Chapter 5 Using an Option's Theoretical Value
第五章 使用期权的理论价值
From a theoretician's point of view, the Black-Scholes Model represents a novel solution to a complex problem. The model requires a limited number of inputs and relatively simple mathematical calculations. These factors have made the Black-Scholes Model the most popular of all option pricing methods.
从理论角度来看，布莱克-肖尔斯模型为解决复杂问题提供了创新的方法。它只需少量输入，且计算过程相对简单。这些特点使得布莱克-肖尔斯模型成为最广泛使用的期权定价工具。
While a trader might appreciate the elegance of the mathematics, it is still the actual performance of the model in the marketplace which is his primary consideration. Is it really possible to profit from price discrepancies between model-generated values and the market prices of options?
虽然交易员可能会欣赏其数学上的优雅，但他们更关注的是该模型在实际市场中的表现。通过模型生成的理论价值与市场期权价格之间的差异，是否真的能够实现盈利呢？
To see how the model is intended to function, let us make two assumptions:
- The price distribution of an underlying contract is accurately represented by a lognormal distribution.
- We actually know the future volatility of an underlying contract.
Clearly, the second assumption is not very realistic since we obviously don't know the future. As we shall see later, the first assumption is also highly questionable. But for the time being, rather than focus on the assumptions in the model, we want to focus on its use by a trader. If we make these assumptions, and the Black-Scholes Model does indeed work, then we ought to be able to turn any difference between an option's price and its theoretical value into a profit. How should we go about doing this?
为了理解模型的运作原理，我们可以做两个假设：
- 标的资产的价格分布完全符合对数正态分布。
- 我们已经准确知道标的资产的未来波动率。
显然，第二个假设并不现实，因为未来波动率是未知的。而且正如我们稍后会探讨的，第一个假设同样存在问题。不过，现在的重点不是这些假设本身，而是交易员如何运用该模型。如果接受这些假设，并假定布莱克-肖尔斯模型确实有效，那么我们应该能够利用期权市场价格和其理论价值之间的差异获利。那么，具体应该如何操作呢？
Suppose that a series of options are available on a certain futures contract with the following conditions:
June futures price = 101.35
Interest rate = 8.00%
Time to June expiration = 10 weeks
Suppose also that the options are subject to stock-type settlement, requiring full payment of the premium, as they currently are on U.S. markets.
假设某期货合约有一系列期权可供交易，条件如下：
6 月期货价格 = 101.35
利率 = 8.00%
距离 6 月到期时间 = 10 周
此外，假设这些期权采用股票式结算方式，需要全额支付期权溢价，类似于当前美国市场的做法。
Given that we can choose any exercise price we want, and that we can choose the type of option we want (call or put), the only input we still need for accurate evaluation of options on this futures contract is volatility. Since we have made the assumption that we actually know the future volatility of the underlying futures contract, let's imagine that we have a crystal ball which will tell us the future volatility. When we look into the crystal ball we see a volatility figure of 18.3% over the next 10 weeks. Now we have all the necessary inputs for theoretical evaluation, and it only remains to choose a specific option.
假设我们可以自由选择任何行权价和期权类型（看涨或看跌），那么唯一剩下需要输入的参数就是期货合约的波动率。既然我们假设已经知道标的期货合约的未来波动率，那不妨想象我们有一个水晶球，它能告诉了我们未来波动率。透过水晶球，我们发现接下来 10 周的波动率是 18.3%。现在所有理论评估的必要输入都已经具备，只剩下选择一个具体的期权了。
The June 100 call, being close to at-the-money, is likely to be actively traded, so let's focus on that option. Feeding our inputs into the Black Model (the futures variation of the Black-Scholes Model), we find that the June 100 call has a theoretical value of 3.88. When we check its price in the marketplace, we find that it is being offered at 3.25. How can we profit from this discrepancy?
由于 6 月行权价 100 的看涨期权接近平值，预计会有较活跃的交易，因此我们将重点关注这一期权。将我们的输入数据代入布莱克模型（即布莱克-肖尔斯模型的期货变体），我们发现 6 月行权价100 的看涨期权的理论价值为 3.88。当我们查看市场价格时，发现它的报价为 3.25。那么，我们该如何从这个价差中获利呢？
Clearly, our first move will be to purchase options since they are underpriced by .63. Can we now walk away from the position and come back at expiration to collect our money?
显然，我们的第一步是购买这些期权，因为它们的价格低估了 0.63。那么，我们可以在购买后就离开这个头寸，等到到期时再回来收钱吗？
We noted in Chapter 3 that the purchase or sale of a theoretically mispriced option requires us to establish a hedge by taking an opposing position in the underlying contract. When this is done correctly, for small changes in the price of the underlying, the increase (decrease) in the value of the option position will exactly offset the decrease (increase) in the value of the opposing position in the underlying contract. Such a hedge is unbiased, or neutral, as to the direction of the underlying contract.
我们在第三章中提到，购买或出售理论上定价错误的期权时，需要通过在标的合约中采取相反头寸来建立对冲。如果对冲操作正确，对于标的资产价格的微小变动，期权头寸价值的增加（减少）将恰好抵消在标的合约中相反头寸价值的减少（增加）。这种对冲在标的合约的方向上是中立的和没有偏向的。
最后一句是什么意思？这种对冲策略不受标的合约价格变化方向的影响，无论标的合约是上涨还是下跌，这种对冲方式都能有效地保护投资者的利益，使得因标的合约价格变动导致的损失和收益能够相互抵消。因此，投资者不需要预测市场的具体走势，只需关注对冲策略的有效性即可。所以这种策略是中立的，或者说，是没有偏向的。
The number which enables us to establish a neutral hedge under current market conditions is a by-product of the theoretical pricing model and is known as the hedge ratio or, more commonly, the delta. We will discuss the delta in greater detail in the next chapter, but certain characteristics will be important in our present example:
- The delta of a call option is always somewhere between 0 and 1.00.
- The delta of an option can change as market conditions change.
- An underlying contract always has a delta of 1.00.
使我们能够在当前市场条件下建立中性对冲的数字是理论定价模型的副产品，称为对冲比率，或更常见的名称——德尔塔。我们将在下一章中更详细地讨论德尔塔，但在我们当前的例子中，某些特征将很重要：
- 看涨期权的德尔塔始终介于 0 和 1.00 之间。
- 期权的德尔塔可以随着市场条件的变化而变化。
- 标的合约的德尔塔始终为 1.00。
Option traders commonly drop the decimal point when discussing deltas, a convention which we will also follow. Therefore, the delta of a call will fall in the range of 0 to 100, and an underlying contract will always have a delta of 100. (footnote 1: This convention originated in the U.S. stock option market where an underlying contract typically consisted of 100 shares of stock. It became common to equate one delta with each share of stock.)
期权交易者通常省略小数点来讨论德尔塔，这一惯例我们也将遵循。因此，看涨期权的德尔塔范围为 0 到 100，而标的合约的德尔塔始终为 100。（脚注 1：这一惯例起源于美国股票期权市场，通常一个标的合约包含 100 股股票，因此一个德尔塔通常被视为每股股票。）
Going back to our example, in order to make correct use of the option's theoretical value, we also need to know the delta, in this case 57 (.57). This means that for each option which we purchase we must sell 57% of an underlying contract to establish an unbiased or neutral hedge. Since the purchase or sale of fractional futures contracts is not permitted, we intend to purchase 100 June 100 calls and sell 57 June futures contracts. This maintains the proper neutral ratio but allows us to deal in whole numbers of contracts. We have established the following position:
| Contract | Contract Delta | Delta Position |
|---|---|---|
| Buy 100 June 100 calls | 57 | +5700 |
| Sell 57 June futures 100 | -100 | -5700 |
回到我们的例子，为了正确使用期权的理论价值，我们还需要知道德尔塔，此例中为 57（0.57）。这意味着对于每一个我们购买的期权，我们必须卖出 57% 的标的合约，以建立一个无偏或中性的对冲。由于不允许买卖部分期货合约，我们计划购买 100 份 6 月行权价 100 的看涨期权，并卖出 57 份 6 月期货合约。这保持了适当的中性比率，同时使我们能够以整数合约进行交易。我们已建立以下头寸：
| 合约 | 合约德尔塔 | 德尔塔头寸 |
|---|---|---|
| 购买 100 份 6 月 100 号看涨期权 | 57 | +5700 |
| 卖出 57 份 6 月期货合约 | -100 | -5700 |
Note that the delta position on each side of the hedge is the number of contracts multiplied by the contract delta, keeping in mind that the purchase of a contract is represented by a positive sign and the sale of a contract is represented by a negative sign. The option delta position is therefore +100 × 57 = +5700 and the futures position is -57 × 100 = -5700. When we add these two numbers together, +5700 and -5700, the total delta position is zero, and we say that the position is delta neutral. Within a small range, a delta neutral position has no particular preference for either upward or downward movement in the underlying market. If the total delta position is a positive number, this indicates an upward bias; if the total delta position is a negative number, this indicates a downward bias.
注意，对冲每一方的德尔塔头寸是合同数量与合同德尔塔的乘积。购买合同用正号表示，出售合同用负号表示。因此，期权的德尔塔头寸为 +100 × 57 = +5700，而期货头寸为 -57 × 100 = -5700。当我们将这两个数字相加，+5700 和 -5700，得到的总德尔塔头寸为零，这意味着该头寸是德尔塔中性。在小范围内，德尔塔中性头寸对标的市场的上涨或下跌没有特别偏好。如果总德尔塔头寸是正数，说明存在上涨偏见；如果总德尔塔头寸是负数，说明存在下跌偏见。
Having established a delta neutral hedge we must still deal with the fact that a theoretical value is based on probability. The roulette player who is able to purchase a roulette bet for less than its theoretical value can only expect to show a profit if he is allowed to play many, many times. On any one bet he will most likely lose, since there is only one way for him to win but 37 ways for him to lose. The same is true of our hedge. The odds may be on our side because we have purchased undervalued options. But in the short run our hedge might very well result in a loss. Is there anything we can do to offset the possibility of short-term bad luck?
建立德尔塔中性对冲后，我们仍需面对理论值基于概率的事实。这就像轮盘赌玩家，即使以低于理论值的价格下注，也只有在多次游戏后才能期望获利。毕竟，赢的方式只有一种，而输的方式却有 37 种。我们的对冲也是如此。由于我们购买了被低估的期权，因此胜算可能对我们有利。但短期内，极端情况可能导致损失。那么，如何应对短期不利情况呢？
We know that in the long run the laws of probability are on our side. We can be fairly certain of making a profit if we are given the chance to make many bets at the same favorable odds. The more bets we make, the better our chances of making a profit identical to that predicted by the theoretical pricing model. One way to accomplish this is to approach the initial hedge as a continuing series of small bets. We can replicate the long-term probability by reassessing our position at regular intervals, and then making appropriate changes in the position so that each new interval represents a new bet.
从长远来看，概率法则会发挥作用。如果我们能多次以相同的有利赔率交易，就能更确信获得理论定价模型预测的收益。将初始对冲视为一系列连续的小额交易，通过定期重新评估仓位并进行调整，我们就能模拟长期概率。每个新的交易区间都代表一次新的下注。
Suppose that one week later the price of the June futures contract has moved up to 102.26. At this point we can feed the new market conditions into our theoretical pricing model:
June futures price = 102.26
Interest rate = 8.00%
Time to June expiration = 9 weeks
Volatility = 18.3%
Note that we have made no change in the interest rate or volatility. The theoretical pricing model we are using, the Black model, assumes that these two inputs remain constant over the life of the option. Based on the new inputs we can calculate the new delta for the 100 call, in this case 62. The delta position is now:
| Contract | Contract Delta | Delta Position |
|---|---|---|
| Long 100 June 100 calls | 62 | +6200 |
| Short 57 June futures 100 | -100 | -5700 |
Our total delta position is now +500. This is the end of one bet, with another about to begin.
假设一周后，6 月期货合约的价格上涨至 102.26。我们将新的市场数据输入模型：
6 月期货价格 = 102.26
利率 = 8.00%
6 月到期时间 = 9 周
波动率 = 18.3%
请注意，我们没有改变利率或波动率。我们使用的理论定价模型，即布莱克模型，假设这两个输入在期权的生命周期内保持不变。基于新的输入，我们计算出 100 看涨期权的新的德尔塔为 62。德尔塔敞口明细如下：
| 合约 | 合约德尔塔 | 德尔塔头寸 |
|---|---|---|
| 多头 100 份 6 月 100 看涨期权 | 62 | +6200 |
| 空头 57 份 6 月期货 100 | -100 | -5700 |
我们的总德尔塔头寸现在是 +500。这是一次交易的结束，另一次交易即将开始。
Whenever we begin a new bet we are required to return to a delta neutral position. In our example it will be necessary to reduce our position by 500 deltas. As we shall see later, there are a number of different ways to do this. But to keep our present calculations as simple as possible we will make the necessary trades in the underlying futures market, since an underlying contract always has a delta of 100.
Here we wish to sell off 500 deltas, and we can do this by selling 5 futures contracts. Our delta position is now:
| Contract | Contract Delta | Delta Position |
|---|---|---|
| Long 100 June 100 calls | 62 | +6200 |
| Short 62 June futures 100 | -100 | -6200 |
We are again delta neutral and about to begin a new bet. As before, our new bet depends only on the volatility of the underlying futures contract, and not on its direction.
每次我们开始新的交易时，都需要调整至德尔塔中性。在我们的例子中，需要减少 500 个德尔塔的头寸。我们稍后会看到，有多种方法可以做到这一点。但为了使目前的计算尽可能简单，我们将对标的期货市场进行必要的交易，因为标的合约的德尔塔始终为 100。
在本例中，我们希望卖出 500 个德尔塔来对冲，可以通过卖出 5 份期货合约来实现。调整后的德尔塔头寸如下：
| 合约 | 合约德尔塔 | 德尔塔头寸 |
|---|---|---|
| 多头 100 份 6 月 100 看涨期权 | 62 | +6200 |
| 空头 62 份 6 月期货 100 | -100 | -6200 |
我们再次处于德尔塔中性状态，并即将开始新的交易。和以前一样，我们的新交易仅取决于标的期货合约的波动率，而不取决于其方向。
The extra five futures contracts we sold were an adjustment to our position. Adjustments are trades which are not necessarily made for the purpose of adding to the theoretical edge, although they may also have that effect. Rather, they are made primarily to ensure that a position remains delta neutral. In our case, the sale of the five extra futures contracts has no effect on our theoretical edge since futures contracts have no theoretical value. (footnote 2: While a futures trader might talk about the theoretical value of a futures contract, from an option trader's point of view the underlying contract has no theoretical value. The theoretical value is whatever price the trader thinks he can trade the contract at.) The trade was made solely for the purpose of adjusting our hedge to remain delta neutral.
我们额外卖出的五份期货合约是对我们仓位的调整。调整交易并不一定是为了增加理论优势，尽管它们也可能产生这种效果。相反，它们主要用于确保仓位保持德尔塔中性。在我们的例子中，额外卖出五份期货合约对我们的理论优势没有影响，因为期货合约没有理论价值。（脚注 2：虽然期货交易员可能会谈论期货合约的理论价值，但从期权交易员的角度来看，标的合约没有理论价值。理论价值是交易员认为自己可以交易合约的价格。）该交易纯粹是为了调整我们的对冲以保持德尔塔中性。
The steps we have thus far taken illustrate the correct procedure in using an option's theoretical value:
- Purchase (sell) undervalued (overvalued) options.
- Establish a delta neutral hedge against the underlying contract.
- Adjust the hedge at regular intervals to remain delta neutral.
到目前为止，我们采取的步骤说明了使用期权理论价值的正确程序：
- 购买低估或卖出高估的期权
- 建立针对标的合约的德尔塔中性对冲
- 定期调整对冲以保持德尔塔中性
Since volatility is assumed to compound continuously, theoretical pricing models assume that adjustments are also made continuously, and that the hedge is being adjusted at every moment in time. Such continuous adjustments are not possible in the real world, since a trader can only trade at discrete intervals. By making adjustments at regular intervals, we are conforming as best we can to the principles of the theoretical pricing model.
由于波动率假设是持续累加的，理论定价模型也假设对冲会被持续调整，即在每一刻都进行调整。然而，在现实世界中，交易者只能在离散的时间间隔内进行交易，因此无法实现连续调整。通过在固定的时间间隔进行调整，我们尽可能地遵循理论定价模型的原则。
Using the proper procedure, how would the adjustment process look if we were to carry the hedge to expiration? The results are shown in Figure 5-1. In our example adjustments were made at weekly intervals. At the end of each interval the delta of the June 100 call was recalculated from the time remaining to expiration, the current price of the underlying futures contract, the fixed interest rate of 8.00%, and the known volatility of 18.3%. Note that we have not changed the volatility, even though other market conditions may have changed. Volatility, like interest rates, are assumed to be constant over the life of the option. In practice, a trader may, and often does, change his opinion about volatility.
如果按照正确的操作流程将对冲持续到期权到期，调整过程会是什么样的呢？结果如图 5-1 所示。在这个例子中，调整是按周进行的。每次调整时，会根据剩余的到期时间、期货合约的当前价格、固定的 8.00% 利率，以及已知的 18.3% 波动率重新计算 6 月 100 看涨期权的德尔塔。需要注意的是，尽管其他市场条件可能发生变化，波动率并未改变。波动率和利率一样，假定在期权存续期间保持不变。不过，交易员在实际操作中往往会调整对波动率的判断。
Figure 5-1
| Week | Futures Price | Delta of 100 Call | Total Delta Position | Adjustment In Futures | Total Futures Adjustment | Variation | Interest on Variation |
|---|---|---|---|---|---|---|---|
| 0 | 101.35 | 57 | 0 | 0 | 0 | 0 | 0 |
| 1 | 102.26 | 62 | +500 | sell 5 | -5 | -51.87 | -.72 |
| 2 | 99.07 | 46 | -1,600 | buy 16 | +11 | +197.78 | +2.43 |
| 3 | 100.39 | 53 | +700 | sell 7 | +4 | -60.72 | -.65 |
| 4 | 100.76 | 56 | +300 | sell 3 | +1 | -19.61 | -.18 |
| 5 | 103.59 | 74 | +1,800 | sell 18 | -17 | -158.48 | -1.22 |
| 6 | 99.26 | 45 | -2,900 | buy 29 | +12 | +320.42 | +1.97 |
| 7 | 98.28 | 35 | -1,000 | buy 10 | +22 | +44.10 | +.20 |
| 8 | 99.98 | 50 | +1,500 | sell 15 | +7 | -59.50 | -.18 |
| 9 | 103.78 | 93 | +4,300 | sell 43 | -36 | -190.00 | -.29 |
| 10 | 102.54 | buy 36 |
What will we do with our position at the end of ten weeks when the options expire? At that time we plan to close out the position by:
- Letting any out-of-the-money options expire worthless.
- Selling any in-the-money options at parity (intrinsic value) or, equivalently, exercising them and offsetting them against the underlying futures contract.
- Liquidating any outstanding futures contracts at the market price.
十周后，当期权到期时，我们将如何处理持仓？届时，我们计划通过以下方式平仓：
- 让虚值期权（未盈利的期权）作废，失去价值。
- 将实值期权（有内在价值的期权）按平价（内在价值）出售，或者等效地行使期权并对冲相应的期货合约。
- 按市场价格平掉任何未结清的期货合约。
Let's go through this procedure step by step and see what the complete results of our hedge were.
让我们一步一步地检查这个对冲操作，看看最终的结果是什么。
Original Hedge: At June expiration (week 10) we can liquidate our June 100 calls by either selling them at 2.54, or by selling futures at 102.54 and exercising the calls. Either method will result in a credit of 2.54 to our account. Since we originally paid 3.25 for each option, we have effectively lost .71 per option, and our total loss on the options is 100 × -.71 = -71.00.
原始对冲部分：在六月到期（第 10 周）时，我们可以通过以 2.54 的价格出售六月 100 的看涨期权，或以 102.54 的价格出售期货并行使期权来清算这些看涨期权。这两种方式都会使我们的账户增加 2.54。由于我们最初为每个期权支付了 3.25，因此每个期权实际上损失了 0.71，总损失为 100 × -0.71 = -71.00。
As part of our original hedge we also sold 57 June futures at 101.35. At expiration we had to buy them back at 102.54, for a loss of 1.19 per contract. Our total loss on the futures is therefore 57 x -1.19 = -67.83. Adding this to our option loss, the total loss on the original hedge is -71.00 -67.83 = -138.83. This certainly does not appear to have been successful. We expected to make money on the hedge, and yet we have a sizable loss.
作为我们原始对冲的一部分，我们还在 101.35 的价格出售了 57 张六月期货。在到期时，我们不得不以 102.54 的价格买回，导致每份合约损失 1.19。因此，期货的总损失为 57 × -1.19 = -67.83。将此加上期权损失，总损失为-71.00 - 67.83 = -138.83。这显然没有取得成功。我们原本希望通过对冲盈利，但结果却出现了相当大的损失。
Adjustments: The original hedge was not our only transaction. In order to remain delta neutral, we were forced to buy and sell futures contracts over the ten week life of the option. At the end of week one we were 500 deltas long, so we had to sell 5 futures at 102.26. At the end of week two we were 1600 deltas short, so we had to buy 16 futures at 99.07; and so on each week until the end of week ten. At expiration we were short an extra 36 futures contracts, and we bought these in at the closing price of 102.54 Note that each time the futures price rose our delta position became positive, so that we were forced to sell futures; and each time the futures price fell our delta position became negative, so that we were forced to buy futures. Because our adjustments depended on the delta position, we were forced to do what every trader wants to do: buy low and sell high.
调整部分：最初的对冲并不是我们唯一的交易。为了保持 德尔塔 中性，我们在期权的十周生命周期内不得不不断买卖期货合约。在第一周结束时，我们的德尔塔为 500，需卖出 5 手期货，成交价为 102.26。在第二周结束时，我们的德尔塔为-1600，需买入 16 手期货，成交价为 99.07；依此类推，直到第十周结束。在到期时，我们短期合约多了 36 手，最后以 102.54 的收盘价买回。每次期货价格上涨时，我们的德尔塔位置变为正，因此不得不卖出期货；而每次期货价格下跌时，德尔塔位置变为负，我们则不得不买入期货。由于调整依赖于德尔塔头寸，我们被迫做出每个交易者都想做的事情：低买高卖。
What was the result of all the adjustments required to maintain a delta neutral position? In fact the result was a profit of 205.27. (The reader may wish to confirm this by adding up the cash flow from all the trades in the adjustment column in Figure 5-1.) This profit more than offset the losses incurred from the original hedge.
保持德尔塔中性所需的所有调整结果如何？实际上，结果是盈利 205.27。（读者可以通过加总图 5-1 中调整列的所有现金流来验证。）这一利润已超过最初对冲所带来的损失。
Carrying Costs: What else will effect our final profit or loss? Originally we bought calls and sold futures. While futures are subject to futures-type settlement and require no initial cash outlay, the options are subject to stock-type settlement and require full payment. We bought 100 calls for 3.25 each, for a total outlay of 325.00. Based on our assumed 8.00% interest rate, the carrying cost on 325.00 for 10 weeks is .08 x 70/365 × 325.00 = 4.99. We will have to include this debit of 4.99 in our final calculations.
持有成本部分：还有哪些因素会影响我们的最终利润或损失？最初，我们买入了期权并卖出了期货。期货交易不需要初始现金支出，而期权则需要全额支付。我们以每份 3.25 的价格买入 100 份期权，总支出为 325.00。根据假设的 8.00% 利率，325.00 在 10 周的持有成本为 0.08 × 70/365 × 325.00 = 4.99。我们必须将这一支出 4.99 纳入最终计算中。
Variation Costs: Finally, we must take into consideration the variation costs required to maintain our futures position. As futures move up or down in price, cash is either credited to, or debited from, a trader's account. In theory, a trader can earn interest on a cash credit, but must pay interest on a cash debit. For example, we initially sold 57 futures contracts at 101.35. One week later the futures price rose to 102.26, so our account was debited 57 × (101.35 - 102.26) = -51.87. Financing this debit for nine weeks at 8.00% cost us -51.87 x .08 × 63/365 = -.72. In order to remain delta neutral at the end of week one we sold an additional five futures contracts, for a new total of short 62 futures. One week later (week 2) the futures price fell to 99.07, so our account was credited 62 × (102.26 - 99.07) = +197.78. The amount of interest we were able to earn on this credit for 8 weeks at 8.00% was +197.78 x .08 × 56/365 = +2.43. The cash flow resulting from futures variation is shown in Figure 5-1 under the "variation" column, and the resulting interest is shown under the "interest on variation" column.The total interest is +1.36.
变动成本部分：最后，我们还需考虑维持期货头寸所需的变动成本。期货价格上涨或下跌时，现金会分别增加或减少到交易者的账户中。理论上，交易者可以对现金余额赚取利息，但需要为现金借款支付利息。例如，我们最初以 101.35 的价格卖出了 57 份期货合约。一周后，期货价格上涨至 102.26，因此我们的账户被扣除 57 × (101.35 - 102.26) = -51.87。在 8.00% 的利率下，维持这一扣款 9 周的费用为 -51.87 × 0.08 × 63/365 = -0.72。为了在第一周结束时保持德尔塔中性，我们又卖出了 5 份期货合约，使空头总数达到 62 份。一周后（第 2 周），期货价格降至 99.07，因此我们的账户增加了 62 × (102.26 - 99.07) = +197.78。在 8.00% 的利率下，我们能对这一信用在 8 周内赚取的利息为 +197.78 × 0.08 × 56/365 = +2.43。由期货变动产生的现金流显示在图 5-1 的 “变动” 列中，所产生的利息则显示在 “变动利息” 列中。总利息为 +1.36。
We can now summarize all the profit and loss components resulting from our position:
| Original Hedge | Adjustments | Option Carrying Costs | Variation Costs |
|---|---|---|---|
| -138.83 | +205.27 | -4.99 | +1.36 |
The total profit is -138.83 +205.27 -4.99 +1.36 = +62.81. How much did the theoretical pricing model predict we would make? We bought 100 calls worth 3.88 for 3.25 each, for a total theoretical edge of 100 × (3.88 - 3.25) = 100 x .63 = +63.00. In other words, the theoretical model came very close to predicting the actual profit from the position.
现在我们可以总结一下我们持仓产生的所有损益成分：
| 初始对冲 | 调整 | 期权持有成本 | 变动成本 |
|---|---|---|---|
| -138.83 | +205.27 | -4.99 | +1.36 |
总利润为 -138.83 +205.27 -4.99 +1.36 = +62.81。理论定价模型预测我们能赚多少钱？我们以 3.25 的价格购买了 100 个看涨期权，每个期权价值 3.88，总理论优势为 100 × (3.88 - 3.25) = 100 × 0.63 = +63.00。换句话说，理论模型非常接近于预测的实际的仓位利润。
In our example, the profit and loss was made up of four components. Two of these were profitable (the adjustments and the variation costs) while two were unprofitable (the original hedge and the option carrying costs). Is this always the case? It is impossible to determine beforehand which components will be profitable and which will not. One could just as easily construct an example where the original hedge was profitable and the adjustments were not. The important point here is that if a trader's inputs are correct, in some combination he can expect to show a profit (or loss) approximately equal to that predicted by the theoretical pricing model. Of all the inputs, volatility is the only one which is not directly observable. Where did our volatility figure of 18.3% come from? Obviously, one can't know the future volatility, but in this case the author took the ten price changes in Figure 5-1 and calculated the annualized standard deviation of the logarithmic changes (the volatility). Hence, the volatility of 18.3% was the correct volatility for this series of price changes. The complete calculations are given in Appendix B.
在我们的例子中，损益由四个部分组成。其中两个是盈利的（调整和变动成本），而另外两个则是亏损的（原始对冲和期权持有成本）。这种情况总是如此吗？我们无法事先确定哪些部分会盈利，哪些不会。也可以构造一个例子，使原始对冲盈利，而调整部分不盈利。重要的是，如果交易者的输入是正确的，在某种组合下，他们可以期望实现与理论定价模型预测相当的盈利（或亏损）。在所有输入中，波动率是唯一无法直接观察的参数。那么，我们的 18.3% 的波动率数据来源于哪里呢？显然，未来的波动率是无法预测的，但在这个案例中，作者利用了图 5-1 中的十个价格变动，计算了对数变动的年化标准差（即波动率）。因此，18.3% 的波动率是这组价格变动的正确波动率。完整的计算过程见附录 B。
In the foregoing example we assumed that the market was frictionless, that no external factors affected the total profit or loss. This assumption is basic to many economic models, including the Black-Scholes Model. In a frictionless market we assume that:
- Traders can freely buy or sell the underlying contract without restriction
- All traders can borrow and lend money at the same rate.
- Transaction costs are zero.
- There are no tax considerations.
在上述例子中，我们假设市场是无摩擦的，外部因素不会影响总的盈亏。这一假设是许多经济模型的基础，包括布莱克-肖尔斯模型。在无摩擦市场中，我们假设：
- 交易者可以自由买卖标的合约，不受限制。
- 所有交易者以相同的利率借款和放贷。
- 交易成本为零。
- 不考虑税务因素。
A trader will immediately realize that option markets are not frictionless, since each of the above assumptions is violated to a greater or lesser degree in the real world. For example, in certain futures markets there is a daily limit on the amount of allowable price movement. When this limit is reached, the market is said to be locked, and no further trading can take place until the price of the futures contract comes off its limit. (footnote 3: It is sometimes possible to circumvent the problem of a locked market by either buying or selling the physical commodity rather than the futures contract, or by trading spreads between futures months if one of the months is not locked.) Clearly, in such a market the underlying contract cannot always be freely bought or sold.
交易者会立即意识到，期权市场并非无摩擦，因为上述假设在现实世界中不同程度地被违反。例如，在某些期货市场，价格波动有每日限制。当达到这个限制时，市场会被锁定，直到期货合约的价格回落，才能进行进一步交易。（脚注 3：有时可以通过买卖实物商品而不是期货合约，或者在某个月未锁定的情况下进行期货月间的套利来规避锁定市场的问题。）显然，在这样的市场中，标的合约并不能总是自由买卖。
Additionally, individual traders cannot generally borrow or lend money at the same rate as large financial institutions. If a trader has a debit balance, it will cost him more to carry that debit; if he has a credit balance, he will not earn as much on that credit. There is a spread, and perhaps a fairly large one, between a trader's borrowing and lending rate. Fortunately, as we discussed in Chapter 3, the interest rate component is usually the least important of the inputs into a theoretical pricing model. Even though the applicable interest rate may vary from trader to trader, in general it will only cause small changes in the total profit or loss in relation to the profit or loss resulting from other inputs.
此外，个人交易者通常无法以与大型金融机构相同的利率借贷。如果交易者有借款余额，持有该余额的成本会更高；如果有存款余额，获得的利息也会更低。交易者的借款和贷款利率之间存在一定的差距，且可能相当大。幸运的是，正如我们在第三章中讨论的，利率因素通常是理论定价模型中影响最小的输入。尽管适用的利率因交易者而异，但一般来说，它对总盈亏的影响相对于其他因素造成的盈亏变化很小。
Transaction costs, on the other hand, can be a very real consideration. If such costs are high, the hedge in Figure 5-1 might not be a viable strategy; all the profits could be eaten up by brokerage fees. In theory, the desirability of a strategy will depend not only on the trader's initial transaction costs, but also on the subsequent costs of making adjustments. The adjustment cost is a function of a trader's desire to remain delta neutral. A trader who wants to remain delta neutral at every moment will have to adjust more often, and more adjustments mean more transaction costs.
另一方面，交易成本是一个非常重要的因素。如果成本过高，图 5-1 中的对冲策略可能就不可行，因为所有的利润可能会被佣金费用吞噬。理论上，一个策略的可行性不仅取决于交易者的初始交易成本，还取决于后续调整的成本。调整成本与交易者保持德尔塔中性（delta neutral）的意愿有关。希望随时保持德尔塔中性的交易者需要更频繁地进行调整，而更多的调整意味着更高的交易成本。
Suppose a trader initiates a hedge but adjusts less frequently, or does not adjust at all. How will this affect the outcome? Since theoretical evaluation of options is based on the laws of probability, a trader who initiates a theoretically profitable hedge still has the odds on his side. Although he may lose on any one individual hedge, if given a chance to initiate the same hedge repeatedly at a positive theoretical edge, on average he should profit by the amount predicted by the theoretical pricing model, assuming, of course, that his inputs are correct. The adjustment process is simply a way of smoothing out the winning and losing hedges by forcing the trader to make more bets, always at the same favorable odds. A trader's disinclination to adjust simply means that there is greater risk of not realizing a profit on any one hedge. Adjustments do not in themselves alter the expected return; they simply reduce the possibility of short-term bad luck.
假设一位交易者发起对冲，但调整的频率较低，甚至不进行调整。这将如何影响结果？由于期权的理论评估基于概率法则，即使交易者发起了一个理论上有利可图的对冲，他的胜算仍然在一边。尽管他可能在某一次对冲中亏损，但如果有机会在正向理论边际下反复发起同样的对冲，假设他的输入是正确的，平均而言，他应该能按理论定价模型预测的金额获利。调整过程只是通过迫使交易者更多地下注、保持相同的有利赔率来平滑赢利和亏损的对冲。交易者不愿意调整只意味着在某一次对冲中获利的风险更大。调整本身并不会改变预期收益，而是降低了短期不利的可能性。
Based on the foregoing discussion, a retail customer and a professional trader are likely to approach option trading in a slightly different manner, even though both understand and use the values generated by a theoretical pricing model. A professional trader, particularly if he is an exchange member, has relatively low transaction costs. Since adjustments cost him practically nothing in relation to the expected theoretical profit from a hedge, he is willing to make frequent adjustments. However, a retail customer who establishes the same hedge will probably not adjust, or will adjust less frequently, because any adjustments are likely to significantly reduce the profitability of a hedge. But if he understands the laws of probability, the retail customer will realize that his position has the same favorable odds as the professional trader's position. At the same time he should realize that his position is more sensitive to short-term bad luck. Even though the retail customer may occasionally experience larger losses than the professional trader, he will also occasionally experience larger profits. In the long run, on average, both should end up with approximately the same profit. (footnote 4: This of course ignores the very real advantage the professional trader often has from being able to buy at the bid price and sell at the ask price. A retail customer can never hope to match the profit resulting from this advantage, nor should he try to do so.)
根据前面的讨论，散户客户和专业交易者在期权交易上的方法可能略有不同，尽管两者都理解并使用理论定价模型生成的数值。专业交易者，尤其是交易所会员，通常交易成本较低。由于调整几乎不影响他从对冲中预期的理论利润，他更愿意频繁进行调整。然而，散户客户如果建立了相同的对冲，可能不会调整，或者调整频率较低，因为调整会显著降低对冲的盈利性。但如果他理解概率法则，就会意识到他的头寸与专业交易者的头寸有相同的有利赔率。同时，他也应意识到自己的头寸对短期不利因素更敏感。尽管散户客户有时可能遭遇比专业交易者更大的损失，但他也有机会获得更大的收益。从长期来看，二者的平均利润大致相同。（脚注 4：当然，这忽略了专业交易者能够以买入价买入并以卖出价卖出的实质性优势。散户客户无法指望达到这种优势带来的利润，也不应该尝试。）
Taxes may also be a factor in evaluating an option strategy. When positions are initiated, when they are liquidated, how the positions overlap, and the relationship between different instruments (options, stock, futures, physical commodities, etc.) may have different tax consequences. Such consequences can often have a significant impact on the value of a diversified portfolio, and for this reason portfolio managers must be sensitive to the tax ramifications of a strategy. Since each trader has unique tax considerations, and this book is intended as a general guide to option evaluation and strategies, we will simply assume that each trader wishes to maximize his theoretical pre-tax profits and that he will worry about taxes afterward.
税收也可能影响期权策略的评估。头寸的建立、平仓、重叠情况以及不同工具（期权、股票、期货、实物商品等）之间的关系可能会产生不同的税务后果。这些后果往往对多元化投资组合的价值产生重大影响，因此投资组合经理必须关注策略的税务影响。由于每位交易者的税务考虑各不相同，本书作为期权评估和策略的一般指南，将假设每位交易者希望最大化理论税前利润，并在之后再考虑税务问题。
Returning to our example in Figure 5-1, note that after the hedge was initiated no subsequent trades were made in the option market. The trader's only concern was the volatility, or price fluctuations, in the underlying market. These price fluctuations determined the size and the frequency of the adjustments, and in the final analysis it was the adjustments which determined the profitability of the hedge. We might think of the hedge as a race between the loss in time value of the June 100 calls and the cash flow resulting from the adjustments, with the theoretical pricing model acting as the judge. The model says that if options are purchased at less than theoretical value, the adjustments will win the race; if options are purchased at more than theoretical value, the loss in time value will win the race. The conditions of the race are determined by the inputs into the theoretical pricing model.
回到图 5-1 的例子，注意到对冲建立后，期权市场没有进行后续交易。交易者唯一关心的是标的市场的波动率或价格波动。这些波动决定了调整的幅度和频率，最终调整的结果决定了对冲的盈利性。我们可以将对冲视为六月行权价 100 的看涨期权的时间价值损失与调整带来的现金流之间的竞赛，理论定价模型充当裁判。模型指出，如果以低于理论价值的价格购买期权，调整将胜出；反之，如果以高于理论价值的价格购买期权，时间价值损失将胜出。这场比赛的条件由理论定价模型的输入决定。
While we assumed in our example that the future volatility was known to be 18.3%, we might ask what the outcome would have been had volatility been other than 18.3%? Suppose, for example, it turned out to be higher than 18.3%. Higher volatility means greater price fluctuations, resulting in more and larger adjustments. In our example, more adjustments mean more profit. This is consistent with the principle that options increase in value as volatility increases.
在我们的例子中，我们假设未来的波动率已知为 18.3%。如果波动率高于 18.3% 会怎样？假设波动率确实高于 18.3%，这意味着价格波动更大，从而导致更多和更大的调整。在我们的例子中，更多的调整意味着更多的利润。这与期权随着波动率增加而增值的原则一致。
What about the reverse, if volatility had been less than 18.3%? Lower volatility means smaller price fluctuations, resulting in fewer and smaller adjustments. This would have reduced the profit. If the volatility were low enough, the adjustment profit would just be enough to offset the other components, so that the total profit from the hedge would be exactly zero. This break-even volatility is identical to the option's implied volatility at the trade price. Using the Black model, we find that the implied volatility of the June 100 call at a price of 3.25 is 14.6%. At that volatility the race between profits from the adjustments and the loss in the option's time value will end in an exact tie. Above a volatility of 14.6% we expect the hedge, including adjustments, to show a profit; below 14.6% we expect the hedge to show a loss.
那么，如果波动率低于 18.3% 呢？较低的波动率意味着较小的价格波动，从而导致较少且较小的调整。这将减少利润。如果波动率足够低，调整带来的利润将恰好抵消其他组件的损失，使得对冲的总利润正好为零。这一平衡波动率与期权在交易价格下的隐含波动率相同。根据布莱克模型，六月份行权价 100 的看涨期权在价格为 3.25 时的隐含波动率为 14.6%。在这一波动率下，调整带来的利润与期权时间价值的损失将恰好相等。波动率超过 14.6% 时，我们预计对冲（包括调整）将显示盈利；低于 14.6% 时，则预计将出现亏损。
Since we needed to make adjustments in order to realize a profit, it may seem that every profitable hedge requires us to maintain the position until expiration. In practice, however, this may not be necessary. Suppose, for example, that immediately after we established the hedge the implied volatility in the option market began to increase. Suppose that it increased from 14.6%, the implied volatility when we bought the June 100 calls, to 18.3%, the future volatility over the life of the option. What would happen to the price of the June 100 call? Its price would rise from 3.25 (an implied volatility of 14.6% to 3.88 (an implied volatility of 18.3%). We could then sell our calls for an immediate profit of .63 per option. Of course, if we wanted to close out the hedge we would also have to buy back the 57 June futures contracts which we originally sold. What effect would the change in implied volatility have on the price of the futures contracts? Implied volatility is a characteristic associated with options, not with underlying contracts. For this reason we would expect to see the underlying futures contract continue to trade at its original price of 101.35. By purchasing our 57 outstanding futures contracts at 101.35 we would realize an immediate total profit from the hedge of 63.00, exactly the amount predicted by the theoretical pricing model. If we could do all this, there would be no reason to hold the position for the full ten weeks.
由于我们需要进行调整才能实现利润，似乎每个盈利的对冲都需要我们持仓直到到期。然而，实际上这并不一定必要。举个例子，假设在我们建立对冲后，期权市场的隐含波动率开始上升。从我们购买六月行权价 100 的看涨期权时的隐含波动率 14.6% 上升到期权存续期间的未来波动率 18.3%。这时，六月行权价 100 的看涨期权的价格会从 3.25 上涨到 3.88。我们可以以每份期权 0.63 的利润出售看涨期权。当然，如果我们想平仓，也需要买回原先卖出的 57 份六月期货合约。那么，隐含波动率的变化对期货合约的价格有什么影响呢？隐含波动率是与期权相关的特性，而非标的合约。因此，我们预计标的期货合约仍会以原价 101.35 交易。以 101.35 的价格购买 57 份未平仓的期货合约，我们将从对冲中实现 63.00 的总利润，这正是理论定价模型所预测的金额。如果我们能做到这一点，就没有理由将仓位持有满十周。
How likely is an immediate reevaluation in implied volatility from 14.6% to 18.3%? While violent changes in implied volatility are possible, they are the exception rather than the rule. Changes usually occur gradually over a period of time, and are the result of equally gradual changes in the volatility of the underlying contract. As the volatility of the underlying contract changes, option demand either rises or falls, and this demand is reflected in a corresponding rise or fall in the implied volatility. In our example, if market participants realized that the price of the underlying futures contract were fluctuating at a volatility greater than 14.6%, we would expect implied volatility to begin to rise. If implied volatility ever reached our target of 18.3%, we could sell out our calls and buy in our futures, thereby realizing our expected profit of 63.00 without having to hold the position for the full ten weeks. However, option prices are subject to a wide variety of market forces, not all of them theoretical. There is no guarantee that implied volatility will ever reevaluate up to 18.3%. In that case, we will have to hold the position for the full ten weeks and continue to adjust in order to realize our profit.
隐含波动率从 14.6% 立即上升到 18.3% 的可能性有多大？虽然隐含波动率的剧烈变化是可能的，但这种情况比较少见。变化通常是逐渐发生的，源于标的合约波动率的同样渐进的变化。随着标的合约波动率的变化，期权需求会相应上升或下降，这种需求会反映在隐含波动率的相应波动上。在我们的例子中，如果市场参与者意识到标的期货合约的价格波动超过 14.6%，我们就会预计隐含波动率开始上升。如果隐含波动率达到了我们的目标 18.3%，我们可以卖出看涨期权并买入期货，从而在不必持仓满十周的情况下实现预期的 63.00 利润。然而，期权价格受到多种市场力量的影响，并非所有的都是理论性的。不能保证隐含波动率会一直升高到 18.3%。在这种情况下，我们将不得不持仓满十周，并继续进行调整以实现利润。
Every trader hopes for quick reevaluation of implied volatility towards his volatility forecast. It not only enables him to realize his profits more quickly, but it eliminates the risk of holding a position for an extended period of time. The longer a position is held, the greater the possibility of error from the inputs into the model.
每个交易者都希望隐含波动率能快速朝其预期方向重新评估。这不仅可以更快地实现利润，还能消除长期持仓的风险。持仓时间越长，模型输入错误的可能性就越大。
Not only might implied volatility not reevaluate favorably, it might actually move against us, even if the true volatility of the underlying contract moves in our favor. Suppose that after initiating our hedge, implied volatility immediately falls to 13.5% from 14.0%. The price of the June 100 call will fall from 3.25 to 3.06. Now we have a paper loss of 100 x -19 = -19.00. Does this mean we made a bad trade and should close out the position? Not necessarily. If the volatility forecast of 18.3% turns out to be correct, the options will still be worth 3.88 by expiration. If we hold the position and adjust, we can eventually expect a profit of 03.00. Realizing this, we ought to maintain the position as we originally intended. Even though an adverse move in implied volatility is unpleasant, it is something with which all traders must learn to cope. Just as a speculator can rarely hope to pick the exact bottom or top at which to take a long or short position, an option trader can rarely hope to pick the exact bottom or top in implied volatility. He must try to establish positions when market conditions are favorable. But he must also realize that conditions might become even more favorable. If they do, his initial trade may show a temporary loss. This is something a trader learns to accept as a practical aspect of trading.
隐含波动率不仅可能不会向有利方向调整，反而可能朝不利方向变化，即使标的合约的实际波动率是有利的。假设在我们建立对冲后，隐含波动率从 14.0% 立即降至 13.5%。这时，六月 100 号看涨期权的价格会从 3.25 降到 3.06。我们就会有一笔纸面损失：100 x -19 = -19.00。这是否意味着我们交易失误，应该平仓？不一定。如果 18.3% 的波动率预测是正确的，到期时期权仍然值 3.88。如果我们持仓并进行调整，最终可以期待 0.03 的利润。意识到这一点后，我们应该坚持原计划持仓。尽管隐含波动率的不利变动令人不快，但这是所有交易者必须学会应对的。正如投机者很难准确选择建仓的底部或顶部，期权交易者也很难准确选择隐含波动率的底部或顶部。他必须在市场条件有利时建立仓位，但也要意识到条件可能会变得更加有利。如果是这样，初始交易可能会出现暂时亏损。这是交易者需要接受的实际交易方面。
Let's look at a somewhat more complex hedge, this time in the form of mispriced stock options. Suppose current market conditions are as follows:
Stock price = 48½/
Interest rate = 8.00%
Time to March expiration = 10 weeks
Expected dividend = 50¢ in 40 days
让我们来看一个稍微复杂一点的对冲，这次是对错误定价的股票期权进行对冲。假设当前市场条件如下：
股票价格：48½
利率：8.00%
到 3 月到期的时间：10 周
预期股息：40 天内 50 美分
Note that we now have an additional input, the expected dividend. Of course, we still need the volatility over the life of the option. Again looking into our crystal ball, we see that the volatility over the next 10 weeks will be 32.4%. Again, we decide to look at a call which is close to at-the-money, specifically the March 50 call. Feeding all of our inputs into the Black-Scholes model, we find that the March 50 call has a theoretical value of 2.17 and a delta of 46.
请注意，我们现在增加了一个输入项，即预期股息。当然，我们仍然需要期权有效期内的波动率。再次查看我们的水晶球，我们看到未来 10 周的波动率将为 32.4%。这次我们选择一个接近平值的看涨期权，具体来说是 3 月行权价 50 的看涨期权。将所有输入项代入布莱克-肖尔斯模型，我们发现 3 月行权价 50 的看涨期权的理论价值为 2.17，德尔塔为 46。
Having determined the theoretical value of the call, we still need its price in order to see if there is any profit opportunity available. In this case, it turns out that the call is trading for 3 (an implied volatility of 42.2%). Since the option is overpriced, we want to sell it and establish a delta neutral hedge against the underlying contract. We might, for example, sell 100 March 50 calls and simultaneously buy 46 stock contracts. Assuming this option is trading on a U.S. stock options exchange where each underlying contract consists of 100 shares of stock, it will be necessary to buy 4600 shares of stock.
确定了看涨期权的理论价值后，我们仍需了解它的市场价格，以判断是否存在获利机会。在这种情况下，发现该期权的交易价格为 3（隐含波动率为 42.2%）。由于该期权被高估，我们希望卖出它，并建立一个与标的合约的德尔塔中性对冲。我们可以选择卖出 100 份 3 月行权价 50 的看涨期权，同时买入 46 份股票合约。假设该期权在美国股票期权交易所交易，每个标的合约由 100 股股票组成，那么我们需要购买 4600 股股票。
As in all hedges based on a theoretically mispriced option, it is necessary to maintain a delta neutral position throughout the life of the option. As before, we will make our adjustments at weekly intervals, but now the underlying contract is shares of stock. Figure 5-2 shows the adjustment process for this hedge. Let's go through the hedge step by step and see what the final results are.
与所有基于理论上的错误定价期权的对冲一样，必须在期权有效期内保持德尔塔中性。与之前一样，我们将在每周进行调整，但这次标的合约为股票。图 5-2 显示了这一对冲的调整过程。接下来，我们逐步分析对冲的结果。
Figure 5-2
| Week | Stock Price | Delta of 50 call | Total Delta Position | Adjustment in Shares | Total Share Adjustment | Cash Flow | Interest on Cash Flow |
|---|---|---|---|---|---|---|---|
| 0 | 48½ | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 49⅝ | 52 | -600 | buy 600 | +600 | -29,775 | -411.14 |
| 2 | 52⅛ | 66 | -1400 | buy 1,400 | +2000 | -72,975 | -895.69 |
| 3 | 51¾ | 64 | +200 | sell 200 | +1800 | +10,350 | +111.16 |
| 4 | 50 | 52 | +1200 | sell 1,200 | +600 | +60,000 | +552.33 |
| 5 | 47 | 28 | +2400 | sell 2,400 | -1800 | +112,800 | +865.32 |
| ex-dividend 50¢ | |||||||
| 6 | 48⅛ | 38 | -1000 | buy 1,000 | -800 | -48,125 | -295.34 |
| 7 | 52 | 73 | -3500 | buy 3,500 | +2700 | -182,000 | -837.70 |
| 8 | 52¼ | 78 | -500 | buy 500 | +3200 | -26,125 | -80.16 |
| 9 | 50⅛ | 55 | +2300 | sell 2,300 | +900 | +115,288 | +176.88 |
| 10 | 52⅜ | 38 | sell 900 | +47,138 | |||
Original Hedge: At expiration, with the stock at 52⅜, the 50 calls were worth 23. So on the option we showed a profit of $300 - $237.50, or $62.50. The 4600 shares of stock we purchased at 48½ we were able to sell at 52⅜, for a profit of 3/ ($3.875) per share. The total hedge profit was therefore + (100 x $62.50) +(4600 x $3.875) = +$24,075.
初始对冲：到期时，股票价格为 52⅜，50 号看涨期权价值 23。因此，我们在期权上的利润为 $300 - $237.50，即 $62.50。我们以 48½的价格购买的 4600 股股票，能够以 52⅜的价格出售，每股利润为 3/($3.875)。因此，总对冲利润为 +(100 x $62.50) + (4600 x $3.875) = +$24,075。
Adjustments: The adjustment process forced us to buy and sell stock to remain delta neutral. At week one we bought 600 shares of stock at 49; at week two we bought 1400 shares at 52%; at week three we sold 200 shares at 51¾; and so on. At week ten we sold out our remaining 900 shares of stock at 52⅜. The result of all these adjustments was a loss of $13,425. (Again, the reader may wish to confirm this by adding up all the numbers in the "cash flow" column.)
调整过程：为了保持 delta 中性，我们不得不进行股票的买卖。在第一周，我们以 49 的价格购买了 600 股；第二周以 52 的价格购买了 1400 股；第三周以 51¾的价格出售了 200 股，依此类推。在第十周，我们以 52⅜的价格卖出了剩余的 900 股。所有这些调整的结果是损失 $13,425。（读者可以通过将 “现金流” 列中的所有数字相加来确认这一点。）
Carrying Costs on the Initial Position: Originally we sold 100 options at $300 each, and bought 4600 shares of stock at $48.50 each. This resulted in a total debit of (100 × $300) - (4600 x $48.50) = $193,100. The carrying cost on this debit for ten weeks at an annual rate of 8.00% (our interest rate assumption) was $193,100 x .08 x 70/365 = $2,962.63. Note that this is somewhat different than the futures example, where only the option contracts entailed a carrying cost. Unlike futures trades, the purchase or sale of stock results in an immediate cash flow, and this cash flow becomes part of the carrying cost on the initial hedge.
初始头寸的持有成本：我们最初以每个 $300 的价格卖出 100 份期权，购买 4600 股股票，每股价格为 $48.50。这导致总支出为 (100 × $300) - (4600 x $48.50) = $193,100。以年利率 8.00%（我们的利率假设）计算，这笔支出的十周持有成本为 $193,100 x .08 x 70/365 = $2,962.63。请注意，这与期货例子有所不同，在期货例子中，只有期权合约涉及持有成本。与期货交易不同，股票的买卖会立即产生现金流，而这部分现金流成为初始对冲的持有成本的一部分。
Interest on the Adjustments: Whenever we bought or sold stock to make an adjustment, the trade resulted in a cash flow, either a debit or a credit. We were able to earn interest on any credit, and we were required to pay interest on any debit, at a rate of 8.00 percent. For example, at week one we bought 600 shares of stock at 49⅝, for a total cash outlay of $29,775. The cost of carrying this debit to expiration (nine weeks hence) was -$29,775 × 63/365 x .08 = -$411.14. The total interest cost from our adjustments is the sum of all the individual interest calculations, or -$814.34.
调整的利息：每当我们买卖股票进行调整时，交易会导致现金流，可能是支出或收入。我们可以对任何收入部分赚取利息，对支出部分则需支付利息，利率为 8.00%。例如，在第一周，我们以 49⅝的价格购买 600 股，现金支出总计为 $29,775。这笔支出到期的持有成本（九周后）为-$29,775 × 63/365 x .08 = -$411.14。调整过程中的总利息成本为所有单项利息计算的总和，即-$814.34。
Dividends: We also assumed that 30 days prior to expiration between week five and six the stock paid a dividend of 50c. What was the dividend earnings or loss to our position? We initially bought 4,600 shares of stock. At the end of week five we were short 1800 shares of stock as part of the adjustment process. This means that on the ex-dividend date we were long a total of 2800 shares of stock. At a dividend of 50¢ per share, we received a dividend payout $1,400.
股息：我们还假设在到期前 30 天，即第五周到第六周之间，股票支付了 50 美分的股息。我们的头寸获得了多少股息收入或损失？我们最初购买了 4600 股股票。在第五周结束时，由于调整过程我们卖空了 1800 股。这意味着在除息日，我们持有 2800 股股票。每股 50 美分的股息，我们获得的股息总额为 $1,400。
Interest on Dividends: We were also able to earn interest on the $1,400 dividend payout for the remaining 30 days to expiration at a rate of 8.00%. The proceeds from this were $1,400 x 30/365 x .08 = $9.21.
股息利息：我们还可以在剩余 30 天到期内对这 $1,400 的股息收入赚取利息，利率为 8.00%。因此收益为 $1,400 x 30/365 x .08 = $9.21。
Summarizing the results, we have:
| Original Hedge | Adjustments | Interest on Hedge | Interest on Adjustments | Dividends | Interest on Dividends |
|---|---|---|---|---|---|
| +$24,075 | -$13,425 | -$2,962.03 | -$814.34 | +$1,400 | +$9.21 |
The total profit on the hedge is therefore:
+$24,075 -$13,425 -$2,962.63 -$814.34 +$1,400 + $9.21 = +$8,282.24
versus a theoretical profit of 100 × ($300 - $217) = 100 x $83 = +$8,300.
总结结果如下：
| 初始对冲 | 调整 | 对冲利息 | 调整利息 | 股息 | 股息利息 |
|---|---|---|---|---|---|
| +$24,075 | -$13,425 | -$2,962.03 | -$814.34 | +$1,400 | +$9.21 |
因此，对冲的总利润为：
+$24,075 -$13,425 -$2,962.63 -$814.34 +$1,400 + $9.21 = +$8,282.24
而理论利润为 100 × ($300 - $217) = 100 x $83 = +$8,300。
As in our last example, our profit depended on our knowing the volatility of the underlying contract over the life of the option. This "known" volatility of 32.4% represents the actual volatility associated with the ten stock price changes in Figure 5-2.
与上一个例子一样，我们的利润依赖于对期权存续期间标的合约波动率的了解。这一 “已知” 的 32.4% 波动率代表了图 5-2 中十个股票价格变化的实际波动率。
While we continue to assume that markets are frictionless, we noted in our futures option example that this is not necessarily true. In a locked futures market it is not always possible to freely buy and sell the underlying contract. A similar type of restriction can occur in stock option trading. In our example we established the initial hedge by selling calls and purchasing stock. If, however, the calls had been underpriced, we might have chosen to buy calls and sell stock, stock which we might not necessarily own. This type of short sale, the sale of stock which is borrowed rather than owned, is prohibited in some markets. This makes it difficult to hedge certain types of option positions. As we shall see later, if we choose to sell puts it is necessary to hedge the position by selling stock. If we are not able to sell stock to hedge the sale of puts, we might hesitate to sell puts, even at a price greater than the theoretical value. Indeed, in markets where short sales are prohibited, puts tend to trade at inflated prices compared to calls.
虽然我们假设市场是无摩擦的，但在期货期权的例子中，我们注意到这并不一定成立。在封闭的期货市场中，并不总是能自由买卖标的合约。类似的限制也可能出现在股票期权交易中。在我们的例子中，我们通过卖出看涨期权并购买股票建立了初始对冲。然而，如果这些期权被低估，我们可能选择买入看涨期权并卖出股票，而这股票不一定是我们所拥有的。这种卖空，即借入股票而非自有股票的卖出，在某些市场中是被禁止的。这使得对冲某些类型的期权头寸变得困难。稍后我们将看到，如果选择卖出看跌期权，必须通过卖出股票来对冲该头寸。如果无法卖出股票来对冲卖出看跌期权，我们可能会犹豫，即使卖价高于理论价值。实际上，在卖空被禁止的市场中，看跌期权的价格往往高于看涨期权的价格。
The short sale of stock is not totally prohibited in U.S. markets, but it is subject to an up-tick rule. This rule specifies that a short sale is always prohibited at a price lower than the price at which the previous trade took place (a down-tick). A short sale is always permitted at a price higher than the price at which the previous trade took place (an up-tick). Finally, a short sale may take place at the same price at which the previous trade took place if the previous trade took place on an up-tick (also an up-tick). The up-tick rule was instituted following the market crash of 1929, with the intent of preventing a similar crash by prohibiting the sale, at continuously lower prices, of stock not actually owned. Below are ten consecutive trade prices (reading from left to right) for a stock with the accompanying ticks (a positive sign for an up-tick; a negative sign for a down-tick).
48½ +48⅝ +48⅝ -48½ -48⅜ -48¼ -48¼ +48⅜ +48⅜ +48⅜
Not only might a short sale not be possible because of the up-tick rule, but many brokerage firms which execute short sales of stock for their customers do not pay full interest on the proceeds from a short sale. This can further distort the interest component used in a theoretical pricing model.
在美国市场，股票的卖空并不完全被禁止，但受到涨停规则的限制。该规则规定，卖空必须在上一个交易价格的基础上进行（上涨时允许，下降时禁止）。卖空在上一个交易价格之上是始终被允许的；若在上涨时进行的交易，卖空则可以在相同价格进行。涨停规则是在 1929 年市场崩溃后实施的，旨在防止类似崩溃，禁止以不断下降的价格出售未实际拥有的股票。以下是某股票连续十个交易价格（从左到右），以及相应的涨跌标记（上涨标记为正，下降标记为负）。
48½ +48⅝ +48⅝ -48½ -48⅜ -48¼ -48¼ +48⅜ +48⅜ +48⅜
由于涨停规则，卖空可能无法进行。此外，许多为客户执行卖空的券商不会对卖空所得全额支付利息，这可能进一步扭曲理论定价模型中使用的利息组成部分。
Taking one last look at our examples, what enabled us to make a profit approximately equal to that predicted by the theoretical pricing model? A simple way of interpreting the results is to realize that, according to the model, the option's cash flow can be replicated through the adjustment process. If we know the conditions of an option contract and the characteristics of the underlying contract, we can replicate the characteristics of the option, and therefore replicate the cash flow resulting from a position in the option, through an adjustment process in the underlying contract. In our examples, because we knew the exact market conditions which would prevail over the life of the options (primarily, this means knowing volatility), we knew we could replicate the option by continuously calculating the delta and taking an appropriate offsetting position in the underlying contract. According to the model, at expiration the total cash flow from this dynamic hedge should exactly equal the value of the option. But in our examples we either bought the option at less than theoretical value (our futures option example) or sold the option at more than theoretical value (our stock option example). Since the cash flow from the adjustment process exactly replicated the option's theoretical value, we were left with a profit equal to the difference between the option's price and its theoretical value.
最后，我们回顾一下这两个例子，是什么让我们获得了与理论定价模型预测的利润大致相等的结果？简单来说，可以理解为根据模型，期权的现金流可以通过调整过程进行复制。如果我们了解期权合约的条件和标的合约的特性，就可以通过标的合约的调整过程复制期权的特性，从而复制期权头寸所产生的现金流。在我们的例子中，由于我们知道期权存续期间将保持的确切市场条件（主要是波动率），我们知道可以通过不断计算德尔塔并在标的合约中采取适当的对冲头寸来复制期权。根据模型，到期时，该动态对冲的总现金流应恰好等于期权的价值。但在我们的例子中，我们要么以低于理论价值的价格买入期权（期货期权例子），要么以高于理论价值的价格卖出期权（股票期权例子）。由于调整过程中的现金流恰好复制了期权的理论价值，我们获得的利润正好等于期权价格与其理论价值之间的差额。
This type of option replication, using the cash flow from continuous hedging in the underlying contract, is the basis for many strategies which make use of option characteristics but do not actually Involve options. We will discuss the best known of these strategies, portfolio insurance, in Chapter 13.
这种通过标的合约的持续对冲现金流进行期权复制的方法，是许多利用期权特性但不实际涉及期权的策略的基础。我们将在第 13 章讨论这些策略中最著名的——投资组合保险。
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