Chapter 3 Introduction to Theoretical Pricing Models

Chapter 3 Introduction to Theoretical Pricing Models
第三章 理论定价模型概述

In the last chapter we looked at some of the simple option strategies a trader might initiate given an opinion on an underlying contract's likely price movement. Whatever the basis for the opinion, it will probably be expressed with terms such as "good chance," "highly likely," "possible," "improbable," etc. The problem with this approach is that opinions cannot easily be expressed in numerical terms. What do we really mean by "good chance"? Or by "highly unlikely"? If we want to approach option markets logically we will need some method of quantifying our opinions about price movements.

在上一章中，我们探讨了交易者在对标的合约价格走势有特定看法时，可能采用的一些简单期权策略。不论这些看法基于什么，通常会用“很大可能”、“高度可能”、“可能性较小”等词语来表达。然而，这类看法难以用具体的数字来量化。究竟“很大可能”意味着什么？或者“高度不可能”该如何理解？如果我们想更理性地分析期权市场，就需要找到一种方法，对价格走势的看法进行量化。

From the elementary strategies discussed in the previous chapter it is clear that the direction in which the underlying market moves can have a significant effect on the profitability of an option strategy. Consequently, option traders are sensitive to the direction in which the underlying market moves. But an option trader has an additional problem: the speed of the market. A commodity trader who believes a commodity will rise in price within a specified period can be reasonably certain of making a profit if he is right. He simply buys the commodity, waits for it to reach his target price, then sells the commodity for a profit.

从上一章讨论的初级策略可以看出，标的市场的方向对期权策略的盈利性有显著影响。因此，期权交易者对市场方向极为敏感。然而，期权交易者还面临另一个挑战：市场的变化速度。一位商品交易者如果相信某商品价格会在指定时间内上涨，只需买入该商品，待其达到目标价后卖出即可获利，过程相对简单。而对于期权交易者来说，时间与价格变化的速度则更加关键。

The situation is not quite so simple for an option trader. Suppose a trader believes a commodity will rise in price from $100, its present price, to $120 within the next five months. Suppose also that a $110 call expiring in three months is available at a price of $4. If the commodity rises to $120 by expiration, the purchase of the $110 will result in a profit of $6 ($10 intrinsic value less the $4 cost of the option). But is this profit a certainty? What will happen if the price of the commodity remains below $110 for the next three months and only reaches $120 after the option expires? Then the option will expire worthless and the trader will lose his $4 investment.

对于期权交易者而言，情况并没有那么简单。假设某交易者预计某商品的价格将在未来五个月内从当前的100元涨至120元。与此同时，市场上有一份执行价为110元、三个月到期的看涨期权，价格为4元。如果商品价格在期权到期时上涨至120元，购买这份110元的看涨期权将带来6元的利润（10元的内在价值减去4元的期权成本）。然而，这笔利润是确定的吗？如果商品价格在接下来的三个月内一直低于110元，直到期权到期后才涨至120元，那么期权将失效，交易者将损失4元的投资。

Perhaps the trader would do better to purchase a $110 call which expires in six months rather than three months. Now he can be certain that when the commodity reaches $120, the call will be worth at least $10 in intrinsic value. But what if the price of the six-month option is $12? In that case the trader might still show a loss. Even though the underlying commodity reaches the target price of $120, there is no guarantee that the $110 call will ever be worth more than its $10 intrinsic value.

也许交易者更适合购买一份6个月到期的110元看涨期权，而不是3个月到期的。这样，他可以确保当商品价格涨到120元时，期权至少会有10元的内在价值。但如果这份6个月到期的期权价格为12元呢？在这种情况下，交易者可能仍然会亏损。即使标的商品达到了120元的目标价，也无法保证110元的看涨期权的价值会超过其10元的内在价值。

A trader in an underlying market is almost exclusively interested in the direction in which the market will move. While the option trader is also sensitive to directional considerations, he must also give careful consideration to how fast the market is likely to move. If a futures trader and an option trader take long market positions in their respective instruments, and the market does in fact move higher, the futures trader is assured of a profit while the option trader may show a loss. If the market fails to move sufficiently fast, the favorable directional move may not be enough to offset the option's loss in time value. This is the primary reason speculators generally lose in option markets. A speculator usually buys options for their seemingly favorable risk/reward characteristics (limited risk/unlimited reward). But if he purchases options, not only must he be right about market direction, he must also be right about market speed. Only if he is right on both counts can he expect to make a profit. If predicting the correct market direction is difficult, correctly predicting direction and speed is probably beyond most traders' capabilities.

在标的市场中，交易者几乎只关心市场的方向。虽然期权交易者同样关注市场方向，但他还必须仔细考虑市场变动的速度。如果期货交易者和期权交易者都持有多头头寸，且市场确实上涨，期货交易者几乎可以确定获利，而期权交易者则可能面临亏损。如果市场上涨的速度不够快，那么有利的方向性变动可能不足以弥补期权的时间价值损失。这是投机者在期权市场上通常亏损的主要原因。投机者通常因为期权看似有利的风险/回报特性（有限风险/无限回报）而买入期权。但如果他买了期权，不仅要正确预测市场方向，还必须准确预测市场的速度。只有当这两点都预测正确时，才有可能获得利润。如果正确预测市场方向已经很难，那么同时预测方向和速度对大多数交易者来说更是难上加难。

The concept of speed is vital in trading options. It is so important that there are many option strategies which depend only on the speed of the underlying market and not at all on its direction. Indeed, if a trader is highly proficient at predicting directional moves in the underlying market, he is probably better advised to stick to the underlying instrument. It is only when he has some feel for the speed component that a trader can hope to intelligently enter the option market.

市场速度的概念在期权交易中至关重要。它的重要性如此之高，以至于很多期权策略完全依赖标的市场的速度，而不关注其方向。事实上，如果交易者非常擅长预测标的市场的方向，他可能更适合直接交易标的资产。只有当交易者对市场的速度有所把握时，才可能明智地进入期权市场。

The option trader who wants to intelligently evaluate the potential profitability of an option trade is faced with the task of analyzing several different factors. At a minimum he must consider:

1. The price of the underlying contract
2. The exercise price
3. The amount of time remaining to expiration
4. The direction in which he expects the underlying market to move
5. The speed at which he expects the underlying market to move

想要理性评估期权交易潜在盈利性的期权交易者，必须分析多个关键因素。至少，他需要考虑以下几项：

1. 标的合约的价格
2. 行权价
3. 距离到期的剩余时间
4. 他对标的市场方向的预期
5. 他对标的市场速度的预期

Ideally, he would like to express each of these factors numerically, feed the numbers into a formula, and derive a value for the option. By comparing the value to its price in the marketplace, the trader would then know whether the purchase or sale of the option was likely to be profitable. This is essentially the goal of option evaluation: to analyze an option based on the terms of the option contract, as well as current market conditions and future expectations.

理想情况下，交易者希望能将这些因素用数字量化，输入公式，得出期权的价值。通过将这个价值与市场价格进行比较，交易者可以判断期权的买卖是否可能盈利。这就是期权评估的核心目标：通过期权合约的条款、当前市场状况以及对未来的预期，综合分析期权的价值。

EXPECTED RETURN
预期收益

Suppose we are given the opportunity to roll a six-sided die, and each time we roll we will receive a dollar amount equal to the number which comes up. If we roll a one, we get $1; if we roll a two, we get $2; and so on up to six, in which case we get $6. If we were to roll the die an infinite number of times, on average, how much would we expect to receive per roll?

假设我们有机会掷一个六面的骰子，每次掷骰子都会得到与骰子面点数相等的金额。如果掷出1点，我们得到1元；掷出2点，我们得到2元；以此类推，掷出6点则得到6元。如果我们无限次地掷骰子，平均而言，每次掷骰子我们期望得到多少收益？

We can calculate the answer using some simple arithmetic. There are six numbers which can come up, each with equal probability. If we add up the six possible outcomes 1+2+3+4+5+6=21 and divide this by the six faces of the die we get 21/6 = 3½. That is, on average we can expect to get back $3½ each time we roll the die. This is the average, or expected, return. If someone were to charge us for the privilege of rolling the die, what might we be willing to pay? If we purchased the chance to roll the die for less than $3½, in the long run we would expect to be winners. If we paid more than $3½, in the long run we would expect to be losers. And if we paid exactly $3½, we would expect to break even. Note the qualifying phrase "in the long run." The expected return of $3½ is a realistic goal only if we are allowed to roll the die many, many times. If we are allowed to roll only once, we cannot count on getting back $3½. Indeed, on any one roll it is impossible to get back $3½ since no face of the die has exactly 3½ spots. Nevertheless, if we pay less than $3½ for even one roll of the die, the laws of probability are on our side because we have paid less than the expected return.

我们可以通过一些简单的算术来计算答案。骰子有六个面，每个面出现的概率相等。如果我们将六个可能的结果相加，即1+2+3+4+5+6=21，然后将其除以骰子的六个面，我们得到21/6=3.5。也就是说，平均来说，每次掷骰子我们期望获得3.5元的回报。这就是平均收益或预期收益。如果有人收费让我们掷骰子，我们愿意支付多少呢？如果掷骰子的成本低于3.5元，长期来看我们预计会获利；如果支付超过3.5元，长期来看我们预计会亏损；如果支付正好是3.5元，我们预计将持平。请注意“长期来看”这一限定词。每次掷骰子获得3.5元的预期收益只有在我们能够进行大量掷骰子的情况下才是现实的。如果我们只被允许掷一次骰子，我们无法指望获得3.5元的回报。事实上，在一次掷骰中获得3.5元是不可能的，因为骰子上没有3.5这个面。然而，即使我们只掷一次骰子，如果我们支付的金额低于3.5元，根据概率法则，我们仍有利可图，因为我们支付的金额低于预期收益。

In a similar vein, consider a roulette bet. A roulette wheel has 38 slots, numbered 1 through 36, 0 and 00. (Footnote 1: As is customary in the U.S., we assume a roulette wheel with 38 slots. In some parts of the world the roulette wheel may have no slot 00. This of course changes the odds.) Suppose a casino allows a player to choose a number. If the player's number comes up, he receives $36; if any other number comes up, he receives nothing. What is the expected return from this proposition? There are 38 slots on the roulette wheel, each with equal probability, but only one slot will return $36 to the player. If we divide the one way to win $36 by the 38 slots on the wheel, the result is $36/38 = $.9474, or about 95¢. A player who pays 95¢ for the privilege of picking a number at the roulette table can expect to break about even in the long run.

类似地，考虑一下轮盘投注。一个轮盘有38个格子，编号从1到36，还有0和00。（脚注1：美国轮盘通常有38个格子。某些地区的轮盘中可能没有00格子，这会改变赔率。）假设赌场允许玩家选择一个数字。如果玩家选择的数字出现，他将获得36元；如果出现其他任何数字，他将一无所获。这个投注的预期收益是多少？轮盘上有38个格子，每个格子出现的概率相等，但只有一个格子能为玩家带来36元。如果我们将赢得36元的方式除以轮盘的38个格子，结果是36/38=0.9474元，约等于95分。一个玩家如果支付95分来进行这个投注，长期来看可以期望持平。

Of course, no casino will let a player buy such a bet for 95¢. Under those conditions the casino would make no profit. In the real world, a player who wants to purchase such a bet will have to pay more than the expected return, typically $1. The 5¢ difference between the $1 price of the bet and the 95¢ expected return represents the profit potential, or edge, to the casino. In the long run, for every dollar bet at the roulette table, the casino can expect to keep about 5¢.

当然，赌场不会让玩家以95分的价格买下这个投注。在这种情况下，赌场就不会有利润。在现实中，玩家要进行这样的投注通常需要支付1元。1元的价格与95分的预期收益之间的5分差额代表了赌场的利润或优势。长期来看，每在轮盘赌桌上下注1元，赌场可以预期赚取大约5分。

Given the above conditions, any player interested in making a profit would rather switch places with the casino so that he could be the house. Then he would have a 5¢ edge on his side by selling bets worth 95¢ for $1. Alternatively, the player would like to find a casino where he could purchase the bet for less than its expected return of 95c, perhaps 88c. Then the player would have a 7¢ edge over the casino. 

基于上述条件，任何想要盈利的玩家都会希望与赌场交换位置，这样他就可以成为庄家，利用这5分的优势来出售价值95分的赌注，收取1元。或者，玩家希望找到一个可以以低于预期收益95分的价格进行投注的赌场，比如88分。在这种情况下，玩家就会拥有7分的优势。

THEORETICAL VALUE
理论价值

The theoretical value of a proposition is the price one would expect to pay in order to just break even in the long run. Thus far the only factor we have considered in determining the value of a proposition is the expected return. We used this concept to calculate the 95¢ fair price for the roulette bet. There may, however, be other considerations.

一个命题的理论价值是指一个人期望为此命题支付的价格，以便在长期内实现盈亏平衡。到目前为止，我们在确定命题的价值时考虑的唯一因素是预期收益。我们使用这个概念计算了轮盘赌的公平价格是95分。然而，可能还有其他考虑因素。

Suppose that in our roulette example the casino decides to change the conditions of the bet slightly. The player may now purchase the roulette bet for its expected return of 95¢ and, as before, if he loses the casino will immediately collect his 95¢. Under the new conditions, however, if the player wins the casino will send him his $36 winnings in two months. Will both the player and the casino still break even on the proposition?

假设在我们的轮盘赌例子中，赌场决定稍微改变赌注的条件。玩家现在可以以预期收益95分的价格购买轮盘赌注，和之前一样，如果他输，赌场会立即收回他的95分赌注。然而，在新条件下，如果玩家赢，赌场将在两个月后将他的36元奖金寄给他。那么，在这种情况下，玩家和赌场是否仍能实现盈亏平衡呢？

Where did the player get the 95¢ he used to place his bet at the roulette wheel? In the immediate sense he may have taken it out of his pocket. But a closer examination may reveal that he withdrew the money from his savings account prior to visiting the casino. Since he won't receive his winnings for two months, he will have to take into consideration the two months interest he would have earned had he left the 95¢ in his savings account. If interest rates are 12% annually (1% per month), the interest loss is 2% × 95¢, or about 2c. If the player purchases the bet for its expected return of 95c, he will still be a 2¢ loser because of the cost of carrying a 95¢ debit for two months. The casino, on the other hand, will take the 95¢, put it in an interest-bearing account, and at the end of two months collect 2¢ in interest.

玩家用来在轮盘赌桌下注的95分是从哪里来的？从直接的角度来看，他可能是从口袋里拿出来的。但仔细观察会发现，他是在光顾赌场之前从储蓄账户中取出的这笔钱。由于他将在两个月后才能收到奖金，他必须考虑如果将95分留在储蓄账户中能获得的两个月利息。如果年利率为12%（每月1%），那么两个月的利息损失为2%×95分，约为2分。如果玩家以预期收益95分的价格购买赌注，由于持有95分的债务两个月的成本，他将仍然有2分的亏损。另一方面，赌场将收回95分，将其存入一个计息账户，并在两个月结束时收到2分的利息。

Under these new conditions the theoretical value of the bet is the expected return of 95¢ less the 2¢ carrying cost on the bet, or about 93¢. If a player pays 93¢ for the roulette bet today and collects his winnings in two months, neither he nor the casino can expect to make any profit in the long run.

在这些新条件下，赌注的理论价值是预期收益95分减去2分的持有成本，约为93分。如果玩家今天以93分购买轮盘赌注并在两个月后收回他的奖金，那么在长期内，他和赌场都无法期望获得任何利润。

The two most common considerations in a financial investment are the expected return and carrying costs. There may, however, be other considerations. For example, suppose the casino decided to send the player a 1¢ bonus over the next two months. He could then add this additional payment to the previous theoretical value of 93¢ to get a new theoretical value of 94¢. This is similar to the dividend paid to owners of stock in a company. And, in fact, dividends are an additional consideration in evaluating options on stock.

金融投资中最常见的两个考虑因素是预期收益和持有成本。然而，可能还有其他考虑因素。例如，假设赌场决定在接下来的两个月内给玩家派送1分的奖金。他可以将这笔额外支付加入到之前的理论价值93分，从而得到新的理论价值94分。这类似于公司向股东支付的股息。事实上，股息在评估股票期权时也是一个额外的考虑因素。

Exchanges will perhaps object to the casino analogy. They prefer that option trading not be thought of as gambling. There is certainly no desire here to assess the moral implications of either gambling or option trading. The fact remains that the same laws of probability which enable a casino to set the odds for different games of chance are the same laws of probability which enable a trader to evaluate an option.

交易所可能会对赌场的类比表示反对。他们更希望将期权交易视为非赌博行为。这里并没有评估赌博或期权交易的道德影响的意图。事实是，使赌场能够为不同机会游戏设定赔率的概率法则与使交易者能够评估期权的概率法则是相同的。

The concept of theoretical value based on probability is common in many aspects of business. For those uncomfortable with the gambling analogy, one can go back to the original justification for options and think of them as insurance policies which require the payment of a premium. Through the use of statistical data and probability theory, an actuary at an insurance company will attempt to calculate the likelihood that the insurance company will have to make good on an insurance policy. He can then factor into the equation what the insurance company expects to earn on premium payments, and thereby arrive at a theoretical value for the insurance policy. The policy can then be offered to prospective customers at an additional cost, which represents the theoretical edge to the insurance company.

基于概率的理论价值概念在商业的许多方面都是普遍存在的。对于那些对赌博类比感到不舒服的人，可以回到期权的原始理由，将其视为需要支付保费的保险单。通过使用统计数据和概率理论，保险公司的精算师会试图计算保险公司需要履行保险单的可能性。然后，他可以将保险公司在保费上的预期收益纳入方程中，从而得出保险单的理论价值。然后，在添加额外费用之后的保险单可以提供给潜在客户，这代表了保险公司所拥有的理论优势。

In the same way, the goal of option evaluation is to determine, through the use of theoretical pricing models, the theoretical value of an option. The trader can then make an intelligent decision whether the option is overpriced or underpriced in the marketplace, and whether the theoretical edge is sufficient to justify going into the marketplace and making a trade.

同样，期权评估的目标是通过使用理论定价模型来确定期权的理论价值。交易者可以在此基础上明智地决定该期权在市场中是被高估还是低估，从而判断理论优势是否足以证明进入市场进行交易的合理性。

A WORD ON MODELS
关于模型的说明

Before continuing, a few observations on models in general will be worthwhile.

在继续之前，对模型的一些洞察是有必要关注的。

A model is a scaled down or more easily managed representation of the real world. The model may be a physical one, such as a model airplane or building, or it may be a mathematical one, such as a formula. In each case, the model is constructed to help us better understand the world in which we live. However, it is unwise, and sometimes dangerous, to assume that the model and the real world which it represents are identical in every way. They may be very similar, but the model is unlikely to exactly duplicate every feature of the real world.

模型是对现实世界的一种缩小或更易管理的表示。模型可以是物理模型，例如模型飞机或建筑物，也可以是数学模型，例如公式。在每种情况下，模型的构建旨在帮助我们更好地理解我们所生活的世界。然而，假定模型与其所代表的现实世界在各方面都是相同的，这既不明智，有时也很危险。它们可能非常相似，但模型不太可能完全复制现实世界的每个特征。

All models, if they are to be effective, require us to make certain prior assumptions about the real world. Mathematical models require the input of numbers which quantity these assumptions. If we feed incorrect data into the model, we can expect an incorrect representation of the real world. Every model user must be aware: garbage in, garbage out.

所有模型若要有效运作，都需要我们对现实世界做出一些前提假设。数学模型需要输入量化这些假设的数字。如果向模型输入错误数据，那么就会得到对现实世界的不准确呈现。每个模型的使用者都应意识到：输入垃圾，输出也会是垃圾。

These general observations about models are no less true for option pricing models. An option model is only someone's idea of how options might be evaluated under certain conditions. Since either the model itself, or the data which we feed into the model, might be incorrect, there is no guarantee that model generated values will be accurate, nor can we be sure that these values will bear any logical resemblance to actual prices in the marketplace.

这些关于模型的一般洞察对于期权定价模型同样适用。期权模型只是某人关于在特定条件下如何评估期权的设想。由于模型本身或我们输入模型的数据可能不正确，因此无法保证模型生成的值是准确的，也不能确保这些值与市场上的实际价格有任何逻辑上的相似性。

There is in fact a great deal of disagreement among traders as to the usefulness of option pricing models. Some traders feel that models are so much hocus-pocus, and have no relationship to what goes on in the real world. Other traders feel that once they have a sheet of theoretical values in hand all their problems are solved. The reality lies somewhere in between.

事实上，交易者之间对期权定价模型的实用性存在很大分歧。一些交易者认为模型不过是胡说八道，与现实世界的运行没有关系。另一些交易者则认为一旦手中掌握了一张理论价值表，所有问题就解决了。现实往往介于两者之间。

A new option trader is like someone entering a dark room for the first time. Without any guidance he will grope in the dark and may eventually find what he is looking for. The trader who is armed with a basic understanding of theoretical pricing models enters the same room with a small candle. He can make out the general layout of the mom, but the dimness of the candle prevents him from distinguishing every detail. Moreover, some of what he does see may be distorted by the flickering candle. In spite of these limitations, a trader is more likely to find what he is looking for with a small candle than with no illumination at all.

新手期权交易者就像第一次进入黑暗房间的人。没有任何指导，他会在黑暗中摸索，最终可能找到他想要的东西。而对基本的理论定价模型有一定了解的交易者，就像带着一小根蜡烛进入同一个房间。他可以辨别房间的大致布局，但蜡烛的微弱光线使他无法分辨每一个细节。此外，他所看到的一些东西可能会受到闪烁蜡烛的扭曲。尽管存在这些局限性，但有小蜡烛的交易者比完全没有光亮的交易者更有可能找到他所寻找的东西。

The real problems with theoretical pricing models arise after the trader has acquired some sophistication. As he gains confidence he may begin to increase the size of his trades. When this happens, his inability to make out every detail in the room, as we as the distortions caused by the flickering candle flame, take on increased importance. Now a misinterpretation of what he thinks he sees can lead to financial disaster, since any error in judgement will be greatly magnified.

理论定价模型的真正问题出现在交易者获得一定的专业知识之后。随着他信心的增强，他可能会开始增加交易的规模。当这种情况发生时，他无法识别房间中每个细节的能力，以及闪烁蜡烛火焰造成的扭曲，变得更加重要。此时，误解他认为自己所看到的东西可能导致财务灾难，因为任何判断错误都会被大大放大。

The sensible approach is to make use of a model, but with a full awareness of what it can and cannot do. Option traders will find that theoretical priding models are invaluable tools to understanding the pricing of options. Because of the insights gained from a model, the great majority of successful option traders rely on some type of theoretical pricing model. However, an option trader, if he is to make the best use of a theoretical pricing model, must be aware of its limitations as well as its strengths. Otherwise he may be no better off than the trader groping in the dark. (
footnote 2:
Two interesting articles discuss these limitations:
Paderewski, Stephen; "What Does an Option Pricing Model Tell Us about Option Prices?," Financial Analysts Journal, September/October 1989, pages 12-15.
Black, Fischer; "Living Up to the Model," Risk, Vol. 3, No. 3., March 1990, pages 11-13.)

合理的做法是使用模型，但要充分意识到它的能力和局限性。期权交易者会发现，理论定价模型是理解期权定价的重要工具。由于从模型中获得的洞见，绝大多数成功的期权交易者依赖于某种类型的理论定价模型。然而，期权交易者如果想要充分利用理论定价模型，必须了解它的局限性以及优点。否则，他的处境可能与在黑暗中摸索的交易者无异。（
脚注2：
两篇有趣的文章讨论了这些局限性：
Figlewski, Stephen; "What Does an Option Pricing Model Tell Us about Option Prices?," Financial Analysts Journal, September/October 1989, pages 12-15.
Black, Fischer; "Living Up to the Model," Risk, Vol. 3, No. 3., March 1990, pages 11-13.）

A SIMPLE APPROACH
简单的方法

How might we adapt the concepts of expected return and theoretical value to the pricing of options? We might begin by calculating the expected return for an option. Let's take a simple example.

我们如何将预期收益和理论价值的概念应用于期权定价呢？我们可以从计算期权的预期收益开始。让我们来看一个简单的例子。

Suppose an underlying contract is trading at $100 and that on a certain date in the future, which we will call expiration, the contract can take on one of five different prices: $80, $90, $100, $110, or $120. Assume, moreover, that each of the five prices is equally likely with 20% probability. The prices and probabilities might be represented by the line in Figure 3-1.

假设某个标的合约的交易价格为100元，并且在未来某个到期日，该合约可能会有五个不同的价格：80元、90元、100元、110元或120元。此外，假设这五个价格都是同样可能的，每个价格的概率为20%。这些价格和概率可以用图3-1中的线来表示。

Figure 3-1

$80	$90	$100	$110	$120
20%	20%	20%	20%	20%

If we take a long position in the underlying contract at today's price of $100, what will be the expected return from this position at expiration? 20% of the time we will lose $20 when the contract ends up at $80. 20% of the time we will lose $10 when the contract ends up at $90. 20% of the time we will break even when the contract ends up at $ 100. 20% of the time we will make $10 when the contract ends up at $110. And 20% of the time we will make $20 when the contract ends up at $120. We can write the arithmetic:

-(20% x $20) -(20% x $10) + (20% x 0) + (20% x $10) + (20% x $20) = 0

如果我们在以今天的100元的价格做多标的合约，那么到期时这一头寸的预期收益将是多少呢？在20%的情况下，当合约价格为80元时，我们将损失20元；在20%的情况下，当合约价格为90元时，我们将损失10元；在20%的情况下，当合约价格为100元时，我们将不赚不亏；在20%的情况下，当合约价格为110元时，我们将获得10元；在20%的情况下，当合约价格为120元时，我们将获得20元。我们可以用算式表示：

−(20%×20)−(20%×10)+(20%×0)+(20%×10)+(20%×20)=0

Since the profits and losses exactly offset each other, the expected return to the long position is zero. The same reasoning will show that the expected return to a short position taken at the current price of $100 is also zero. Given the prices and probabilities, If we take either a long or short position we can expect to just break even in the long run.

由于利润和损失恰好相抵消，做多头寸的预期收益为零。同样的推理也表明，在当前价格100元下，做空头寸的预期收益也是零。考虑到价格和概率，无论我们选择做多还是做空，长期来看我们都可以预期只会持平。

Now suppose that we take a long position in a $100 call. Forgetting for a moment about what we might pay for the call, what will be the expected return given the prices and probabilities in Figure 3-1? If the underlying contract finishes at $80, $90, or $100 the call will expire worthless. If the underlying contract finishes at $110 or $120 the call will be worth $10 and $20, respectively. The arithmetic is:

(20% x 0) +(20% × 0) + (20% x 0) + (20% x $10) +(20% × $20) = +$6

现在假设我们持有一个执行价为100美元的看涨期权。暂且不考虑购买该期权的成本，根据图3-1中的价格和概率，预期回报将如何？如果标的合约最终价格为80、90或100美元，则期权将变得毫无价值；若标的合约价格最终为110或120美元，期权将分别价值10美元和20美元。计算如下：

(20%×0)+(20%×0)+(20%×0)+(20%×10)+(20%×20)=+6

The call can never be worth less than zero, so the expected return from the call position is always a non-negative number, in this case $6.

看涨期权的价值永远不会低于零，因此看涨期权头寸的预期收益始终是非负数，在此情况下为6元。

If we want to develop a theoretical pricing model using this approach, we might propose a series of possible prices and probabilities for the underlying contract at expiration. Then, given an exercise price, we can calculate the value of the option at each price outcome, multiply the value by its associated probability, add up all these numbers, and thereby obtain an expected return for the option.

如果我们想利用这种方法开发一个理论定价模型，我们可以提出一系列可能的到期价格和其概率。然后，在给定行使价格的情况下，我们可以计算每个价格结果的期权价值，将该价值乘以其相关的概率，最后将所有这些数字相加，从而获得期权的预期收益。

In the foregoing example we took a very simple situation with only five possible price outcomes, each with identical probability. What changes might we make in order to develop a more realistic model? For one thing, we would have to know the settlement procedure for the option. In the United States, all options are subject to stock-type settlement, which requires full payment for the option. If the $100 call will have an expected return of $6 at expiration, we will have to deduct the carrying costs to get its value today. If interest rates are 12% annually (1% per month) and the option will expire in two months, we will have to discount the $6 expected return by the 2% carrying cost, or about 12¢. The theoretical value of the option will then be $5.88.

在上述例子中，我们考虑了一个非常简单的情况，仅有五个可能的价格结果，每个结果的概率相同。我们可能会作出哪些调整以制定一个更为现实的模型呢？首先，我们需要知道期权的结算程序。在美国，所有期权都采用股票型结算，这要求对期权进行全额支付。如果100元的看涨期权到期时的预期收益为6元，我们需要扣除持有成本才能得到今天的价值。如果年利率为12%（每月1%），而该期权将在两个月后到期，我们需要将6元的预期收益折现2%的持有成本，约为12美分。该期权的理论价值将为5.88元。

What other factors might we have to consider? We assumed that all five price outcomes were equally likely. Is this a realistic assumption? Suppose you were told that only two possible prices were possible at expiration, $110 and $250. With the underlying contract at $100 today, which do you think is more likely? Based on experience, most traders would probably agree that extreme price changes which are far away from today's price are less likely than small changes which remain close to today's price. For this reason, $110 is more likely than $250. To take this into consideration, perhaps our price outcomes, in terms of probability, ought to be concentrated around the present price of the underlying contract. Such a distribution is shown in Figure 3-2. Now the expected return from a $100 call is:

(10% x 0) +(20% x 0) +(40% × 0) +(20% x $10) +(10% × $20) = $4.00

我们还需要考虑其他因素吗？我们假设这五个价格结果的可能性是相等的。这是一个现实的假设吗？假设您被告知到期时只有两个可能的价格：110元和250元。在今天标的合约为100元的情况下，您认为哪个更可能？根据经验，大多数交易者可能会同意，与今天的价格相距较远的极端价格变化的可能性小于保持在今天价格附近的小幅变化。因此，110元的可能性高于250元。为了考虑这一点，我们或许应该将价格结果的概率集中在标的合约当前价格附近。这种分布在图3-2中显示。现在，100元看涨期权的预期收益为：

(10%×0)+(20%×0)+(40%×0)+(20%×10)+(10%×20)=4.00

If, as before, the option is subject to stock-type settlement and carrying costs are 2%, the theoretical value of the option will now be $3.92.

如果像之前一样，该期权受股票型结算且持有成本为2%，那么该期权的理论价值将为3.92元。

Figure 3-2

$80	$90	$100	$110	$120
10%	20%	40%	20%	10%

Figure 3-3

$80	$90	$100	$110	$120
10%	20%	30%	25%	15%

Note that in Figure 3-2 all outcomes and probabilities are arranged symmetrically. Even though the new probabilities altered the expected return for the $100 call, the expected return from any position taken in the underlying contract is still zero. For each upward price move, there is a downward move of equal magnitude and probability. We might, however, believe that the expected return to an underlying contract is not zero, that there is a greater chance that the contract will move one direction rather than another. Look at the price outcomes and probabilities in Figure 3-3. Using these new probabilities, the expected return from a long position in the underlying contract is:

-(10% x $20) - (20% x $10) +(30% x 0) +(25% x $10) +(15% × $20) = +$1.50

and the expected return for the $100 call is:

(10% x 0) +(20% x 0) +(30% × 0) +(25% × $10) + (15% × $20) = +$5.50

请注意，在图3-2中，所有结果和概率都是对称排列的。尽管新概率改变了100元看涨期权的预期收益，但在标的合约中进行的任何头寸的预期收益仍然为零。对于每次上涨的价格变动，都有一个相等大小和概率的下跌变动。然而，我们可能认为标的合约的预期收益不是零，而是有更大的机会朝一个方向移动而不是另一个方向。看看图3-3中的价格结果和概率。使用这些新概率，从标的合约的多头头寸的预期收益为：

−(10%×20)−(20%×10)+(30%×0)+(25%×10)+(15%×20)=+1.50

而100元看涨期权的预期收益为：

(10%×0)+(20%×0)+(30%×0)+(25%×10)+(15%×20)=+5.50

Note that the underlying contract now has a positive expected return, so it may seem that there is money to be made simply by purchasing the underlying contract. This would be true if there were no other considerations. But suppose the underlying contract is a stock, and therefore subject to stock-type settlement. If we purchase the stock at today's price of $100 and hold it for some period, there is a carrying cost associated with the investment. If the carrying cost is exactly equal to the expected return of $1.50, we will just break even. For a long stock position to be profitable, the stock must appreciate by at least the amount of carrying costs over the holding period. Therefore, the expected return from the stock must be some positive number, If we assume that any stock trade will just break even, the expected return must be equal to the carrying costs.

注意，标的合约现在有了正的预期收益，因此看起来似乎仅通过购买标的合约就能获利。如果没有其他考虑，这确实是正确的。但是，假设标的合约是一只股票，因此受股票型结算的影响。如果我们以今天的价格100元购买该股票并持有一段时间，则与投资相关的持有成本是存在的。如果持有成本恰好等于预期收益1.50元，我们将仅仅持平。为了使股票的多头头寸获利，股票的增值必须至少超过持有成本。因此，股票的预期收益必须是某个正数。如果我们假设任何股票交易仅持平，则预期收益必须等于持有成本。

Some stocks also pay dividends. If the dividend is paid during the holding period, it will affect the expected return. A trader who buys stock will have to pay out carrying costs, but he will receive the dividends. If we again assume that a stock trade will break even, the expected return at the end of the holding period must be identical to the carrying costs less the dividend. If the carrying cost for the stock over some period is $3.50, and a $1.00 dividend is expected during this period, the expected return at the end of the period must be $2.50. A trader who purchases the stock today will incur an interest debit of $3.50 at the end of the holding period, but this will be exactly offset by the $1.00 dividend which he receives during the holding period, (footnote 3: The trader can also earn interest on the dividend from the time he receives it until the end of the holding period. Since this will usually be a very small amount in relation to the other factors, we will ignore it.) as well as the $2.50 expected return at the end of the period.

一些股票还支付股息。如果在持有期间支付股息，将影响预期收益。购买股票的交易者需要支付持有成本，但他们将收到股息。如果我们再次假设股票交易将持平，则在持有期结束时的预期收益必须等于持有成本减去股息。如果在某段时间内股票的持有成本为3.50元，而在此期间预期支付1.00元的股息，则在该期间结束时的预期收益必须为2.50元。今天购买股票的交易者将在持有期结束时产生3.50元的利息支出，但这将正好被在持有期内收到的1.00元股息抵消，以及在期间结束时的2.50元预期收益。（脚注3：交易者还可以在收到股息后，直到持有期结束时，赚取股息的利息。由于这通常相对于其他因素是一个很小的数额，因此我们将忽略它。）

In an arbitrage-free market, where no profit can be made by either buying or selling a contract, all credits and debits, including the expected return, must exactly cancel out. If we assume an arbitrage-free market, we must necessarily assume that the forward price, the average price of the contract at the end of the holding period, is the current price, plus an expected return which will exactly offset all other credits and debits. If the holding costs on a $100 stock over some period are $4, the forward price must be $104. If the stock also pays a $1 dividend, the forward price must be $103. In both cases the credits and debits will exactly cancel out.

在无套利市场中，无法通过买卖合约获利，因此所有的收入和支出，包括预期收益，必须完全抵消。如果我们假设市场是无套利的，那么我们必须假设远期价格，即在持有期结束时合约的平均价格，等于当前价格，加上将完全抵消所有其他收入和支出的预期收益。如果某段时间内，100元股票的持有成本为4元，则远期价格必须为104元。如果该股票还支付1元的股息，则远期价格必须为103元。在这两种情况下，收入和支出将完全抵消。

The calculation of a forward price depends on the characteristics of the contract as well as market conditions. In the case of a stock, the considerations are the price of the stock, the length of the holding period, interest rates, and dividends. In the case of a futures contract, the situation is much simpler. A futures contract requires no initial cash outlay, since it is subject to futures-type settlement. Moreover, a futures contract does not pay dividends. This means that the forward price of a futures contract in an arbitrage-free market is simply the current price of the futures contract. If a trader buys a futures contract at $100, the break even price for the contract at the end of the holding period is $100.

远期价格的计算依赖于合约的特性以及市场状况。在股票的情况下，考虑的因素有股票价格、持有成本、股息、预期收益以及股票的波动率。在期货合约的情况下，情况要简单得多。期货合约不需要初始现金支出，因为它受期货型结算的影响。此外，期货合约不支付股息。这意味着，在无套利市场中，期货合约的远期价格仅是期货合约的当前价格。如果交易者以100元购买期货合约，那么该合约在持有期结束时的盈亏平衡价格就是100元。

Going back to our very simple pricing model, we might make the assumption that the underlying market is arbitrage-free (footnote 4: We need not necessarily assume an arbitrage-free underlying market. But we shall see that this is an important assumption in most theoretical pricing models.), that there is no money to be made from trading the underlying contract. The expected return must then be equal to the difference between the current price of the underlying market and its forward price. In the case of stock, the expected return will be carrying costs less dividends. In the case of futures, the expected return will be zero.

回到我们非常简单的定价模型，我们可以假设标的市场是无套利的（脚注4：我们不必假设标的市场完全无套利，但我们应该看到，在大多数理论定价模型中，这一假设是重要的。），即通过交易标的合约无法获利。那么，预期收益必须等于标的市场的当前价格与其远期价格之间的差额。对于股票而言，预期收益将是持有成本减去股息；而对于期货，预期收益则为零。

Even if we assume an arbitrage free market in the underlying, with appropriate probabilities associated with each price outcome, we still have one major problem. In our simplified model there were only five possible price outcomes, while in the real world there are an infinite number of possibilities. To enable our model to more closely approximate real world conditions we will have to construct a probability line with every possible price outcome and its associated probability. This may seem an impossible task, but it is the basis for all theoretical pricing models.

即使我们假设标的市场是无套利的，并且与每个价格结果相关的概率是合适的，我们仍然面临一个主要问题。在我们简化的模型中，只有五个可能的价格结果，而在现实世界中，存在无数的可能性。为了使我们的模型更接近现实条件，我们必须构建一个包含所有可能价格结果及其相关概率的概率线。这看起来似乎是一个不可能的任务，但它是所有理论定价模型的基础。

We can now summarize the necessary steps in developing a model:

1. Propose a series of possible prices at expiration for the underlying contract
2. Assign an appropriate probability to each possible price
3. Maintain an arbitrage-free underlying market
4. From the prices and probabilities in steps 1, 2, and 3, calculate the expected return for the option
5. From the option's expected return, deduct the carrying cost

If we can accomplish all this, we will finally have a theoretical value from which we can begin to trade.

我们现在可以总结出开发模型所需的步骤：

1. 提出标的合约到期时的一系列可能价格
2. 为每个可能的价格分配一个合适的概率
3. 维持无套利的标的市场
4. 根据步骤1、2和3中的价格和概率计算期权的预期收益
5. 从期权的预期收益中扣除持有成本

如果我们能够完成这些步骤，我们将最终得到一个理论价值，作为交易的起点。

Prior to 1973, evaluation of options required the solution of complex mathematical equations. Since such methods were slow and tedious, a trader who tried to use them quickly found that profit opportunities disappeared faster than the evaluation methods could identify them. In 1973, concurrent with the opening of the Chicago Board Options Exchange, Fischer Black and Myron Scholes introduced the first practical theoretical pricing model for options. The Black-Scholes Model, with its relatively simple arithmetic and limited number of inputs, most of which were easily observable, proved an ideal tool for traders in the newly opened U.S. option market. Although other models have since been introduced to overcome some of its original deficiencies, the Black-Scholes Model remains the most widely used of all option pricing models.

在1973年之前，期权的评估需要解决复杂的数学方程。由于这些方法既缓慢又繁琐，试图使用这些方法的交易者很快发现，获利机会消失的速度往往快于评估方法能够识别的速度。1973年，伴随着芝加哥期权交易所的开幕，费希尔·布莱克（Fischer Black）和麦伦·肖尔斯（Myron Scholes）提出了第一个实用的期权理论定价模型——布莱克-肖尔斯模型。该模型采用相对简单的算术运算和有限数量的输入，其中大部分输入易于观察，因此成为新开设的美国期权市场交易者的理想工具。尽管后来出现了其他模型以克服其一些原始缺陷，但布莱克-肖尔斯模型仍然是所有期权定价模型中使用最广泛的模型。

In its original form, the Black-Scholes Model was intended to evaluate European options (no early exercise permitted) on non dividend paying stocks. Shortly after its introduction, realizing that most stocks do pay dividends, Black and Scholes added a dividend component. In 1976, Fischer Black made slight modifications to the model to allow for the evaluation of options on futures contracts, And in 1983, Mark Garman and Steven Kohlhagen made several other modifications to allow for the evaluation of options on foreign currencies, (footnote 5: We are speaking here of options on a physical foreign currency, rather than options on a foreign currency futures contract. The latter may be evaluated using the Black Model for futures options.) The futures version and the foreign currency version are known officially as the Black Model and the Garman-Kohlhagen Model, respectively.

在其最初形式中，布莱克-肖尔斯模型旨在评估不支付股息的欧洲期权（不允许提前行权）。模型推出后不久，布莱克和肖尔斯意识到大多数股票确实支付股息，因此添加了股息因素。1976年，布莱克对模型进行了小幅修改，以便评估期货合约上的期权；1983年，马克·加曼（Mark Garman）和史蒂文·科尔哈根（Steven Kohlhagen）又进行了若干修改，以允许评估外汇期权（脚注5：这里所指的是实物外币的期权，而不是外币期货合约的期权。后者可以使用期货期权的布莱克模型进行评估。）期货版本和外汇版本分别被正式称为布莱克模型和加曼-科尔哈根模型。

But the evaluation method in each version, whether the original Black-Scholes Model for stock options, the Black Model for futures options, or the Garman-Kohlhagen Model for foreign currency options, is so similar that they have all come to be known as simply the Black-Scholes Model. The various forms of the model differ primarily in how they calculate the forward price of the underlying contract, and an option trader will simply choose the form appropriate to the underlying instrument.

在每个版本的评估方法中，无论是原始的布莱克-肖尔斯模型用于股票期权、布莱克模型用于期货期权，还是加曼-科尔哈根模型用于外汇期权，它们的评估方法都非常相似，因此它们都被统称为布莱克-肖尔斯模型。模型的不同主要在于它们如何计算标的合约的远期价格，期权交易者会选择适合标的工具的模型形式。

The great majority of options currently traded are American options, carrying with them the right of early exercise. For this reason, it may seem that the Black-Scholes model, with its assumption of no early exercise, is poorly suited for use in most markets. However, the Black-Scholes Model has proven so easy to use that many traders do not believe the more accurate values derived from an American option pricing model, which allows for the possibility of early exercise, is worth the additional effort. In some markets, particularly futures options markets, the additional early exercise value is so small that there is virtually no difference between values obtained from the Black-Scholes model and values obtained from an American pricing model.

目前交易的大多数期权都是美式期权，具有提前行使的权利。因此，布莱克-肖尔斯模型在没有提前行使假设的情况下，似乎不太适合大多数市场。然而，布莱克-肖尔斯模型的易用性使得许多交易者认为，从美式期权定价模型中得出的更准确的值并不值得额外的努力，尽管该模型允许提前行使。在某些市场，特别是期货期权市场中，额外的提前行使价值非常小，以至于从布莱克-肖尔斯模型和美式定价模型得到的值几乎没有差异。

Due to its widespread use and its importance in the development of other pricing models, we will for the moment restrict ourselves to a discussion of the Black-Scholes model and its various forms. In later chapters we will consider the question of early exercise. We will also look at alternative methods for pricing options when we question some of the basic assumptions in the Black-Scholes Model.

由于布莱克-肖尔斯模型的广泛应用及其在其他定价模型发展中的重要性，我们暂时将讨论限制在布莱克-肖尔斯模型及其各种形式。在后面的章节中，我们将考虑提前行权的问题。同时，当我们质疑布莱克-肖尔斯模型中的一些基本假设时，也会探讨期权定价的其他方法。

The reasoning which led to the development of the Black-Scholes Model depends on the five steps we listed earlier in this chapter when we proposed a simple method for evaluating options. Black and Scholes worked originally with call values, but put values can be derived in much the same way. Alternatively, we will see in Chapter 11 that in an arbitrage-free market there is a unique relationship between an underlying contract, and a call and put with the same exercise price and expiration date. This relationship enables us to derive a put value simply by knowing the associated call value.

布莱克-肖尔斯模型的推导依赖于我们在本章早些时候列出的五个步骤，这些步骤为期权评估提供了一种简单的方法。布莱克和肖尔斯最初是以看涨期权的价值为基础进行工作的，但看跌期权的价值也可以通过类似的方式得出。此外，在第11章中，我们将看到，在无套利市场中，标的合约、相同执行价格和到期日的看涨期权与看跌期权之间存在独特的关系。这一关系使我们能够仅通过了解相关的看涨期权价值，简单地推导出看跌期权的价值。

In order to calculate an option's theoretical value using the Black-Scholes Model, we need to know at a minimum five characteristics of the option and its underlying contract. These are:

1. The option's exercise price
2. The amount of time remaining to expiration
3. The current price of the underlying contract
4. The risk-free interest rate over the life of the option
5. The volatility of the underlying contract

为了使用布莱克-肖尔斯模型计算期权的理论价值，我们至少需要了解期权及其标的合约的五个特征。这些特征包括：

1. 期权的行使价格
2. 距到期的剩余时间
3. 标的合约的当前价格
4. 期权有效期内的无风险利率
5. 标的合约的波动率

The last input, volatility, may be unfamiliar to the new trader. While we will put off a detailed discussion of this input to the next chapter, from our previous discussion one can reasonably infer that volatility is related to the speed of the market.

最后一个输入——波动率，可能对新手交易者来说比较陌生。虽然我们将在下一章详细讨论这一输入量，但根据之前的讨论，可以合理推测波动率与市场的波动速度相关。

If we know each of the required inputs, we can feed them into the theoretical pricing model and thereby generate a theoretical value.

如果我们知道所有所需的输入，就可以将它们代入理论定价模型，从而生成理论价值。

Black and Scholes also incorporated into their model the concept of the riskless hedge. For every option position there is a theoretically equivalent position in the underlying contract such that, for small price changes in the underlying contract, the option position will gain or lose value at exactly the same rate as the underlying position. To take advantage of a theoretically mispriced option, it is necessary to establish a hedge by offsetting the option position with this theoretically equivalent underlying position. That is, whatever option position we take, we must take an opposing market position in the underlying contract. The correct proportion of underlying contracts needed to establish this riskless hedge is known as the hedge ratio.

布莱克和肖尔斯在他们的模型中引入了无风险对冲的概念。对于每一个期权头寸，理论上都存在一个等价的标的合约头寸，这样在标的合约价格发生小幅变动时，期权头寸的价值将以与标的头寸相同的速度变化。为了利用理论上定价错误的期权，必须通过用这个等价的标的头寸抵消期权头寸来建立对冲。也就是说，无论我们采取何种期权头寸，都必须在标的合约中采取相反的市场头寸。建立这一无风险对冲所需的标的合约比例被称为对冲比率。

Why is it necessary to establish a riskless hedge? Recall that in our simplified approach an option's theoretical value depended on the probability of various price outcomes for the underlying contract. As the underlying contract changes in price, the probability of each outcome will also change. If the underlying price is currently $100 and we assign a 25% probability to $120, we might drop the probability for $120 to 10% if the price of the underlying contract falls to $80. By initially establishing a riskless hedge, and then by adjusting this hedge as market conditions change, we are taking into consideration these changing probabilities.

为什么有必要建立无风险对冲？回想一下，在我们简化的方法中，期权的理论价值依赖于标的合约各种价格结果的概率。随着标的合约价格的变化，各种结果的概率也会随之改变。如果标的价格当前为100元，而我们将120元的概率赋值为25%，那么当标的合约价格降至80元时，我们可能会将120元的概率降低到10%。通过最初建立一个无风险对冲，并根据市场条件的变化调整这一对冲，我们就考虑了这些变化的概率。

In this sense an option can be thought of as a substitute for a similar position in the underlying contract. A call is a substitute for a long position; a put is a substitute for a short position. Whether it is better to take the position in the option or in the underlying contract depends on the theoretical value of the option and its price in the marketplace, If a call can be purchased (sold) for less (more) than its theoretical value, it will, in the long run, be more profitable to take a long (short) market position by purchasing (selling) calls than by purchasing (selling) the underlying contract. In the same way, if a put can be purchased (sold) for less (more) than its theoretical value, it will, in the long run, be more profitable to take a short (long) market position by purchasing (selling) puts than by selling (buying) the underlying contract.

从这个意义上说，期权可以被视为标的合约中类似头寸的替代品。看涨期权是多头头寸的替代品；看跌期权是空头头寸的替代品。选择在期权中还是在标的合约中建立头寸，取决于期权的理论价值及其在市场中的价格。如果看涨期权的购买价格低于其理论价值，或者其卖出价格高于理论价值，从长远来看，通过购买（出售）看涨期权而不是购买（出售）标的合约来采取多头（空头）市场头寸将更有利。同样，如果看跌期权的购买价格低于其理论价值，或者其卖出价格高于理论价值，从长远来看，通过购买（出售）看跌期权而不是卖出（购买）标的合约来采取空头（多头）市场头寸也将更有利。

Since the theoretical value obtained from a theoretical pricing model is no better than the inputs into the model, a few comments on each of the inputs will be worthwhile.

由于从理论定价模型获得的理论价值与模型的输入密切相关，因此对每一个输入进行一些评论是很有必要的。

EXERCISE PRICE
行权价

There ought never be any doubt about the exercise price of an option, since it is fixed in the terms of the contract and does not vary over the life of the contract, (footnote 6: It is true that an exchange may adjust the exercise price of a stock option if there is a stack split. In practical terms this is not really a change in the exercise price because the exercise price retains the same relationship to the stock price. The characteristics of the option contract remain essentially unchanged.) A Deutschemark March 58 call traded on the Chicago Mercantile Exchange cannot suddenly turn into March 59 call or a March 57 call. An IBM July 55 put traded on the Chicago Board Options Exchange cannot turn into a July 50 put or a July 60 put.

期权的行权价应当毫无疑问，因为它在合约条款中是固定的，并且在合约有效期内不会发生变化。（脚注6：确实，如果发生股票拆分，交易所可能会调整股票期权的行权价。但从实际角度来看，这并不构成行权价的真正变化，因为行权价与股票价格之间的关系保持不变，期权合约的特征本质上也保持不变。）在芝加哥商品交易所交易的德意志马克三月58号看涨期权不能突然变为三月59号看涨期权或三月57号看涨期权。在芝加哥期权交易所交易的IBM七月55号看跌期权也不能变为七月50号看跌期权或七月60号看跌期权。

TIME TO EXPIRATION
到期时间

Like the exercise price, the option's expiration date is fixed and will not vary. Our DM March 58 call will not suddenly turn into an April 58 call, nor will our IBM July 55 put turn into a June 55 put. Of course, each day that passes brings us closer to expiration, so in that sense the time to expiration is constantly growing shorter. However, the expiration date, like the exercise price, is fixed by the exchange and will not change.

与行使价格一样，期权的到期日期是固定的，不会发生变化。我们的DM三月58号看涨期权不会突然变成四月58号期权，而我们的IBM七月55号看跌期权也不会变成六月55号期权。当然，随着每一天的过去，我们距离到期日越来越近，从这个意义上说，到期时间在不断缩短。然而，到期日期和行使价格一样，是由交易所固定的，不会改变。

Time to expiration, like all inputs in the Black-Scholes Model, is entered as an annualized number. If we are entering raw data directly into the model we must make the appropriate annualization. With 91 days remaining to expiration, we would enter an input of ,25 (91/365=,25). With 36 days remaining, we would enter .10 (36/365=.10). However, most option evaluation computer programs already have this transformation incorporated into the software so that we need only enter the correct number of days remaining to expiration.

到期时间与布莱克-肖尔斯模型中的所有输入一样，以年化数字的形式输入。如果我们直接将原始数据输入模型，我们必须进行适当的年化处理。如果距离到期还有91天，我们将输入0.25（91/365=0.25）。如果还有36天，则输入 0.10（36/365=0.10）。然而，大多数期权评估计算机程序已经将这种转换纳入软件中，因此我们只需输入剩余的确切天数即可。

It may seem that we have a problem in deciding what number of days to enter into the model. We need the amount of time remaining to expiration for two purposes, to calculate the interest considerations and to calculate the likelihood of movement in the underlying contract. For volatility purposes in assessing the "speed" of the market we are only interested in trading days. Only on those days can the price of the underlying contract actually change. This might lead us to drop weekends and holidays from our calculations. On the other hand, for interest rate purposes we must include every day. If we borrow or lend money we expect the interest to accrue every day, no matter that some of the days are not business days.

看起来我们在决定输入模型的天数时可能会遇到问题。我们需要剩余到期时间的量有两个目的：计算利息和评估标的合约价格变动的可能性。出于波动率评估市场“速度”的目的，我们只关心交易日。只有在那些日子里，标的合约的价格才能实际发生变化。这可能导致我们在计算时忽略周末和节假日。另一方面，出于利率的考虑，我们必须包括每一天。如果我们借入或贷出资金，我们希望每天都能产生利息，无论某些日子是否为非营业日。

It turns out that this is not really a problem. In calculating the "speed" of the market we observe only the price changes that occur on business days. But we can make slight changes to this observed value and annualize the number before feeding it into the theoretical pricing model. The result is that we can feed into our model the actual number of days remaining to expiration knowing that the model will interpret the number correctly.

事实上，这并不是一个问题。在计算市场“速度”时，我们只观察营业日发生的价格变化。但我们可以对观察到的数值进行轻微调整，并在输入理论定价模型之前进行年化处理。结果是，我们可以输入剩余到期的实际天数，而模型将正确解读这个数字。

PRICE OF THE UNDERLYING
标的资产价格

Unlike the exercise price and time to expiration, the correct price of the underlying is not always obvious. At any one time there is usually a bid price and an asked price, and it may not be clear whether we ought to use one or the other of these prices, or perhaps some price in between.

与行使价格和到期时间不同，标的资产的正确价格并不总是显而易见的。在任何时刻，通常会存在买入价和卖出价，而我们可能不清楚应该使用其中的哪个价格，或者是否应该使用两个价格之间的某个价格。

We have noted that the correct use of an option's theoretical value requires us to hedge the option position with an opposing trade in the underlying contract. Therefore the underlying price we feed into our theoretical pricing model ought to be the price at which we believe we can make the opposing trade. If we intend to purchase calls or sell puts, both of which are long market positions, we will have to hedge by selling the underlying contract. In that case we ought to use the bid price since that is the price at which we can sell the underlying. On the other hand, if we intend to sell calls or buy puts, both of which are short market positions, we will have to hedge by purchasing the underlying contract. Now we ought to use the asked price since that is the price at which we can buy the underlying.

我们已提到，正确使用期权的理论价值需要通过在标的合约中进行对立交易来对冲期权头寸。因此，输入到理论定价模型中的标的价格应是我们认为可以进行对立交易的价格。如果我们打算购买看涨期权或卖出看跌期权，这两者都是多头市场头寸，那么我们将需要通过卖出标的合约来对冲。在这种情况下，我们应该使用买入价，因为这是我们能够出售标的资产的价格。相反，如果我们打算卖出看涨期权或购买看跌期权，这两者都是空头市场头寸，我们将需要通过购买标的合约来对冲。在这种情况下，我们应该使用卖出价，因为这是我们能够购买标的资产的价格。

In practice, the bid and offer are constantly changing, and many traders will simply use the last trade price as the basis for theoretical evaluation. But the last trade price may not always reflect the present market. Even the settlement price quoted in a newspaper may not accurately reflect the market at the close of business. The last trade price may show 75¼ for a contract, but the market at the close may have been 75¼ bid, 75/2 offered. A trader who hoped to buy at 75¼ would have very little chance of being filled because of the difficulty of buying at the bid price. Even a purchase at some middle price, say 75⅜, may be unlikely if the market is very unbalanced with many more contracts being bid for at 75¼ than offered at 75½. For all of these reasons, an experienced trader will rarely enter an option market without knowing the exact bid and offer in the underlying market.

在实际操作中，买入价和卖出价会不断变化，许多交易者通常会将最后交易价格作为理论评估的基础。然而，最后的交易价格并不总能反映当前市场情况。即使报纸上引用的结算价格也可能无法准确反映收盘时的市场。例如，最后交易价格可能显示一个合约为75¼，但在收盘时市场可能为75¼买入、75½卖出。希望以75¼买入的交易者几乎没有成交的机会，因为在买入价上买入的难度很大。即使以某个中间价格（例如75⅜）进行购买，如果市场严重失衡，即在75¼买入的合约远多于在75½卖出的合约，这种购买也可能不太可能。基于这些原因，有经验的交易者很少会在不了解标的市场的确切买入和卖出价格的情况下进入期权市场。

INTEREST RATES
利率

Since an option trade may result in either a cash credit or debit to a trader's account, the interest considerations resulting from this cash flow must also play a role in option evaluation. This is a function of interest rates over the life of the option.

由于期权交易可能导致交易者账户上出现现金信用或借记，因此由此产生的现金流所涉及的利息因素在期权评估中也必须发挥作用。这与期权有效期内的利率相关。

The interest rate component plays two roles in the theoretical evaluation of options. First, it may affect the forward price of the underlying contract. If the underlying contract is subject to stock-type settlement, as we raise interest rates we raise the forward price, increasing the value of calls and decreasing the value of puts. Secondly, the interest rate may affect the cost of carrying the option. If the option is subject to stock-type settlement, as we raise interest rates we decrease the value of the option. In spite of the fact that the interest rate plays two roles, in most cases the same rate is applicable and we need only input one interest rate into the model. If, however, different rates are applicable, such as would be the case with foreign currency options (the foreign currency interest rate plays one role, the domestic currency interest rate plays a different role) the model will require the input of two interest rates. This is the case with the Garman-Kohlhagen version of the Black-Scholes Model.

利率在期权的理论评估中发挥着两个作用。首先，它可能影响标的合约的远期价格。如果标的合约采用股票类型结算，随着利率的上升，远期价格也会随之提高，从而增加看涨期权的价值并降低看跌期权的价值。其次，利率可能影响期权的持有成本。如果期权采用股票类型结算，随着利率的上升，期权的价值将会降低。尽管利率发挥着两个作用，但在大多数情况下，适用相同的利率，因此我们只需在模型中输入一个利率。然而，如果适用不同的利率，例如在外汇期权的情况下（外币利率和国内货币利率分别发挥不同的作用），模型将需要输入两个利率。这种情况适用于Garman-Kohlhagen版本的黑-肖尔斯模型。

The fact that interest rates play a dual role also means that the relative importance of interest rates will vary, depending on the type of underlying instrument and the settlement procedure. For example, interest rates have a much greater impact on the value of stock options than on futures options. As we raise interest rates, we increase the forward price of stock, but leave the forward price of a futures contract unchanged. At the same time, assuming stock-type settlement, as we raise interest rates we decrease the value of options. The option price, however, is usually very small in relation to the price of the underlying contract.

利率的双重作用也意味着，利率的相对重要性会因标的工具和结算程序的类型而有所不同。例如，利率对股票期权的价值影响显著大于对期货期权的影响。随着利率的上升，我们提高了股票的远期价格，而期货合约的远期价格则保持不变。同时，假设采用股票类型结算，随着利率的上升，我们降低了期权的价值。然而，期权价格通常相对于标的合约的价格来说是非常小的。

What interest rate should a trader use when evaluating options? Most traders cannot borrow and lend at the same rate, so the correct interest rate will, in theory, depend on whether the trade will create a debit or a credit. In the former case the trader will be interested in the borrowing rate, while in the latter case he will be interested in the lending rate. In practice, however, the most common solution is to use the risk-free interest rate, i.e., the most secure rate. In the United States, the government is usually considered the most secure borrower of funds, so that the yield on a government security with a term equivalent to the life of the option is the general benchmark. For a 60-day option, use the yield on a 60-day treasury bill; for a 180-day option, use the yield on a 180-day treasury bill.

在评估期权时，交易者应使用什么利率？大多数交易者无法以相同的利率借入和借出资金，因此理论上，正确的利率将取决于交易是否会产生借记或信用。在借记的情况下，交易者将关注借款利率，而在信用的情况下，他将关注贷款利率。然而，在实践中，最常见的解决方案是使用无风险利率，即最安全的利率。在美国，政府通常被视为最安全的资金借款人，因此与期权有效期相等的政府证券收益率是一般基准。对于60天的期权，使用60天国库券的收益率；对于180天的期权，使用180天国库券的收益率。

DIVIDENDS
股息

We did not list dividends as an input in Figure 3-4 since they are only a factor in the theoretical evaluation of stock options, and then only if the stock is expected to pay a dividend over the life of the option.

我们没有在图3-4中列出股息，因为股息仅在理论评估股票期权时是一个因素，并且只有在期权有效期内，股票预计会支付股息时才适用。

In order to accurately evaluate a stock option, a trader must know both the amount of the dividend which the stock will pay and the ex-dividend date, the date on which a trader must own the stock in order to receive the dividend. The emphasis here is on ownership of the stock. A deeply in the money option may have many of the same characteristics as stock, but only ownership of the stock carries with it the right to collect the dividend.

为了准确评估股票期权，交易者必须了解股票将支付的股息金额以及除息日期，即交易者必须在该日期之前拥有股票才能获得股息。这里强调的是对股票的拥有权。一个深度实值的期权可能与股票有许多相似特性，但只有拥有股票才能享有收取股息的权利。

In the absence of other information, most traders tend to assume that a company will continue the same dividend policy it has had in the past. If the company has been paying a 75¢ dividend each quarter, it will probably continue to do so. However, this is not always a certainty. Companies sometimes increase or decrease dividends, and introduction to Theoretical Pricing Models occasionally omit them completely. If there is the possibility of a change in a company's dividend policy, a trader has to consider its impact on option values. Additionally, if the ex-dividend date is expected just prior to expiration, there is the danger that a delay of several days will cause the ex-dividend date to fall after expiration. For purposes of option evaluation, this is the same as eliminating the dividend completely. In such a situation a trader ought to make a special effort to ascertain the exact ex-dividend date.

在缺乏其他信息的情况下，大多数交易者倾向于假设公司将继续维持其过去的股息政策。如果公司每季度支付75美分的股息，那么它很可能会继续这样做。然而，这并不总是确定的。公司有时会增加或减少股息，理论定价模型的引入有时也会完全忽略股息。如果公司股息政策有可能发生变化，交易者需要考虑其对期权价值的影响。此外，如果预计除息日期恰好在期权到期之前，则可能存在几天的延迟，导致除息日期在到期后。对于期权评估而言，这相当于完全取消股息。在这种情况下，交易者应特别努力确认确切的除息日期。

VOLATILITY
波动率

Of all the inputs required for option evaluation, volatility is the most difficult for traders to understand. At the same time, volatility often plays the most important role in actual trading situations. Changes in our assumptions about volatility can have a dramatic effect on an option's value, and the manner in which the marketplace assesses volatility can have an equally dramatic effect on an option's price. For these reasons, we will devote the next chapter to a detailed discussion of volatility.

在期权评估所需的所有因素中，波动率是交易者最难以理解的，同时波动率在实际交易中往往起着最重要的作用。我们对波动率的假设变化可能会对期权的价值产生显著影响，而市场对波动率的评估方式同样会对期权价格产生显著影响。因此，我们将把下一章专门用于详细讨论波动率。