command high premiums.
如果我们仅假设标的合约的价格运动是随机漫步，并不考虑其方向，那么图 4-2、图 4-3 和图 4-4 中的曲线可能分别代表中等波动、低波动和高波动的市场。在低波动市场中，价格波动受到严格限制，因此期权的溢价会相对较低。而在高波动市场中，价格剧烈波动的概率大大增加，因此期权溢价较高。
Since the different price distributions in Figure 4-5 are symmetrical, it may seem that increased volatility should have no effect on an option's value. After all, increased volatility may increase the likelihood of large upward movement, but this should be offset by the equally greater likelihood of large downward movement. Here, however, there is an important distinction between an option position and an underlying position. Unlike an underlying contract, an option's potential loss is limited. No matter how far the market drops, a call option can only go to zero. In our example, whether the market finishes at $80 or $104 at expiration, the $105 call will be worthless. However, if we buy the underlying contract at $100, there is a tremendous difference between the market finishing at $80 or $104. With an underlying contract all outcomes are important; with an option, only those outcomes which result in the option in-the-money are important. In Figure 4-5, we are only concerned with price outcomes to the right of the exercise price. Everything else is zero.
因为图 4-5 中的价格分布是对称的，似乎波动率增加对期权价值没有影响。毕竟，波动率增加意味着价格大幅上涨的可能性增加，但这会被同样可能的大幅下跌所抵消。然而，期权头寸与标的头寸之间有一个重要区别。与标的合约不同，期权的潜在损失是有限的。无论市场如何下跌，看涨期权的价值最多只能降至零。在我们的例子中，不论到期时市场是 80 美元还是 104 美元，105 美元看涨期权都将毫无价值。然而，如果我们在 100 美元买入标的合约，市场价格是 80 美元还是 104 美元会有巨大的差异。对于标的合约，所有结果都很重要；而对于期权，只有那些使期权有内在价值的结果才是重要的。在图 4-5 中，我们只关注行权价右侧的价格结果，其他结果都为零。
This leads to an important distinction between evaluation of an underlying contract and evaluation of an option. If we assume that prices are distributed along a normal distribution curve, the value of an underlying contract depends on where the peak of the curve is located, while the value of an option depends on how fast the curve spreads out.
这引出了评估标的合约与评估期权之间的重要区别。如果我们假设价格沿着正态分布曲线分布，标的合约的价值取决于曲线峰值的位置，而期权的价值则取决于曲线扩展的速度。
MEAN AND STANDARD DEVIATION
均值与标准差
Suppose we want to use the concept of price movement based on normal distribution curves in a theoretical pricing model. To do this we need a method of describing the characteristics of the curve to the model. Since the model is based on mathematics, we need to describe the curve in numerical terms so that we can feed the numbers into the model.
假设我们想在理论定价模型中使用基于正态分布曲线的价格波动概念。为此，我们需要一种方法来向模型描述曲线的特征。由于模型是基于数学的，因此我们需要用数值来描述曲线，以便将这些数字输入模型。
Fortunately, a normal distribution curve can be fully described with two numbers, the mean and the standard deviation. If we know that a distribution is normal, and we also know these two numbers, then we know all the characteristics of the distribution.
幸运的是，正态分布曲线可以用两个数字完全描述：均值和标准差。如果我们知道一个分布是正态分布，并且知道这两个数字，就可以了解该分布的所有特征。
Graphically, we can interpret the mean as the location of the peak of the curve, and the standard deviation as a measure of how fast the curve spreads out. Curves which spread out very quickly, such as Figure 4-4, have a high standard deviation; curves which spread out very slowly, such as Figure 4-3, have a low standard deviation.
在图形上，均值表示曲线峰值的位置，标准差则衡量曲线扩展的速度。像图 4-4 中扩展较快的曲线，标准差较大；而像图 4-3 中扩展较慢的曲线，标准差较小。
While the mean is nothing more than the average outcome, and therefore a familiar concept for many traders, the standard deviation may not be quite so familiar. Indeed, one need not know how either of these numbers are calculated in order to successfully trade options. (For those who are interested, a more detailed discussion appears in Appendix B.) What is important to an option trader is the interpretation of these numbers, in particular what a mean and standard deviation suggest in terms of likely price movement.
均值是平均结果，这是交易者比较熟悉的概念，但标准差可能没有那么熟悉。实际上，交易者不需要了解如何计算这些数字，就可以成功交易期权（感兴趣的读者可以参见附录 B）。对期权交易者来说，重要的是理解这些数字的含义，尤其是均值和标准差对于预测价格波动的意义。
Let's go back to Figure 4-2 and consider the troughs numbered 0 to 15 at the bottom. We suggested that these numbers might represent the number of heads resulting from 15 flips of a coin. Alternatively, they might also represent the number of times a ball went right at each nail as it dropped down through the maze. The first trough is assigned zero since any ball which ends there must have gone left at every nail. The last trough is assigned 15 since any ball which ends there must have gone right at every nail.
我们回到图 4-2，底部标有 0 到 15 的槽位。这些数字可能代表 15 次掷硬币中正面出现的次数，或是弹球每次经过钉子时向右的次数。第一个槽位编号为 0，表示弹球每次都向左移动；最后一个槽位编号为 15，表示弹球每次都向右移动。
Suppose we are told that the mean and standard deviation in Figure 4-2 are 7.50 and 3.00, respectively. What does this tell us about the distribution? (The actual mean and standard deviation of 7.51 and 2.99 are calculated in Appendix B. Here, for simplicity, we will round to 7.50 and 3.00.) The mean tells us the average outcome. If we add up all the outcomes and divide by the number of occurrences, the result will be 7.50. In terms of the troughs, the average result will fall half-way between troughs 7 and 8. (Of course this is not an actual possibility. However, we noted in Chapter 3 that the average outcome does not have to be an actual possibility for any one outcome.)
假设图 4-2 中的均值为 7.50，标准差为 3.00。那么，这说明了什么呢？（附录 B 中精确计算的均值和标准差分别为 7.51 和 2.99，但这里为了简化，我们将其近似为 7.50 和 3.00。）均值代表平均结果。如果我们将所有的结果加起来并除以出现的次数，结果就是 7.50。也就是说，平均结果会落在 7 号槽和 8 号槽之间。（当然，单个结果不一定会精确落在这个位置。正如我们在第 3 章提到的，平均值并不一定是某个实际可能的结果。）
The standard deviation not only describes how fast the distribution spreads out; it also tells us something about the likelihood of a ball ending up in a specific trough or group of troughs. In particular, the standard deviation tells us the probability of a ball ending up in a trough which is a specified distance from the mean. For example, we may want to know the likelihood of a ball falling down through the maze and ending up in a trough lower than 5 or higher than 10. We can answer the question by asking how many standard deviations the ball must move away from the mean, and then determine the probability associated with that number of standard deviations.
标准差不仅描述分布的扩展速度，还能告诉我们弹球落入某个槽位或一组槽位的概率。特别是，标准差能够告诉我们弹球落入距均值一定范围内槽位的概率。例如，我们可能想知道弹球通过迷宫后落入 5 号槽以下或 10 号槽以上的概率。通过计算弹球偏离均值的标准差数量，我们可以确定相应的概率。
The exact probability associated with any specific number of standard deviations can be found in mathematical tables in most books on statistics. Alternatively, such probabilities can be closely approximated using an appropriate formula (see Appendix B). For option traders the following approximations will be useful:
+-1 standard deviation takes in approximately 68.3% (about ⅔3) of all occurrences
+-2 standard deviations takes in approximately 95.4% (about 1%20) of all occurrences
+-3 standard deviations takes in approximately 99.7% (about 36%370) of all occurences
与特定标准差相关的概率可以在大多数统计书籍的数学表中找到，或者通过公式近似计算（参见附录 B）。对期权交易者来说，以下近似值非常有用：
±1 个标准差涵盖约 68.3% 的所有结果（大约 2/3）
±2 个标准差涵盖约 95.4% 的所有结果（大约 19/20）
±3 个标准差涵盖约 99.7% 的所有结果（几乎全部）
Note that each number of standard deviations is preceded by a plus or minus sign. Because normal distributions are symmetrical, the likelihood of up movement and down movement is identical.
注意，每个标准差前都有正负号。因为正态分布是对称的，上涨和下跌的可能性是相同的。
Now let's try to answer our question about the likelihood of getting a ball in a trough lower than 5 or higher than 10. We can designate the divider between troughs 7 and 8 as the mean of 7½. If the standard deviation is 3, what troughs are within one standard deviation of the mean? One standard deviation from the mean is 7½±3 = 4½ to 10½. Again interpreting ½ as the divider between troughs, we can see that troughs 5 through 10 fall within one standard deviation of the mean. We know that one standard deviation takes in about 2/3 of all occurrences, so we can conclude that out of every three balls we drop into the maze, two should end up in troughs 5 through 10. What is left over, one out of every three balls, will end up in one of the remaining troughs, 0-4 and 11-15. Hence, the answer to our original question about the likelihood of getting a ball in a trough lower than 5 or higher than 10 is about 1 chance in 3, or about 33%. (The exact answer is 100% - 68.3%, or 31.7%.) This is shown in Figure 4-6.
现在，我们来回答弹球落入 5 号槽以下或 10 号槽以上的概率问题。我们可以将 7 号槽和 8 号槽之间的分界点设为均值 7.5。如果标准差为 3，那么一个标准差范围为 7.5±3，即 4.5 到 10.5。因此，5 号槽到 10 号槽都落在一个标准差范围内。我们知道，一个标准差内涵盖了约 2/3 的所有结果，因此我们可以推断，每三次下落的弹球中，约有两次会落入 5 号槽到 10 号槽。剩下的三分之一将落入 0 号槽到 4 号槽以及 11 号槽到 15 号槽。因此，弹球落入 5 号槽以下或 10 号槽以上的概率约为 1/3，或大约 33%。（确切的概率是 100% – 68.3%，即 31.7%。这在图 4-6 中有所展示。）
Let's try another calculation, but this time we can think of the problem as a wager. Suppose someone offers us 30 to 1 odds that we can't drop a ball into the maze and get it specifically in troughs 14 or 15. Is this bet worth making? One characteristic of standard deviations is that they are additive. In our example, if one standard deviation is 3, then two standard deviations are 6. Two standard deviations from the mean is therefore 7½ 6 = 1½2 to 13½. We can see in Figure 4-6 that troughs 14 and 15 lie outside two standard deviations. Since the probability of getting a result within two standard deviations is approximately 19 out of 20, the probability of getting a result beyond two standard deviations is 1 chance in 20. Therefore 30 to 1 odds may seem very favorable. Recall, however, that beyond two standard deviations also includes troughs 0 and 1. Since normal distributions are symmetrical, the chances of getting a ball specifically in troughs 14 or 15 must be half of 1 chance in 20, or about 1 chance in 40. At 30 to 1 odds the bet must be a bad one since the odds do not sufficiently compensate us for the risk involved.
接下来我们再做一个计算，假设有人以 30 比 1 的赔率打赌我们无法将弹球准确落入 14 号槽或 15 号槽。这个赌注值得参与吗？标准差是可加的。在我们的例子中，一个标准差是 3，那么两个标准差就是 6。因此，两个标准差范围是 7.5±6，即 1.5 到 13.5。从图 4-6 中可以看出，14 号槽和 15 号槽位于两个标准差之外。由于两个标准差内的结果概率约为 19/20，因此超出两个标准差的概率为 1/20。30 比 1 的赔率看起来似乎很有吸引力，但请记住，超出两个标准差的还包括 0 号槽和 1 号槽。由于正态分布是对称的，弹球准确落入 14 号槽或 15 号槽的概率为 1/20 的一半，约为 1/40。因此，30 比 1 的赔率不足以弥补我们承担的风险。
In Chapter 3 we said that one logical approach to option evaluation is to assign a probability to an infinite number of possible price outcomes for an underlying contract. Then, if we multiply each possible price outcome by its associated probability we can use the results to calculate an option's theoretical value. The problem is in dealing with an infinite number of price outcomes and probabilities, since an infinite number of anything is not easy to work with. Fortunately, the characteristics of normal distributions have been so closely studied that formulas have been developed which facilitate the computation of both the probabilities associated with every point along a normal distribution curve, as well as the area under various portions of the curve. If we assume that prices of an underlying instrument are normally distributed, these formulas represent a unique set of tools with which we can solve for an option's theoretical value. This is one of the reasons Black and Scholes adopted the normal distribution assumption as part of their model.
在第 3 章中我们提到，评估期权的一种逻辑方法是为标的合约的无限可能价格结果分配概率。然后，如果我们将每个可能的价格结果乘以其对应的概率，就可以用这些结果来计算期权的理论价值。问题在于，处理无限多个价格结果和概率是非常复杂的。幸运的是，正态分布的特性已经被深入研究，开发出了一些公式，这些公式可以帮助我们计算正态分布曲线上的每个点的概率，以及曲线下不同部分的面积。如果我们假设标的合约价格是正态分布的，这些公式为我们提供了独特的工具来解决期权的理论定价问题。这也是 Black-Scholes 模型采用正态分布假设的原因之一。
UNDERLYING PRICE AS THE MEAN OF A DISTRIBUTION
标的资产价格作为分布的均值
Now that we have decided to describe prices in terms of a normal distribution, how do we feed this distribution into a theoretical pricing model? Since all normal distributions can be described by a mean and the standard deviation, in some way we must feed these two numbers into our pricing model.
现在我们决定用正态分布来描述价格，接下来如何将这种分布输入到理论定价模型中呢？正态分布可以通过均值和标准差来描述，因此我们必须将这两个数字输入到定价模型中。
When we enter the present price of an underlying instrument we are actually entering the mean of a normal distribution curve. An important assumption in the Black-Scholes Model is that, in the long run, a trade in the underlying instrument will just break even. It will neither make money nor lose money. Given this assumption, the mean of the normal distribution curve assumed in the model must be the price at which a trade in the underlying instrument, either a purchase or a sale, would just break even. What is that price? The answer depends on the type of underlying instrument.
当输入标的资产的现价时，实际上是在输入正态分布曲线的均值。在布莱克-肖尔斯模型中，一个重要假设是，长期来看，标的资产的交易将恰好收支平衡，不会产生盈利或亏损。基于这一假设，模型中假设的正态分布曲线的均值必须是交易价格，只有在这个价格下，购买或出售标的资产才能实现收支平衡。这个价格是什么？答案取决于标的资产的类型。
Suppose a trader purchases a futures contract at 100 and holds the position for three months. Where does the price of the futures contract have to be at the end of three months for the trader to break even? Since futures contracts entail no carrying costs, nor do they pay dividends, the break even price three months from now is exactly the original trade price of 100.
假设一位交易者以 100 的价格购买了一个期货合约，并持有三个月。为了收支平衡，期货合约在三个月末的价格需要达到多少？由于期货合约没有持有成本，也不支付股息，因此三个月后的收支平衡价格就是原始交易价格 100。
Now suppose that a trader purchases a $100 stock and holds it for three months. Where does the stock price have to be at the end of the holding period for the trader to break even? Since the purchase of stock requires immediate payment, the break even price will have to include the cost of carrying a $100 debit for three months. If interest rates are 8% annually the carrying cost on $100 for three months is 3/12 x 8% x $100 = $2. Therefore, the stock price must be $102 at the end of three months for the trade to break even. If the stock will pay a dividend of $1 during the holding period, then the stock price need only be $101 for the trade to break even.
现在假设一位交易者购买了一只价值 100 的股票，并持有三个月。为了在持有期结束时收支平衡，股票价格需要达到多少？由于购买股票需要立即付款，收支平衡价格必须包括持有 100 美元的借款成本。如果年利率为 8%，那么持有 100 美元三个月的成本为 3/12 × 8% × $100 = $2 美元。因此，股票在三个月末必须达到 102 美元才能收支平衡。如果在持有期内股票将支付 1 美元的股息，那么股票价格只需达到 101 美元即可收支平衡。
Note that this is exactly how we calculated the forward price of a contract in Chapter 3. And indeed this is exactly the type of calculation built into the various forms of the Black-Scholes Model. When we enter an underlying price into the Black-Scholes model, based on the type of underlying instrument, Interest rates, and dividends, the model calculates the forward distribution curve.
这正是我们在第三章计算合约远期价格的方式。实际上，这也是布莱克-肖尔斯模型的各种形式中所内置的计算。当我们根据标的资产的类型、利率和股息将标的价格输入布莱克-肖尔斯模型时，模型将计算出远期分布曲线。
VOLATILITY AS A STANDARD DEVIATION
波动率作为标准差
In addition to the mean, we also need a standard deviation to fully describe a normal distribution curve. This is entered in the form of a volatility. With some slight modifications, which we will discuss shortly, we can define the volatility number associated with an underlying instrument as a one standard deviation price change, in percent, at the end of a one-year period.
在阐述正态分布时，除了均值，我们还需要标准差，这通常以波动率的形式呈现。我们可以将标的资产的波动率定义为一年期末的一个标准差价格变化，单位为百分比。
For example, suppose that an underlying futures contract is currently trading at 100 and has a volatility of 20%. Since this represents a one standard deviation price change, one year from now we expect the same futures contract to be trading between 80 and 120 (100 ± 20%) approximately 68% of the time, between 60 and 140 (100 ± (2 × 20%)) approximately 95% of the time, and between 40 and 160 (100 ± (3 × 20%)) approximately 99.7% of the time.
例如，假设一个期货合约当前价格为 100，波动率为 20%。这表示一个标准差的价格变化，因此我们预计一年后该期货合约的价格将在 80 到 120 之间（100 ± 20%）约 68% 的时间内交易，在 60 到 140 之间（100 ± (2 × 20%)）约 95% 的时间内交易，以及在 40 到 160 之间（100 ± (3 × 20%)）约 99.7% 的时间内交易。
If the underlying contract is a stock currently trading at $100, then the 20% volatility will have to be based on the forward price of the stock at the end of one year. If interest rates are 8% and the stock pays no dividends, the one-year forward price will be $108. Now a one standard deviation price change is 20% x $108 = $21.60. So one year from now we would expect the same stock to be trading between $86.40 and $129.60 ($108 $21.60) approximately 68% of the time, between $64.80 and $151.30 ($108 $ (2 x $21.60)) approximately 95% of the time, and between $43.20 and $172.90 ($108 ‡ (3 x $21.60)) approximately 99.7% of the time.
如果标的合约是一只当前交易价格为 $100 的股票，那么 20% 的波动率应基于该股票在一年后的远期价格。如果年利率为 8%，且该股票不支付股息，则一年后的远期价格为 $108。此时，一个标准差的价格变化为 20% × $108 = $21.60。因此，一年后，我们预计这只股票将在 $86.40 到 $129.60 之间（$108 ± $21.60）约 68% 的时间内交易，在 $64.80 到 $151.30 之间（$108 ± (2 × $21.60)）约 95% 的时间内交易，以及在 $43.20 到 $172.90 美元之间（$108 ± (3 × $21.60)）约 99.7% 的时间内交易。
Suppose we come back at the end of one year and find our futures contract, which we thought had a volatility of 20%, trading at 35. Does this mean our volatility of 20% was wrong? A price change of more than three standard deviations may be unlikely, but one shouldn't confuse unlikely and impossible. Flipping a perfectly balanced coin 15 times may result in 15 heads, even though the odds against it are more than 32,000 to 1. If 20% was the right volatility, the odds of the futures price going from 100 to 35 one year later is more than 1,500 to 1. But one chance in 1,500 is not impossible, and perhaps this was the one time in 1,500 when the price would indeed end up at 35. Of course, it is also possible that we had the wrong volatility. But we wouldn't know this without looking at price changes of the futures contract over many years so that we have a representative price distribution.
假设一年后我们发现期货合约的价格为 $35，这是否意味着我们的 20% 波动率判断错误？虽然价格变化超过三个标准差不太可能，但也并非不可能。比如，将一个完全平衡的硬币抛 15 次，可能出现 15 次正面，尽管其概率极小。如果 20% 是正确的波动率，那么期货价格从 $100 跌到 $35 的概率超过 1500 比 1。但一千五百分之一的机会并不意味着不可能，或许这正是 1500 次中唯一的一次。当然，也有可能之前的波动率估算不准确，但只有通过观察多年期货合约的价格变化，我们才能得出更具代表性的价格分布。
LOGNORMAL DISTRIBUTIONS
对数正态分布
Is it reasonable to assume that the prices of an underlying instrument are normally distributed? Beyond the question of the exact distribution of prices in the real world, the normal distribution assumption has one serious flaw. A normal distribution curve is symmetrical. Under a normal distribution assumption, for every possible upward move in the price of an underlying instrument there must be the possibility of a downward move of equal magnitude. If we allow for the possibility of a $50 instrument rising $75 to $125, we must also allow for the possibility of the instrument dropping $75 to a price of -$25. Since it is impossible for traditional stocks and commodities to take on negative prices, the normal distribution assumption is clearly flawed. What can we do about this?
假设标的资产的价格呈正态分布是否合理？在现实世界中，价格的确切分布问题之外，正态分布假设有一个明显的缺陷。正态分布曲线是对称的。在这种假设下，标的资产每一次可能的上涨都必须对应同等幅度的下跌。如果我们允许一个价格为 $50 的资产上涨 $75 至 $125，就必须允许它下跌 $75，跌至-$25。由于传统股票和商品不可能出现负价格，这一假设显然是不合理的。对此，我们该如何应对？
Up to now we have defined volatility in terms of the percent changes in the price of an underlying instrument. In this sense an interest rate and volatility are similar in that they both represent a rate of return. The primary difference between interest and volatility is that interest generally accrues at a positive rate, while volatility represents a combination of positive and negative rates of return. If one invests money at a fixed interest rate, the value of the principal will always grow. But if one invests in an underlying instrument with a volatility other than zero, the instrument may go either up or down in price. Volatility, which is a standard deviation, says nothing about the direction of movement.
目前为止，我们通过标的资产价格的百分比变化来定义波动率。从这个角度看，利率和波动率都代表一种回报率。它们的主要区别在于，利率通常以正的速率增长，而波动率则包含正负回报率的组合。如果在固定利率下投资，资本的价值总是增长的。但如果投资于波动率不为零的标的资产，价格可能会波动。波动率作为标准差，并不能说明运动的方向。
Since volatility represents a rate of return, an important consideration is the manner in which the rate of return is calculated. For example, suppose we were to invest $1,000 for one year at an annual interest rate of 12%. How much would we have at the end of one year? The answer depends on how the 12% interest on our investment is paid out.
由于波动率代表回报率，计算方式也至关重要。例如，假设我们以 12% 的年利率投资 $1000 一年。那么，年底我们将拥有多少钱？答案取决于这 12% 的利息是如何支付的。
| Rate of Payment | Value after One Year | Total Yield |
|---|---|---|
| 12% once a year | $1,120.00 | 12% |
| 6% twice a year | $1,123.60 | 12.36% |
| 3% every three months | $1,125.51 | 12.55% |
| 1% every month | $1,126.83 | 12.68% |
| 支付利率 | 一年后的价值 | 总收益率 |
|---|---|---|
| 每年 12% | $1,120.00 | 12% |
| 每年 6%（每年两次） | $1,123.60 | 12.36% |
| 每三个月 3% | $1,125.51 | 12.55% |
| 每月 1% | $1,126.83 | 12.68% |
As interest is paid more often, even though it is paid at the same rate of 12% per year, the total yield on the investment increases. The yield is greatest when interest is paid continuously. In this case it is just as if interest were paid at every possible moment in time.
随着利息支付频率的增加，即使年利率仍为 12%，投资的总收益率也会提高。当利息连续支付时，收益率达到最大值。这就好像在每一个可能的时刻都在支付利息。
Although less common, we can do the same type of calculation using a negative interest rate. For example, suppose we were to lose 12 percent annually on our $1,000 investment (interest rate = -12%). How much would we have at the end of a year? The answer depends on the frequency at which our losses accrue.
虽然这种情况不常见，但我们也可以使用负利率进行类似的计算。例如，假设我们的 $1000 投资每年损失 12%（利率为 -12%）。那么一年结束时我们会有多少钱？答案取决于损失累积的频率。
| Rate of Loss | Value after One Year | Total Yield |
|---|---|---|
| 12% once a year | $880.00 | -12% |
| 0% twice a year | $883.60 | -11.64% |
| 3% every three months | $885.29 | -11.47% |
| 1% every month | $886.38 | -11.36% |
| 12%/52 every week | $886.80 | -11.32% |
| 12%/365 every day | $886.90 | -11.31% |
| 12% compounded continuously | $886.92 | -11.31% |
| 损失率 | 一年后的价值 | 总收益率 |
|---|---|---|
| 每年 12% | $880.00 | -12% |
| 每半年 0% | $883.60 | -11.64% |
| 每三个月 3% | $885.29 | -11.47% |
| 每月 1% | $886.38 | -11.36% |
| 每周 12%/52 | $886.80 | -11.32% |
| 每天 12%/365 | $886.90 | -11.31% |
| 连续复利 12% | $886.92 | -11.31% |
In the case of a negative interest rate, as losses are compounded more frequently, even though at the same rate of -12% per year, the loss becomes smaller, and so does the negative yield.
在负利率的情况下，由于损失频繁复利，尽管年利率仍为 -12%，损失反而减小，负收益也随之下降。
In the same way that interest can be compounded at different intervals, volatility can also be compounded at different intervals. For purposes of theoretical pricing of options, volatility is assumed to compound continuously, just as if the price changes in the underlying instrument, either up or down, were taking place continuously but at an annual rate corresponding to the volatility number associated with the underlying instrument.
就像利率可以以不同的时间频率进行复利计算一样，波动率也可以以不同的时间频率进行复利计算。为了理论上对期权进行定价，我们假设波动率是连续复利的，就像标的资产的价格变化（无论是上涨还是下跌）都是连续发生的，但年化利率与标的资产的波动率数字相对应。
What would happen if at every moment in time the price of an underlying could go up or down a given percent, and that these up and down movements were normally distributed? When price changes are assumed to be normally distributed, the continuous compounding of these price changes will cause the prices at maturity to be distributed. Such a distribution is skewed toward the upside because upside prices resulting from a positive rate of return will be greater, in absolute terms, than downside prices resulting from a negative rate of return (Figure 4-7). In our interest rate example, a continuously compounded rate of return of +12% yields a profit of $127,50 after one year, while a continuously compounded rate of return of - 12% yields a loss of only $113.08. If the 12% were a volatility, then a one standard deviation upward price change at the end of one year would be +$127.50, while a one standard deviation downward price change would be -$113.08. Even though the rate of return was a constant 12%, the continuous compounding of the 12% yielded different upward and downward moves.
 |
| 图 4-7 对数正态分布和正态分布 |
如果在每个时刻，标的资产的价格都可以上下波动一个特定百分比，并且这些波动是正态分布的，会发生什么？假设价格变化是正态分布的，这些价格变化的连续复利将导致到期价格的分布。这种分布向上偏斜，因为正收益所导致的上涨价格在绝对值上会大于负收益导致的下跌价格。在我们的利率例子中，持续复利的 +12% 回报在一年后将带来 $127.50 的利润，而持续复利的 -12% 回报则仅造成 $113.08 的损失。如果 12% 是波动率，那么一年末一个标准差的上涨价格变化将为 $127.50，而一个标准差的下跌价格变化则为 $113.08。尽管回报率始终为 12%，但 12% 的持续复利带来了不同的上涨和下跌幅度。
The Black-Scholes Model is a continuous time model. It assumes that the volatility of an underlying instrument is constant over the life of the option, but that this volatility is continuously compounded. These two assumptions mean that the possible prices of the underlying instrument at expiration of the option are lognormally distributed. It also explains why options with higher exercise prices carry more value than options with lower exercise prices, where both exercise prices appear to be an identical amount away from the price of the underlying instrument. For example, suppose a certain underlying contract is trading at exactly 100. If there are no interest considerations and we assume a normal distribution of possible prices, then the 110 call and the 90 put, both being 10% out-of-the-money, ought to have identical theoretical values. But under the lognormal assumption in the Black-Scholes Model, the 110 call will always have a greater value than the 90 put. In absolute terms, the lognormal distribution assumption allows for greater upside price movement than downside price movement. Consequently, the 110 call will have a greater possibility of price appreciation than the 90 put. (footnote 1: Of course, this is only theory. There is no law that says the price of the 90 put in the marketplace cannot be greater than the price of the 110 call.)
布莱克-肖尔斯模型是一个连续时间模型。它假设标的资产的波动率在期权的有效期内是恒定的，但这种波动率是持续复利的。这两个假设意味着到期时标的资产的可能价格呈对数正态分布。这也解释了为何行权价更高的期权比行权价更低的期权更有价值，即使这两个行权价在标的资产的价格上是相同的。例如，假设某个标的资产的交易价格恰好为 100。如果不考虑利息因素，并且假设可能价格的分布是正态的，那么行权价为 110 的看涨期权和行权价为 90 的看跌期权，都是 10% 虚值，理论价值应该是相同的。但根据布莱克-肖尔斯模型中的对数正态假设，110 的看涨期权的价值将始终高于 90 的看跌期权。在绝对值上，对数正态分布假设允许上涨价格变化大于下跌价格变化。因此，110 的看涨期权更有可能实现价格升值，而 90 的看跌期权则较少。（脚注 1：当然，这只是理论。在市场上，并没有法律规定 90 的看跌期权价格不能高于 110 的看涨期权价格。）
Finally, the lognormal assumption built into the Black-Scholes Model overcomes the logical problem we initially posed. If we were to allow for the possibility of unlimited upside price movement of an underlying instrument, a normal distribution assumption would force us to allow for unlimited downside movement. This would require us to accept the possibility of negative prices for the underlying instrument, clearly not a possibility for most optionable instruments. A lognormal distribution, however, does allow for open ended upside prices (the logarithm of +o is +∞), while bounding downside prices by zero (the logarithm of -∞ is zero). This is a more realistic representation of how prices are actually distributed in the real world.
最后，布莱克-肖尔斯模型中的对数正态假设克服了我们最初提出的逻辑问题。如果我们允许标的资产的上涨价格无限制地上涨，正态分布假设将迫使我们接受下跌价格的无限制下降。这将要求我们接受标的资产的负价格，这显然不适用于大多数可交易期权。然而，对数正态分布允许上涨价格无限制（对数 +∞），而下跌价格则限制为零（对数-∞为零）。这更真实地反映了现实世界中价格的分布情况。
A more complete discussion of logarithmic price changes and probability calculations can be found in Appendix B.
有关对数价格变化和概率计算的更完整讨论，请参见附录 B。
We can now summarize the most important assumptions governing price movement in the Black-Scholes Model:
现在，我们可以总结布莱克-肖尔斯模型中价格变化的最重要假设：
- Changes in the price of an underlying instrument are random and cannot be artificially manipulated, nor is it possible to predict beforehand the direction in which prices will move.
- The percent changes in the price of an underlying instrument are normally distributed.
- Because the percent changes in the price of the underlying instrument are assumed to be continuously compounded, the prices of the underlying instrument at expiration will be lognormally distributed.
- The mean of the lognormal distribution will be located at the forward price of the underlying contract.
- 标的资产价格的变化是随机的，无法人为操控，亦无法预测价格的移动方向。
- 标的资产价格的百分比变化是正态分布的。
- 由于标的资产价格的百分比变化被假设为持续复利，到期时标的资产的价格将呈对数正态分布。
- 对数正态分布的均值将位于标的合约的远期价格。
The first of these assumptions may meet with resistance from some traders. Technical analysts believe that by looking at past price activity it is possible to predict the future direction of prices. One can chart support and resistance points, double tops and bottoms, head and shoulders, and many similar formations which are believed to predict future price trends. We leave debate on this question to others. The important point here is that the Black-Scholes Model makes the assumption that price changes are random and that their direction cannot be predicted. This does not mean that there is no predictive requirement in using the Black-Scholes Model. However, price prediction will focus on the magnitude of the price changes, rather than on the direction of changes.
第一个假设可能会遭到一些交易员的反对。技术分析师认为，通过分析过去的价格活动，可以预测未来价格的方向。人们可以绘制支撑和阻力点、双顶和双底、头肩形等许多被认为可以预测未来价格趋势的图形。我们将这个问题的争论留给其他人。这里重要的一点是，布莱克-肖尔斯模型假设价格变化是随机的，其方向无法预测。这并不意味着在使用布莱克-肖尔斯模型时没有预测的必要。然而，价格预测将专注于价格变化的幅度，而不是变化的方向。
As we shall see later, there is also good reason to question the third assumption, that prices are lognormally distributed at expiration. This may be a reasonable assumption for some markets, but a very poor assumption for other markets. Again, the important point here is for the trader who uses a theoretical pricing model to understand the assumptions on which the theoretical values are based. He can then make his own decision, based on his knowledge of a particular market, as to whether these assumptions, and hence the theoretical values generated by the model, are likely to be accurate.
正如我们稍后将看到的，对到期价格呈对数正态分布这一第三个假设也有理由质疑。这对于某些市场来说可能是合理的假设，但对其他市场来说则可能是一个很差的假设。同样，重要的是，使用理论定价模型的交易员应理解理论价值所基于的假设。这样，他们就可以根据自己对特定市场的了解，决定这些假设以及模型生成的理论值是否可能准确。
DAILY AND WEEKLY STANDARD DEVIATIONS
每日和每周的标准差
As an annual standard deviation, we know what the volatility tells us about the likely price movement of a contract over a one-year period. However, this is a period of time longer than the life of most listed options. We might want to know what a volatility tells us about price changes over a shorter period of time, for example over a month, or a week, or a day.
年标准差反映了合同在一年内价格变动的波动率。然而，这段时间通常超过了大多数上市期权的存续期。因此，我们可能想了解波动率在较短时间内（例如一个月、一周或一天）对价格变动的影响。
An important characteristic of volatility is that it is proportional to the square root of time. As a result of this, we can approximate a volatility over some period of time shorter than a year by dividing the annual volatility by the square root of the number of trading periods in a year.
波动率的一个重要特性是，它与时间的平方根成正比。因此，我们可以通过将年波动率除以一年中的交易周期平方根，来近似计算某一段时间内的波动率。
Suppose we are interested in a daily volatility. While it would take a logarithmic calculation to give us an exact daily volatility, if we ignore the relatively minor effect of continuous compounding over such a short period of time, it is possible to make an estimate of daily volatility. First we must determine the number of daily trading periods in a year. That is, if we look at prices at the end of every day, how many times a year can prices change? If we restrict ourselves to exchange traded options, even though there are 365 days in a year, prices cannot really change on weekends or holidays. This leaves us with about 256 trading days during the year. (footnote 2: Depending on holidays, the number of trading days is usually somewhere between 250 and 255. We use 256 as a reasonable approximation since its square root is a whole number and therefore easier to work with.) Since the square root of 256 is 16, to approximate a daily volatility we can divide the annual volatility by 16.
假设我们关心日波动率。尽管准确计算日波动率需要进行对数计算，但如果忽略连续复利在这样短的时间内的相对微小影响，我们仍然可以估算日波动率。首先，我们需要确定一年中的交易日数量。也就是说，如果我们关注每天结束时的价格变化，一年中价格能变动多少次？如果只考虑交易所交易的期权，尽管一年有 365 天，但周末和节假日并不能算作价格变动的日子。因此，实际的交易日大约为 256 天。（脚注 2：根据假期的不同，交易日通常在 250 到 255 之间。我们使用 256 作为合理的近似值，因为它的平方根是一个整数，便于计算。）由于 256 的平方根是 16，我们可以通过将年波动率除以 16 来近似计算日波动率。
Going back to our futures contract trading at 100 with a volatility of 20%, what is a one standard deviation price change over a day's time? 20%/16 = 1¼%, so a one standard deviation daily price change is 1¼% x 100 = 1.25. We expect to see a price change of 1.25 or less approximately two trading days out of every three, and a price change of 2.50 or less approximately 19 trading days out of every 20. Only one day in 20 would we expect to see a price change of more than 2.50.
回到我们的期货合约，当前价格为 100，波动率为 20%，那么一天内一个标准差的价格变化是多少呢？计算为 20%/16 = 1¼%，所以每日一个标准差的价格变化为 1¼% x 100 = 1.25。我们预计在每三天中大约有两天的价格变化在 1.25 或更少，而在每 20 天中，约有 19 天的价格变化在 2.50 或更少。只有在 20 天中，有一天的价格变化会超过 2.50。
We can do the same type of calculation for a weekly standard deviation. Now we must ask how many times per year prices can change if we look at prices once a week. Unlike trading days, we don't have "holiday" weeks, so we must make our calculations using all 52 trading weeks in a year. Dividing our annual volatility of 20% by the square root of 52, or approximately 7.2, we get 20%/7.2 = 2¾. For our futures contract trading at 100, we would expect to see a price change of 2.75 or less two weeks out of every three, a price change of 5.50 or less 19 weeks out of every 20, and only one week in twenty would we expect to see a price change of more than 5.50.
我们可以同样计算每周的标准差。现在我们需要问，如果我们每周查看一次价格，那么一年中价格能变动多少次。与交易日不同，我们没有 “假期周”，所以必须使用一年中的 52 个交易周进行计算。将年波动率 20% 除以 52 的平方根，约为 7.2，我们得到 20%/7.2 = 2¾。对于当前价格为 100 的期货合约，我们预计在每三周中，有两周的价格变化在 2.75 或更少，在每 20 周中，约有 19 周的价格变化在 5.50 或更少，只有在 20 周中，有一周的价格变化会超过 5.50。
Since we assume that the price of a stock will appreciate by the carrying cost, it may seem that we cannot use the same method (divide by 16 for dally volatility; divide by 7.2 for weekly volatility) to approximate expected movement in an underlying stock. However, over a short period of time, the carrying cost component, like the effect of continuous compounding of volatility, will be relatively small. Therefore, we can use the same method as a reasonable estimate of daily and weekly volatility. For example, suppose a stock is trading at $45 per share and has an annual volatility of 28%. What is an approximate one standard deviation price change over a day's time and over a week's time?
For a daily volatility we calculate:
28%/16 × $45 = 1.75% × $45 = $.79
For a weekly volatility we calculate:
28%/7.2 × $45 = 3.89% × $45 = $1.75
考虑到我们假设股票价格会因持有成本而上涨，似乎无法用同样的方法（日波动率除以 16；周波动率除以 7.2）来近似预期的标的股票变动。然而，在短时间内，持有成本的影响就像波动率连续复利的影响一样，相对较小。因此，我们可以使用相同的方法来合理估算日和周的波动率。例如，假设一只股票的交易价格为每股 45 美元，年波动率为 28%。那么一天和一周内的标准差价格变化分别是多少？
对于日波动率，我们计算：
28%/16 × $45 = 1.75% × $45 = $0.79
对于周波动率，我们计算：
28%/7.2 × $45 = 3.89% × $45 = $1.75
We expect to see a price change of approximately ¾ point or less two days out of every three, 1½ points or less 19 days out of every 20, and only one day in 20 would we expect to see a price change of more than 1½ points. On a weekly basis, we would expect to see a price change of 1¾ points or less two weeks out of every three, a price change of 3½ points or less 19 weeks out of every 20, and only one week in 20 would we expect to see a price change of more than 3½ points.
我们预计在每三天中，有两天的价格变化约为 ¾ 点或更少，在每 20 天中，约有 19 天的价格变化在 1½ 点或更少，只有在 20 天中，有一天的价格变化会超过 1½点。在每周的基础上，我们预计在每三周中，有两周的价格变化约为 1¾ 点或更少，在每 20 周中，约有 19 周的价格变化在 3½ 点或更少，只有在 20 周中，有一周的价格变化会超过 3½ 点。
We have used the phrase "price change" in conjunction with our volatility estimates. Exactly what do we mean by this? Do we mean the high/low during some period? Do we mean open to close price changes? Or is there another interpretation? While various methods have been suggested to estimate volatility, (footnote 3: See:
Parkinson, Michael, "The Extreme Value Method of Estimating the Variance of the Rate of Return," Journal of Business, 1980, vol. 53, no. 1, pp. 61-64.
Garman, Mark B. and Klass, Michael J., "On the Estimation of Security Price Volatilities from Historical Data," Journal of Business, 1980, vol. 53, no. 1, pp. 67-78.
Beckers, Stan, "Variance of Security Price Returns Based on High, Low, and Closing Prices," Journal o Business, 1983, vol. 56, no. 1, pp. 97-112.) the traditional method has been to calculate volatility based on settlement-to-settlement price changes. Using this approach, when we say a one standard deviation daily price change is ¾ point, we mean a ¾ point price change from one day's settlement price to the next day's settlement price. The high/low or open/close price change may have been either more or less than ¾ point, but it is the settlement-to-settlement price change on which we focus.
在我们的波动率估算中，我们使用了 “价格变化” 这一术语。那么，我们究竟是什么意思呢？是指某一时期的最高价与最低价之间的差距？是开盘价与收盘价之间的变化？还是有其他的解释？虽然有多种方法来估算波动率（脚注 3：详见：
Parkinson, Michael, “The Extreme Value Method of Estimating the Variance of the Rate of Return,” Journal of Business, 1980, vol. 53, no. 1, pp. 61-64.
Garman, Mark B. 和 Klass, Michael J., “On the Estimation of Security Price Volatilities from Historical Data,” Journal of Business, 1980, vol. 53, no. 1, pp. 67-78.
Beckers, Stan, “Variance of Security Price Returns Based on High, Low, and Closing Prices,” Journal of Business, 1983, vol. 56, no. 1, pp. 97-112.），但传统方法是基于结算价变化来计算波动率。采用这种方法，当我们说一个标准差的每日价格变化为 ¾ 点时，我们指的是从一天的结算价到下一个交易日的结算价的¾点变化。最高价和最低价或开盘价与收盘价的变化可能多于或少于 ¾ 点，但我们关注的是结算价之间的变化。
VOLATILITY AND OBSERVED PRICE CHANGES
波动率与观察到的价格变化
Why is it important for a trader to be able to estimate daily or weekly price changes from an annual volatility? Volatility is the one input into a theoretical pricing model which cannot be directly observed. Yet many option strategies, If they are to be successful, require an accurate assessment of volatility. Therefore, an option trader needs some method of determining whether his expectations about volatility are indeed being realized in the marketplace. Unlike directional strategies, whose success or failure can be immediately observed from posted prices, there is no posting of volatilities. A trader must determine for himself whether he is using a reasonable volatility input into the theoretical pricing model.
为什么交易者需要根据年化波动率估算每日或每周的价格变化？波动率是理论定价模型中唯一无法直接观察的输入项。许多期权策略的成功都依赖于对波动率的准确评估，因此期权交易者需要有方法判断市场中的波动率是否符合预期。与方向性策略不同，波动率没有直接显示在价格中，交易者必须自己评估使用的波动率是否合理。
For example, suppose a certain underlying contract is trading at 40 and a trader is using a 30% volatility for theoretical evaluation. A one standard deviation daily price change is approximately 30%/16 x 40 = .75. Over five days of trading a trader notes the following five settlement-to-settlement price changes:
+.43, -.06, -.61, +.50, -.28
Are these five price changes consistent with a 30% volatility?
例如，假设某合约的价格为 40，交易者使用 30% 的波动率进行评估。一个标准差的每日价格变化大约是 30%/16 x 40 = 0.75。在五个交易日中，交易者观察到如下结算价格变化：
+0.43, -0.06, -0.61, +0.50, -0.28
这些价格变化与 30% 的波动率一致吗？
The trader expects to see a price change of more than .75 (one standard deviation) about one day in three, or between one and two times over a five-day period. Yet during this five-day period he did not see a price change of this magnitude even once. What conclusions can be drawn from this? (footnote 4: Five days is admittedly a very small sample from which to draw a meaningful conclusion about volatility. The method and reasoning, however, are still valid.) One thing is certain: these five price changes are not consistent with 30% volatility. The trader might explain the discrepancy in one of two ways. On the one hand, perhaps this was expected to be an abnormally quiet week (perhaps it was a holiday week); and next week when trading returns to normal the market will go right back to making moves which are more consistent with a 30% volatility, If the trader comes to this conclusion, perhaps he ought to continue to use a 30% volatility for his calculations. On the other hand, perhaps there is no apparent reason for the market being less volatile than predicted by a 30% volatility. He may simply be using the wrong volatility. If the trader comes to this conclusion, perhaps he ought to consider using a new volatility input which is more consistent with the observed price changes. If he continues to use a 30% volatility in the face of price changes which are significantly less than predicted by that number, he will be assigning the wrong probabilities to the possible price outcomes for the underlying contract. Consequently, he will generate incorrect theoretical values, defeating the purpose of using a theoretical pricing model in the first place.
根据预期，价格变化超过 0.75（一个标准差）的情况大约每三天出现一次，或者在五天中应该出现一到两次。然而，在这五天中，并未出现一次超过 0.75 的价格变化。交易者可以得出什么结论？（脚注 4：五天的数据样本较小，难以得出准确的波动率结论，但推理方法依然有效。）可以确定的是，这五次价格变化与 30% 的波动率不一致。交易者可以通过两种方式解释这种差异。一种可能是，或许这是一个异常平静的交易周（比如假期周），下周市场恢复正常时，价格变化将符合 30% 的波动率。如果交易者得出这个结论，他可能会继续使用 30% 的波动率。另一种可能是，市场没有明显的理由表现得比 30% 的波动率更安静，那么他可能使用了错误的波动率。如果交易者得出这个结论，他应该考虑使用一个更符合观察到的价格变化的新波动率。否则，继续使用 30% 的波动率将导致对合约价格结果的错误概率分配，从而产生不正确的理论价值，失去使用理论定价模型的意义。
Exactly what volatility is associated with the five price changes in the foregoing example? Without some rather involved calculations it is difficult to say. (The answer is actually 18.8%.) However, if a trader has some idea beforehand of what price changes he expects, he can easily see that the changes over the five-day period are not consistent with a 30% volatility.
到底与这五次价格变化相对应的波动率是多少呢？没有复杂的计算很难得出结论。（实际上答案是 18.8%。）然而，如果交易者事先对预期的价格变化有所了解，他可以很容易看出这些变化与 30% 的波动率并不一致。
18.8% 是如何的出来的？请看附录 A 里面的 historical volatility 的计算步骤。
Let's look at another example. Now the underlying contract is trading at 332½ and a trader notes the following five daily price changes:
-5, +2½, +1, -7¾, -4¼
Are these price changes consistent with an 18% volatility? At 18% a one standard deviation price change is approximately 18%/16 x 332½ = 3¾. Over five days we expect to see a price change of more than 3¾ between one and two times. Yet here we have a price change of more than 3¾ three days out of five. And once the price change was 7¾ (more than two standard deviations) which we expect to see only one day in twenty. Again, unless the trader believes that the five price changes occurred during an extraordinary week, then perhaps he ought to consider changing his volatility figure so that it is more consistent with the observed price changes.
再看一个例子。此时，合约价格为 332½，交易者观察到如下五天的每日价格变化：
-5, +2½, +1, -7¾, -4¼
这些价格变化与 18% 的波动率一致吗？对于 18% 的波动率，一个标准差的价格变化大约是 18%/16 x 332½ = 3%。在五天内，预期会有一到两次超过3¾的价格变化。但这里有三次价格变化超过了3¾，其中一次是 7¾（超过了两个标准差），而这种情况理论上每 20 天才会出现一次。如果交易者不认为这是一个特殊的交易周，那么他应考虑调整波动率，使其与观察到的价格变化更一致。
A NOTE ON INTEREST RATE PRODUCTS
关于利率产品的说明
Suppose Eurodollars are at 93.00 and we assume a volatility of 16 percent. We can apply the method previously described to calculate an approximate one standard deviation daily price change: 16%/16 × 93.00 = .93. As any trader familiar with the Eurodollar market will attest, a daily price change of .93 is wildly unlikely. How can we account for this seemingly illogical answer? One might conclude that we simply have the wrong volatility, and some lower number is more accurate. In fact, the 16% volatility is not at all unusual for Eurodollars, so the explanation must lie elsewhere.
假设欧元美元的价格为 93.00，且我们假设波动率为 16%。我们可以用之前提到的方法来计算大致的每日价格变化的一个标准差：16%/16 × 93.00 = 0.93。任何熟悉欧元美元市场的交易者都会告诉你，价格每日波动 0.93 几乎是不可能的。那么，如何解释这个看似不合理的答案呢？有人可能认为我们用了错误的波动率，实际的波动率应更低。事实上，16% 的波动率在欧元美元市场并不罕见，因此问题出在其他地方。
Eurodollars, like many other interest rate contracts (U.S. Treasury Bills, Short Sterling, Euromarks, Euroyen) are indexed from 100. This means that the interest rate associated with a Eurodollar contract is 100 less the value of the contract. It also means that, barring the unlikely advent of negative interest rates, the contract cannot take on a value greater than 100. In this respect, 100 acts as a limiting value for Eurodollars in the same way that zero acts as a limiting value for traditional underlying contracts such as stocks and commodities. We can integrate this characteristic into our calculations by assuming that the value of a Eurodollar contract is actually 100 less its listed price. If the listed price is 93.00, for theoretical evaluation purposes we must use a value of 100 - 93.00, or 7.00, in our pricing model. If we define the value of the contract as 7.00, a one standard deviation price change is 16%/16 x 7.00 = .07. This is certainly a more realistic result than .93.
像许多其他利率合约（如美国国债、短期英镑、欧元马克、欧元日元）一样，欧元美元是以 100 为基准进行标价的。这意味着欧元美元合约对应的利率是 100 减去合约的价格。同时，也意味着除非出现极为罕见的负利率情况，合约价格不可能超过 100。因此，100 对于欧元美元来说就像是一个上限，类似于股票和大宗商品合约中价格不能低于零的限制。我们可以在计算中引入这一特性，即假设欧元美元合约的实际价值为 100 减去其标价。如果标价为 93.00，那么在理论评估中我们应使用 100 - 93.00，也就是 7.00，来代入定价模型。如果我们定义合约的价值为 7.00，那么一个标准差的价格变化为 16%/16 × 7.00 = 0.07。这个结果显然比 0.93 更为合理。
To be consistent, if we index Eurodollar prices from 100 we must also index exercise prices from 100. Therefore, a 93.50 exercise price in our pricing model is really a 6.50 (100 - 93.50) exercise price. This also requires us to reverse the type of option, changing calls to puts and puts to calls. To see why, consider a 93.50 call. For this call to go into-the-money, the underlying contract must rise above 93.50. But this requires that interest rates fall below 6.50 percent. Therefore, a 93.50 call in listed terms is the same as a 6.50 put in interest rate terms. A model which is correctly set up to evaluate options on Eurodollar or other types of indexed interest rate contracts automatically makes this transformation. The price of the underlying contract and the exercise price are subtracted from 100, while listed calls are treated as puts and listed puts are treated as calls.
为了保持一致，如果我们以 100 为基准计算欧元美元的价格，我们也必须以 100 为基准计算行权价。因此，在定价模型中，93.50 的行权价实际上应为 6.50（100 - 93.50）。这也要求我们反转期权的类型，将看涨期权变为看跌期权，反之亦然。举例来说，一个 93.50 的看涨期权要成为实值期权，合约价格必须超过 93.50。但这意味着利率必须低于 6.50%。因此，标价为 93.50 的看涨期权在利率计算中相当于 6.50 的看跌期权。一个正确设置的定价模型会自动进行这种转换，标的合约的价格和行权价都会从 100 中减去，而标价的看涨期权会被视为看跌期权，标价的看跌期权则被视为看涨期权。
Note that this type of transformation is not required for most bonds and notes. Depending on the coupon rate, the prices of these products may range freely without upper limit, often exceeding 100. They are therefore most often evaluated using a traditional pricing model, although interest rate products present other problems that may require specialized pricing models.'
注意，这种转换对于大多数债券和票据是不需要的。根据票息率，这些产品的价格可以自由波动，且没有上限，通常会超过 100。因此，它们大多使用传统的定价模型进行评估，尽管利率产品存在其他问题，可能需要专门的定价模型。
It is possible to take an instrument such as a bond and calculate the current yield based on its price in the marketplace. If we were to take a series of prices and from these calculate a series of yields, we could also calculate the yield volatility, i.e., the volatility based on the change in yield. We might then use this number to evaluate the theoretical value of an option on the bond, although to be consistent we would also have to specify the exercise price in terms of yield. Because it is possible to calculate the volatility of an interest rate product using these two different methods, interest rate traders sometimes refer to yield volatility (the volatility calculated from the current yield on the instrument) versus price volatility (the volatility calculated from the price of the instrument in the marketplace).
我们可以拿一个债券类产品，根据其市场价格计算当前收益率。如果我们计算一系列价格，并由此得出一系列收益率，我们还可以计算收益率的波动率，即基于收益率变化的波动率。然后我们可以用这个数据来评估债券期权的理论价值，但为了保持一致，我们还需要用收益率来指定行权价。因为可以用这两种不同的方法计算利率产品的波动率，利率交易者有时会区分收益率波动率（根据当前收益率计算的波动率）和价格波动率（根据市场价格计算的波动率）。
TYPES OF VOLATILITIES
波动率的类型
When traders discuss volatility, even experienced traders may find that they are not always talking about the same thing. When a trader makes the comment that the volatility of XYZ is 25%, this statement may take on a variety of meanings. We can avoid confusion in subsequent discussions if we first define the various ways in which traders interpret volatility.
当交易者讨论波动率时，即使是经验丰富的交易者，也可能发现他们并不总是在谈论同样的东西。例如，当一位交易者提到 XYZ 的波动率为 25% 时，这个说法可能有多种含义。为了避免混淆，我们首先需要定义交易者理解波动率的不同方式。
Future Volatility
未来波动率
Future volatility is what every trader would like to know, the volatility that best describes the future distribution of prices for an underlying contract. In theory it is this number to which we are referring when we speak of the volatility input into a theoretical pricing model. If a trader knows the future volatility, he knows the right "odds." When he feeds this number into a theoretical pricing model, he can generate accurate theoretical values because he has the right probabilities. Like the casino, he may occasionally lose because of short-term bad luck. But in the long run, with the odds on his side, a trader can be fairly certain of making a profit.
未来波动率是每个交易者都想知道的，它是最能描述标的合约未来价格分布的波动率。理论上，这就是我们在谈论定价模型中的波动率输入时所指的数值。如果交易者知道未来波动率，他就掌握了正确的 “概率”。当他将这个数值输入到理论定价模型中时，能够生成准确的理论价值，因为他使用了正确的概率。就像赌场一样，虽然短期内可能因运气不好而亏损，但从长期来看，概率在他这边，交易者可以较为确定地获得利润。
Of course, traders rarely talk about the future volatility since it is impossible to know what the future holds.
当然，交易者很少讨论未来波动率，因为未来是无法预知的。
Historical Volatility
历史波动率
Even though one cannot know the future, if a trader intends to use a theoretical pricing model he must try to make an intelligent guess about the future volatility. In option evaluation, as in other disciplines, a good starting point is historical data. What typically has been the volatility of this contract over some period in the past? If, over the past ten years the volatility of a contract has never been less than 10% nor greater than 30%, a guess for the future volatility of either 5% or 40% hardly makes sense. This does not mean that either of these extremes is impossible (In option trading the impossible always seems to happen sooner or later), but based on past performance, and in the absence of any extraordinary circumstances, a guess within the historical limits of 10% and 30% Is probably more realistic than a guess outside these limits. Of course, 10% to 30% Is still a huge range, but at least the historical data offers a starting point. Additional information may further narrow the estimate.
尽管无法预测未来，交易者在使用理论定价模型时，仍然需要对未来波动率做出合理的预测。在期权评估中，和其他领域一样，历史数据是一个不错的起点。过去某段时间里，这个合约的波动率通常是多少？如果在过去十年中，某个合约的波动率从未低于 10% 或高于 30%，那么预测未来波动率为 5% 或 40% 显然不太合理。这并不意味着这些极端情况不可能发生（在期权交易中，所谓不可能的事情迟早会发生），但基于过去的表现，并且在没有任何特殊情况下，预测在 10% 到 30% 之间的波动率要比超出这个范围的预测更加现实。当然，10% 到 30% 之间的区间依然很大，但至少历史数据为预测提供了一个起点，额外的信息可能进一步缩小这一范围。
Note that there are a variety of ways to calculate historical volatility, but most methods depend on choosing two parameters, the historical period over which the volatility is to be calculated, and the time interval between successive price changes. The historical period may be ten days, six months, five years, or any period the trader chooses. Longer periods tend to yield an average or characteristic volatility, while shorter periods may reveal unusual extremes in volatility. To become fully familiar with the volatility characteristics of a contract, a trader may have to examine a wide variety of historical time periods.
需要注意的是，计算历史波动率的方法有很多，但大多数方法都依赖于两个参数的选择：用于计算波动率的历史时期和连续价格变化之间的时间间隔。历史时期可以是十天、六个月、五年或任何交易者选择的时间段。较长的时期往往得出一个平均或特征波动率，而较短的时期可能揭示出波动率的极端情况。为了全面了解某个合约的波动率特征，交易者可能需要审视多个不同的历史时间段。
Next, the trader must decide what intervals to use between price changes. Should he use daily price changes? Weekly changes? Monthly changes? Or perhaps he ought to consider some unusual interval, perhaps every other day, or every week and a half. Surprisingly, the interval which is chosen does not seem to greatly affect the result. Although a contract may make large daily moves, yet finish a week unchanged, this is by far the exception. A contract which is volatile from day to day is likely to be equally volatile from week to week, or month to month. This is typified by the graphs in Figure 4-8. The data points on the three graphs represent the volatility of the S&P 500 index over successive 50-day periods. For the solid line dally price changes were used, for the dotted line price changes every two days were used, and for the broken line price changes every 5 days were used. Even though the graphs occasionally diverge, for the most part they exhibit the same general volatility levels and trends.
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| 图 4-8 标普 500 指数的 50 天历史波动 |
As a general rule, services which supply historical volatility data base their calculations on daily settlement-to-settlement price changes. If this is not the case, an explanation of how the volatility was calculated will usually accompany the data. If, for example, a service gave the volatility of a contract for the month of August as 21.6%, it can be assumed that the calculations were made using the daily settlement-to-settlement price changes for all the business days during that month.
通常，提供历史波动率数据的服务是基于每日的结算价格变化进行计算的。如果不是这种情况，数据中通常会附有解释说明波动率的计算方法。例如，如果某服务提供了八月份的某合约波动率为 21.6%，那么可以假定这些计算是基于该月所有交易日的每日结算价格变化。
Historical and future volatility are sometimes referred to as realized volatility.
历史波动率和未来波动率有时也被称为已实现的波动率。
Forecast Volatility
预测波动率
Just as there are services which will attempt to forecast future directional moves in the price of a contract, there are also services which will attempt to forecast the future volatility of a contract. Forecasts may be for any period, but most commonly cover periods identical to the remaining life of options on the underlying contract. For an underlying contract with three months between expirations, a service might forecast volatilities for the next three, six, and nine months. Due to the relatively recent introduction of options, volatility forecasting is still in its infancy, and must be considered an inexact science at best. Nonetheless, a trader's guess about the future volatility of a contract might very well take into consideration any volatility forecast to which he has access.
就像有服务尝试预测合约价格未来的方向性变化一样，也有服务会尝试预测合约未来的波动率。预测的时间范围可以是任何期限，但最常见的是与该合约期权剩余期限相同的时间段。对于一个距离到期还有三个月的标的合约，某服务可能会预测未来三个月、六个月和九个月的波动率。由于期权是相对较新的金融工具，波动率预测仍处于早期阶段，充其量只能算是一门不精确的科学。尽管如此，交易者对合约未来波动率的猜测，很可能会参考他们能够获得的任何波动率预测数据。
Implied Volatility
隐含波动率
Generally speaking, future, historical, and forecast volatility are associated with an underlying contract. We can talk about the future volatility of the S&P 500 index, or the historical volatility of U.S. Treasury Bonds, or a forecast volatility for IBM stock. In each case we are referring to the volatility of the underlying contract. There is, however, a different interpretation of volatility which is associated with an option rather than with the underlying contract.
通常来说，未来波动率、历史波动率和预测波动率是与标的合约相关的。我们可以讨论标普 500 指数的未来波动率，美国国债的历史波动率，或者 IBM 股票的预测波动率。在这些情况下，我们都是在指标的合约的波动率。然而，还有一种与期权相关的不同波动率解释，这就是隐含波动率。
Suppose a certain futures contract is trading at 98.50 with interest rates at 8%. Suppose also that a 105 call with three months to expiration is available on this contract, and that our best guess about the volatility over the next three months is 16%. If we want to know the theoretical value of the 105 call we might feed all these inputs into a theoretical pricing model. Using the Black-Scholes Model, we find that the option has a theoretical value of .96. Having done this we might compare the option's theoretical value to its price in the marketplace. To our surprise, we find that the option is trading for 1.34. How can we account for the fact that we think the option is worth .96, while the marketplace seems to believe it is worth 1.34?
假设某个期货合约的价格为 98.50，利率为 8%，而该合约上带有三个月到期时间的 105 看涨期权正在交易。我们对未来三个月的最佳波动率估计是 16%。如果我们想知道 105 看涨期权的理论价值，可能会将这些输入参数放入一个理论定价模型中。使用布莱克-肖尔斯模型，我们得出该期权的理论价值为 0.96。接着，我们将期权的理论价值与市场价格进行比较，结果发现该期权的市场价格为 1.34。我们认为期权价值应为 0.96，但市场却认为它值 1.34，如何解释这种差异？
One way to answer the question is to assume that everyone in the marketplace is using the same theoretical pricing model that we are, in this case the Black-Scholes Model. If we make this assumption, then the discrepancy between our value of .96 and the marketplace's value of 1.34 must be due to a difference of opinion concerning one or more of the inputs into the model. We can therefore start going down the list of inputs and try to identify the culprits.
一种解释是，假设市场上的每个人都在使用相同的理论定价模型，即布莱克-肖尔斯模型。如果作此假设，那么我们计算出的 0.96 和市场价格 1.34 之间的差异，必定源于输入模型的某些参数存在不同看法。接下来，我们可以逐一排查这些参数。
We know that it can't be either the amount of time to expiration or the exercise price, since these inputs are fixed in the option contract. What about the underlying price of 98.50? Perhaps we think the price of the underlying is at 98.50, but it is really trading at some higher price, say 99.00. Indeed, in such circumstances it is always wise to double check the inputs. But suppose we still find that the underlying is at 98.50. Even given that there is a spread between the bid and ask price, if the market is reasonably liquid it is unlikely that the spread would be wide enough to cause a discrepancy of .38 In the value of the option. Perhaps our problem is the interest rate of 8%. But, as we noted in the last chapter, the interest rate component is usually the least important of the inputs into a theoretical pricing model. And in the case of futures options, the interest rate component is often trivial. This leaves us with only one likely culprit, and that is volatility. In some sense, the marketplace must be using a volatility other than 16% to evaluate the 105 call.
期权到期时间和执行价格是固定的，排除掉它们。标的资产价格为 98.50，是否我们误认为标的资产的价格是 98.50，实际上可能是 99.00？在这种情况下，检查输入参数是明智的，但假设标的资产的价格确实是 98.50。即使考虑买卖差价，如果市场流动性合理，这种差价也不太可能导致期权价值有 0.38 的差异。利率 8% 是否有问题？正如前一章所述，利率通常是理论定价模型中最不重要的因素，尤其在期货期权中，利率的影响常常微不足道。这让我们得出唯一的可能因素：波动率。市场在评估 105 看涨期权时，可能使用了不同于 16% 的波动率。
What volatility is the marketplace using? To find out, we can ask the following question: If we hold all other inputs constant (time to expiration, exercise price, underlying price, interest rates), what volatility must we feed into our theoretical pricing model to yield a theoretical value identical to the price of the option in the marketplace? In our example, we want to know what volatility will yield a value of 1.34 for the 105 call. Clearly the volatility has to be higher than 16%, so we might sit down with a computer programmed with the Black-Scholes Model and start to raise the volatility. If we do, we will find that at a volatility of 18.5% the 105 call has a theoretical value of 1.34. We refer to this volatility as the implied volatility of the 105 call. It is the volatility we must feed into our theoretical pricing model to yield a theoretical value identical to the price of the option in the marketplace. We can also think of it as the volatility being implied to the underlying contract through the pricing of the option in the marketplace.
市场使用的波动率是多少？我们可以问这样一个问题：如果将其他参数保持不变（到期时间、执行价格、标的资产价格、利率），需要输入什么样的波动率，才能让理论定价模型的期权价值与市场价格一致？在我们的例子中，我们希望找到一个波动率，它能得出 105 看涨期权的理论价值为 1.34。显然，这个波动率必须高于 16%，因此我们可以通过增加波动率来求解。最终，我们发现当波动率为 18.5% 时，105 看涨期权的理论价值与市场价格 1.34 一致。这种波动率称为该 105 看涨期权的隐含波动率，即将其输入理论定价模型后所得出的期权价格与市场价格一致。也可以将其视为通过市场期权定价隐含出的标的资产波动率。
When we solve for the implied volatility of an option we are assuming that the theoretical value (the option's price) is known, but that the volatility is unknown. In effect, we are running the theoretical pricing model backwards to solve for this unknown, as shown in Figure 4-9. In fact, this is easier said than done since most theoretical pricing models cannot be reversed. However, a number of computer programs have been written which can quickly solve for the implied volatility when all other inputs are known.
计算期权的隐含波动率时，假设期权的价格（即理论价值）是已知的，而波动率是未知的。我们实际上是在反向运行定价模型来求解这个未知数。然而，大多数理论定价模型并不能简单反转运行，但有很多计算机程序可以在输入已知参数的情况下快速求解隐含波动率。
Note that the implied volatility depends on the theoretical pricing model being used. For some options, a different model can yield a significantly different implied volatility. The accuracy of an implied volatility also depends on the accuracy of inputs into the model. This not only includes the price of the option, but the other inputs as well. In particular, problems can occur when an option has not traded for some time, but market conditions have changed significantly. Suppose in our example the price of 1.34 for the 105 call reflected the last trade, but that trade took place two hours ago when the underlying futures contract was actually at 99.25. If the price of the underlying contract is 99.25, the implied volatility of the option at a price of 1.34 is actually 17.3%, This is a significant difference, and underscores the importance of accurate and timely inputs when calculating implied volatilities.
需要注意的是，隐含波动率依赖于所使用的理论定价模型。对于某些期权，不同的定价模型可能会给出显著不同的隐含波动率。隐含波动率的准确性还取决于模型输入参数的准确性，不仅包括期权价格，还包括其他输入，尤其是在期权未交易一段时间且市场条件发生显著变化时，问题可能会更加突出。例如，假设 105 看涨期权 1.34 的价格反映的是两小时前的最后一笔交易价格，而当时标的期货合约的价格为 99.25。如果标的资产价格为 99.25，期权价格 1.34 的隐含波动率实际上是 17.3%，这就是一个显著的差异，强调了在计算隐含波动率时精确、及时输入数据的重要性。
Services which supply theoretical analysis of options usually include implied volatilities. The implied volatilities may be for each option on an underlying contract, or there may be one implied volatility which is representative of all options on the same underlying in the latter case, the figure will usually represent an average of all the individual implied volatilities. The single implied volatility is weighted according to some criteria, such as volume of options traded, open interest, or, as is most common, by assigning the greatest weight to the at-the-money options.
提供理论分析服务的机构通常也会提供隐含波动率数据。这些隐含波动率可能针对某个标的合约的每个期权，或者可能提供一个代表所有期权的隐含波动率，通常是各个隐含波动率的加权平均数，权重可以基于期权的交易量、未平仓合约量，或更常见的情况是基于平值期权的权重。
The implied volatility in the marketplace is constantly changing because option prices, as well as other market conditions, are constantly changing. It is as if the marketplace were continuously polling all participants to come up with a consensus volatility for the underlying contract. This is not a poll in the true sense, since all traders do not huddle together and eventually vote on the correct volatility. However, as bits and offers are made, the trade price of an option will represent the equilibrium between supply and demand. This equilibrium can be translated into an implied volatility.
市场中的隐含波动率会不断变化，因为期权价格及其他市场条件也在不断变化。我们可以将这种变化视为市场对标的合约隐含波动率的 “共识”。虽然不是真正的投票，但市场上的每笔买卖和交易价格，都是供求关系的平衡点，反映了隐含波动率。
Even though the term premium really refers to an option's price, it is common among traders to refer to the implied volatility as the premium or premium level. If the current implied volatility is high by historical standards, or high relative to the recent historical volatility of the underlying contract, a trader might say that premium levels are high; if implied volatility is unusually low, he might say that premium levels are low.
即便如此，交易员们有时会将隐含波动率称为 “溢价” 或 “溢价水平”。如果当前隐含波动率相较历史标准较高，或者相较近期的历史波动率较高，交易员可能会说 “溢价水平较高”；如果隐含波动率异常低，则会说 “溢价水平较低”。
Assuming a trader had a reliable theoretical pricing model, if he could look into a crystal ball and determine the future volatility of an underlying contract he would be able to accurately evaluate options on that contract. He might then look at the difference between each option's theoretical value and its price in the marketplace, selling any options which were overpriced relative to the theoretical value, and buying any options which were underpriced. If given the choice between selling one of two overpriced options, he might simply sell the one which was most overpriced in total dollars. However, a trader who has access to implied volatilities might use a different yardstick for comparison. He might compare the implied volatility of an option to either a volatility forecast, or to the implied volatility of other options on the same underlying contract. Going back to our example of the 105 call, we might say that with a theoretical value of .96 and a price of 1.34, the 105 call is .38 overpriced. But in volatility terms it is 2.5% overpriced since its theoretical value is based on a volatility of 16% (the trader's volatile estimate) while its price is based on a volatility of 18.5% (the implied volatility). Due to the unusual characteristics of options, it is often more useful for the serious trader to consider an option's price in terms of implied volatility rather than in terms of its total dollar price.
假设交易者有一个可靠的理论定价模型，如果他能够通过水晶球看出标的资产未来的波动率，他就能够准确评估该资产的期权。他可以通过比较每个期权的理论价值与市场价格的差异，卖出相对理论价值定价过高的期权，买入定价过低的期权。如果需要在两个定价过高的期权中选择一个卖出，他可能会选择总金额最高的那个。然而，如果交易者有隐含波动率的参考，他可能会用不同的标准进行比较。他可以将期权的隐含波动率与未来波动率预测或该资产其他期权的隐含波动率进行对比。 例如，假设 105 的认购期权理论价值是 0.96，但市场价格是 1.34，因此这个期权高估了 0.38 美元。然而，按照波动率来衡量，它的价格高估了 2.5%，因为理论价格基于 16% 的波动率（交易者的估计值），而市场价格基于 18.5% 的隐含波动率。由于期权具有独特的特性，对于严肃的交易者来说，考虑期权价格时，用隐含波动率来衡量通常比直接看总金额更有用。
For example, suppose a Treasury Bond 98 call is trading for 3-32 ($3,500) with a corresponding implied volatility of 10.5%. Suppose also that a 102 call with the same expiration date is trading for 1-16 ($1,250) with an implied volatility of 11.5%. In total dollar terms the 102 call is $2,250 cheaper than the 98 call. Yet an experienced trader will probably conclude that in theoretical terms the 98 call is actually less expensive than the 102 call because the implied volatility of the 98 call is a full percentage point less than the implied volatility of the 102 call. Does this mean that one ought to buy the 98 call and sell the 102 call? Not necessarily. If the future volatility of the Treasury Bonds turns out to be 8%, then both options are overpriced; while if volatility turns out to be 14%, both options are underpriced. Moreover, the leverage values of the options may not be the same so that their sensitivity to changes in market conditions may, under some circumstances, make several 102 calls a more desirable purchase than one 98 call. If we ignore these considerations, in relative terms the 98 call is a better value because its implied volatility is lower.
再举个例子，假设 98 的国债认购期权交易价格为 3-32（$3,500），隐含波动率为 10.5%。同时，102 的认购期权交易价格为 1-16（$1,250），隐含波动率为 11.5%。从总金额上看，102 的认购期权比 98 的便宜 $2,250。然而，经验丰富的交易者可能会认为 98 的期权在理论上比 102 的便宜，因为它的隐含波动率比 102 低整整一个百分点。这是否意味着应该买入 98 的期权并卖出 102 的期权呢？不一定。如果未来国债的波动率为 8%，那么两个期权都被高估；如果波动率为 14%，则两个期权都被低估。此外，这两个期权的杠杆值可能不同，因此在某些情况下，买入几个 102 的期权可能比买入一个 98 的期权更有利。如果不考虑这些因素，相对来说，98 的期权因为其隐含波动率较低而更具价值。
While option traders may at times refer to any of the four types of volatilities, two of these stand out in importance, the future volatility and the implied volatility. The future volatility of an underlying contract determines the value of options on that contract. The implied volatility is a reflection of each option's price. These two numbers, value and price, are what all traders, not just option traders, are concerned with. If a contract has a high value and a low price, then a trader will want to be a buyer. If a contract has a low value and a high price, then a trader will want to be a seller. For an option trader this usually means comparing the future volatility with the implied volatility. If implied volatility is low with respect to the expected future volatility, a trader will prefer to buy options; if implied volatility is high, a trader will prefer to sell options. Of course, future volatility is an unknown, so we tend to look at the historical and forecast volatilities to help us make an intelligent guess about the future. But in the final analysis, it is the future volatility which determines an option's value.
期权交易者有时会提到四种波动率中的任意一种，但其中两种最为重要：未来波动率和隐含波动率。标的资产的未来波动率决定了期权的价值，而隐含波动率则反映了每个期权的市场价格。这两个数字——价值和价格，是所有交易者关心的核心问题。如果某个资产的价值高而价格低，交易者会想买入；如果价值低而价格高，交易者则会想卖出。对于期权交易者来说，这通常意味着比较未来波动率与隐含波动率。如果隐含波动率相对未来波动率较低，交易者倾向于买入期权；如果隐含波动率较高，交易者倾向于卖出期权。当然，未来波动率是未知的，因此我们通常会查看历史波动率和预测波动率，以帮助我们对未来做出合理的猜测。但最终决定期权价值的还是未来波动率。
To help the new trader understand the various types of volatility, consider the following analogy to weather forecasting. Suppose a trader living in Chicago gets up on a July morning and must decide what clothes to wear that day. Do you think he will consider putting on a parka? This is not a logical choice because he knows that historically it is not sufficiently cold in Chicago in July to warrant wearing a winter coat. Next, he might turn on the radio or television to listen to the weather forecast. The forecaster is predicting clear skies with temperatures around 90° (32°C). Based on this information, our trader has reached a decision: he will wear a short sleeve shirt with no sweater or jacket, and he certainly won't need an umbrella. However, just to be sure, he decides to take a look out the window to see what the people outside are wearing. To his surprise, everyone is wearing a coat and carrying an umbrella. The people outside, through their clothing, are implying completely different weather. What clothes should the trader then wear? He must make some decision, but whom should he believe, the weather forecaster or the people in the street? There can be no certain answer because the trader will not know the future weather until the end of the day.
为了帮助新手交易者理解各种波动率，我们可以用天气预报来做个类比。假设一个住在芝加哥的交易者在七月的早晨起床，决定今天穿什么衣服。你认为他会穿厚重的防寒服吗？这并不是一个合理的选择，因为他知道根据历史情况，七月的芝加哥不会冷到需要穿冬衣。接着，他可能会打开收音机或电视，听天气预报。预报员预测今天是晴天，气温大约 90°F（32°C）。根据这个信息，交易者决定穿短袖衬衫，不用毛衣或外套，而且肯定不需要带伞。然而，为了确保万无一失，他决定看看窗外的情况，发现外面的人都穿着大衣，拿着伞。外面的人通过他们的穿着暗示了完全不同的天气。那么，交易者应该穿什么衣服呢？他必须做出决定，但他应该相信天气预报员，还是相信街上的人呢？没有确切的答案，因为交易者直到一天结束后才会知道当天的天气如何。
Much depends on the trader's knowledge of local conditions. Perhaps the trader lives in an area far removed from where the weather forecaster is located. Then he must give added weight to local conditions. On the other hand, perhaps the people in the street all listened to a weather forecaster who has a history of playing practical jokes.
最终，很多取决于交易者对当地情况的了解。也许交易者住的地方距离天气预报员的所在地很远，那他就需要更多关注本地的实际情况。另一方面，或许街上的人都听信了一个喜欢开玩笑的天气预报员。
The decision on what clothes to wear, just like every trading decision, depends on a great many factors. Not only must the decision be made on the basis of the best available information, but the decision must also be made with consideration for the possibility of error. What are the benefits of being right? What are the consequences of being wrong? If a trader falls to take an umbrella and it rains, that may be of little consequence if the bus picks him up right outside his residence and drops him off right outside his place of work. On the other hand, If he must walk several blocks in the rain, he might catch the flu and be away from work for a week. The choices are never easy, and one can only hope to make the decision that will turn out best in the long run.
决定穿什么衣服，就像每个交易决定一样，取决于很多因素。不仅需要基于现有的最佳信息做决定，还要考虑到可能的错误。判断正确时有什么好处？判断错误时会有什么后果？如果交易者没带伞却下雨了，可能没什么大不了的，如果巴士就在他家门口接他，送他到工作地点。然而，如果他得在雨中走几条街，他可能会感冒并因此请假一周。选择从来都不简单，只能希望在长远来看，做出的决定是最好的。
Seasonal Volatility
季节性波动率
There is one other type of volatility with which a commodity trader may have to deal. Certain agricultural commodities, such as corn, soybeans, and wheat, are very sensitive to volatility factors arising from severe seasonal weather conditions. Such conditions occur especially in the summer months when drought can destroy major portions of a crop and cause prices to fluctuate wildly. For this reason, grains show a significant increase in volatility during the months of June, July, and August. Conversely, they show a significant decrease during the early spring months, before American planting has begun but after the South American crop has been harvested. Given these factors, one must automatically assign a higher volatility to an option contract which extends through the summer months. If, in February, a trader has assigned a volatility of 18% to a May soybean contract, he will certainly choose some higher volatility, perhaps 22%, for a November contract. He knows that the November contract includes the summer months, while the May contract does not. The effect of seasonal volatility on soybeans is shown in Figure 4-10.
商品交易者可能还需要处理另一种类型的波动率。某些农业商品，如玉米、大豆和小麦，对因严重季节性天气条件引起的波动因素非常敏感。这种天气条件通常发生在夏季，干旱可能会摧毁大部分作物，导致价格剧烈波动。因此，谷物在 6 月、7 月和 8 月的波动率显著增加。相反，在早春月份（即美国种植尚未开始但南美作物已收获后），波动率则显著降低。考虑到这些因素，必须自动为跨越夏季的期权合约分配更高的波动率。例如，如果在 2 月，一位交易者为 5 月的大豆合约指定了 18% 的波动率，他肯定会为 11 月的合约选择更高的波动率，可能是 22%。因为他知道，11 月合约涵盖了夏季，而 5 月合约则没有。图 4-10 显示了季节性波动率对大豆的影响。
A trader who is new to options might initially question whether volatility is really that important. He has probably been pursuing directional strategies where volatility was not a consideration. It is also possible to pursue a variety of directional strategies in the option market. But if a trader has a thorough understanding of volatility, he has an additional variable with which to work. He can, in effect, approach the market from two directions instead of one. Many traders find it easier to work exclusively with volatility, rather than try to guess market direction. Moreover, volatility strategies can be extremely profitable and, when chosen intelligently, can even reduce a trader's risk exposure. These two variables, market direction and volatility, enable an option trader to pursue many strategies not available to the pure stock or futures trader.
对期权交易新手来说，可能会质疑波动率是否真的那么重要。他可能一直在进行方向性策略，而波动率并没有考虑在内。在期权市场中也可以追求多种方向性策略。然而，如果交易者对波动率有深入的理解，他就有了一个额外的变量可以利用。实际上，他可以从两个方向而不是一个方向接近市场。许多交易者发现，专注于波动率比试图猜测市场方向更容易。此外，波动率策略可能非常盈利，并且在明智选择时甚至可以减少交易者的风险敞口。这两个变量——市场方向和波动率，使期权交易者能够追求许多纯股票或期货交易者无法实现的策略。
Changing our assumptions about future volatility can have a dramatic effect on the value of options. To see this, look at the prices, theoretical values, and implied volatilities for ten-week gold options in Figure 4-11. Note the change in theoretical values as volatility is increased from 11% to 14% to 17%. The 360 call and put, which are essentially at-the-money, change by approximately 1.85 ($185) for each three-percent-age-point increase in volatility. While out-of-the-money options do not show as great a dollar change in value, in percent terms their sensitivity to a change in volatility is even greater. As volatility increases from 11% to 14%, the 390 call and 330 put more than double in value, and double again as volatility increases from 14% to 17%. A three-percentage point change in volatility over ten weeks is not at all uncommon. Indeed, the volatility of gold can show swings of six or seven percentage points in a relatively short period of time. This is evident from the historical volatility of gold shown in Figure 4-12.
改变我们对未来波动率的假设可能会对期权的价值产生显著影响。要了解这一点，可以查看图 4-11 中十周期金期权的价格、理论价值和隐含波动率。注意随着波动率从 11% 提高到 14% 再到 17%，理论价值的变化。360 的认购和认沽期权（基本上是平值期权）在每次波动率增加三个百分点时，价值变化约为 1.85（185 美元）。虽然平值期权在美元价值变化上不如平值期权大，但在百分比方面，它们对波动率变化的敏感性甚至更大。当波动率从 11% 提高到 14% 时，390 的认购期权和 330 的认沽期权的价值翻了一番，并在波动率从 14% 提高到 17% 时再次翻倍。十周内波动率的三百分点变化并不罕见。实际上，黄金的波动率在相对较短的时间内可能出现六到七个百分点的波动。这从图 4-12 中显示的黄金历史波动率中可以看出。
Given its importance, it is not surprising that the serious option trader spends a considerable amount of time thinking about volatility. Using his knowledge of historical, forecast, implied, and, in the case of agricultural commodities, seasonal volatility, he must try to make an intelligent decision about future volatility. From this, he will look for option strategies which will be profitable when he is right, but which will not result in a disastrous loss when he is wrong. Because of the difficulty in predicting volatility, a trader must always look for strategies which will leave the greatest margin for error. No trader will survive very long pursuing strategies based on a future volatility estimate of 15% if such a strategy results in a loss when volatility actually turns out to be 16%. Given the shifts that occur in volatility, a one-percentage-point margin for error is no margin for error at all.
鉴于波动率的重要性，严肃的期权交易者会花相当多的时间考虑波动率。利用他对历史、预测、隐含波动率以及在农业商品情况下的季节性波动率的知识，他必须努力做出关于未来波动率的明智决策。由此，他将寻找在正确时能够盈利但在错误时不会导致灾难性损失的期权策略。由于预测波动率的困难，交易者必须始终寻找能够留出最大误差空间的策略。如果一位交易者基于 15% 的未来波动率估计进行策略，但当波动率实际为 16% 时导致亏损，那么他很难长期生存。考虑到波动率的变动，1 个百分点的误差空间根本没有误差空间。
We have not yet concluded our discussion of volatility. But before continuing, it will be useful to look at option characteristics, trading strategies, and risk considerations. We will then be in a better position to examine volatility in greater detail.
我们尚未结束关于波动率的讨论。但在继续之前，查看期权特性、交易策略和风险考虑将是有益的。然后，我们将能更好地详细考察波动率。
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