Chapter 11 Option Arbitrage
第十一章 期权套利
One important characteristic of options is that they can be combined with other options or underlying contracts to create positions with characteristics which are almost identical to some other contract or combination of contracts. This type of replication leads to a new category of trading strategies which are unique to the option market.
期权的一个重要特性在于，它们可以与其他期权或标的合约组合，形成与某些其他合约或合约组合几乎相同特征的头寸。这种复制方式引出了期权市场独有的一类交易策略。
SYNTHETIC POSITIONS
合成头寸
Suppose a trader has the following position where all options are European (no early exercise permitted):
long a June 100 call
short a June 100 put
假设交易者持有以下头寸，所有期权均为欧式（不可提前行权）：
买入一个6月100看涨期权
卖出一个6月100看跌期权
What will happen to this position at expiration? It may seem that one cannot answer the question without knowing where the underlying contract will be at expiration. Surprisingly, the price of the underlying contract does not affect the outcome. If the underlying contract is above 100, the put will expire worthless, but the trader will exercise the 100 call. This results in his buying the underlying contract at 100. Conversely, if the underlying contract is below 100, the call will expire worthless, but the trader will be assigned on the 100 put. This also results in his buying the underlying contract at 100.
到期时，这个头寸会发生什么情况？可能看似无法在不知道标的合约价格的情况下回答这个问题。但出乎意料的是，标的合约的价格并不会影响结果。如果标的合约价格高于100，认沽期权将变得毫无价值，交易者会执行100的认购期权，这导致他以100的价格买入标的合约。反之，如果标的合约价格低于100，认购期权将变得毫无价值，而交易者会被指派执行100的认沽期权，这同样导致他以100的价格买入标的合约。
Ignoring for the moment the unique case when the underlying contract is right at 100, at June expiration the above position will always result in the trader going long the underlying instrument at the exercise price of 100. He will go long, either by choice (the underlying contract is above 100 and he exercises the 100 call) or by force (the underlying contract is below 100 and he is assigned on the 100 put). We refer to this position as a synthetic long underlying. The position has the same characteristics as the underlying contract, but won't actually become an underlying contract until expiration.
暂时忽略标的合约正好在100的特殊情况，到6月到期时，上述头寸将始终使交易者以100的行权价建立标的合约的多头仓位。无论是主动选择（标的合约价格高于100时执行100的认购期权）还是被动接受（标的合约价格低于100时被指派执行100的认沽期权），此头寸都称为“合成多头标的”。它具备与标的合约相同的特性，但要到到期时才真正变为标的合约。
If the trader takes the opposite position, selling a June 100 call and purchasing a June 100 put, he has a synthetic short underlying. This position will always result in his selling the underlying contract at the exercise price of 100, either by choice (the underlying contract is below 100 and he exercises the 100 put) or by force (the underlying contract is above 100 and he is assigned on the 100 call).
如果交易者采取相反的操作，即卖出6月100看涨期权并买入6月100看跌期权，他就拥有一个合成空头标的仓位。无论标的价格在到期时如何变化，这一仓位最终都会以100的行权价卖出标的资产，不论是因标的价格低于100时主动行使看跌期权，还是标的价格高于100时被动被分配看涨期权。
We can express the foregoing relationships as follows:
synthetic long underlying = long call + short put
synthetic short underlying = short call + long put
where all options expire at the same time and have the same exercise price.
我们可以将上述关系表示为：
合成多头标的 = 买入看涨期权 + 卖出看跌期权
合成空头标的 = 卖出看涨期权 + 买入看跌期权
其中所有期权在同一时间到期，且具有相同的执行价格。
A synthetic position acts very much like its real equivalent. For each point the underlying instrument rises, a synthetic long position will gain approximately one point in value and a synthetic short position will lose approximately one point in value. This leads us to conclude, correctly, that the delta of a synthetic underlying position is approximately 100. If the delta of the June 100 call is 75, the delta of the June 100 put will be approximately -25. If the delta of the June 100 put is -60, the delta of the June 100 call will be approximately 40. Ignoring the positive sign associated with a call and the negative sign associated with a put, the deltas of calls and puts with the same underlying contract, expiration date, and exercise price will always add up to approximately 100. As we will see, interest rates, as well as the possibility of early exercise, can cause the delta of a synthetic underlying position to be slightly more or less than 100. But for most practical calculations, traders tend to look at a synthetic underlying position and mentally assign it a delta of 100.
合成头寸的表现类似于实际头寸。每当标的合约上涨一个点，合成多头头寸的价值将增加约一点，而合成空头头寸的价值将下降约一点。因此，我们可以合理地得出结论，合成标的头寸的delta大约为100。例如，如果6月100行权价的看涨期权Delta为75，则6月100看跌期权的Delta约为-25；如果看跌期权的Delta为-60，那么看涨期权的Delta约为40。忽略看涨期权的正号与看跌期权的负号，具有相同标的、到期日及行权价的看涨和看跌期权的Delta总和将约等于100。利率及提前行权可能会导致合成标的头寸的Delta略高或略低于100，但在大多数实际计算中，交易者通常将合成标的头寸的Delta视为100。
Rearranging the components of a synthetic underling position, we can create four other synthetic relationships:
synthetic long call = long an underlying contract + long put
synthetic short call = short an underlying contract + short put
synthetic long put = short an underlying contract + long call
synthetic short put = long an underlying contract + short call
Again, all options must expire at the same time and have the same exercise price. Each synthetic position has a delta approximately equal to its real equivalent and will therefore gain or lose value at approximately the same rate as its real equivalent.
重新排列合成标的合约的组成部分，可以得到四种其他合成关系：
合成多头看涨期权 = 买入标的合约 + 买入看跌期权
合成空头看涨期权 = 卖出标的合约 + 卖出看跌期权
合成多头看跌期权 = 卖出标的合约 + 买入看涨期权
合成空头看跌期权 = 买入标的合约 + 卖出看涨期权
同样，所有期权必须在同一时间到期并具有相同的执行价格。每个合成头寸的delta大约等于其实际头寸，因此它的价值变化率与实际头寸相当。
If the underlying instrument in our examples is a June futures contract, we can create six synthetic positions:
synthetic long June future = long June 100 call + short June 100 put
synthetic short June future = short June 100 call + long June 100 put
synthetic long June 100 call = long June future + long June 100 put
synthetic short June 100 call = short June future + short June 100 put
synthetic long June 100 put = short June future + long June 100 call
synthetic short June 100 put = long June future + short June 100 call
如果我们示例中的标的合约是6月期货合约，那么可以创建六种合成头寸：
合成多头6月期货 = 买入6月100看涨期权 + 卖出6月100看跌期权
合成空头6月期货 = 卖出6月100看涨期权 + 买入6月100看跌期权
合成多头6月100看涨期权 = 买入6月期货 + 买入6月100看跌期权
合成空头6月100看涨期权 = 卖出6月期货 + 卖出6月100看跌期权
合成多头6月100看跌期权 = 卖出6月期货 + 买入6月100看涨期权
合成空头6月100看跌期权 = 买入6月期货 + 卖出6月100看涨期权
We need not create a synthetic position using the 100 exercise price. We can choose any available exercise price. A long June 110 call together with a short June 110 put is still a synthetic long June futures contract; however, at June expiration the June futures contract will be purchased at 110. A short June 95 call together with a long June 95 put is a synthetic short June futures contract. But at expiration the June futures contract will be sold at 95.
我们不必一定要使用100的执行价来构建合成头寸。可以选择任意可用的执行价。例如，买入6月110看涨期权并卖出6月110看跌期权仍是一个合成多头6月期货合约，但到期时将在110执行价买入；卖出6月95看涨期权并买入6月95看跌期权则是合成空头6月期货合约，到期时在95的执行价卖出。
Suppose an underlying contract is trading at 102.00 and we want to take a long position in the market. We can simply go into the underlying market and buy the contract at 102.00. But we now have an additional choice. We can take a long synthetic position by purchasing a June call and selling a June put at the same exercise price. Which of these strategies is best? As with any option strategy, the decision depends on the prices of the options in the marketplace. Suppose the June 100 call is trading at 5.00 and the June 100 put is trading at 3.00. If we buy the June 100 call for 5.00 and sell the June 100 put for 3.00, we will show an immediate debit of 2.00. If at expiration the underlying contract is at 110.00, we will show a credit of 10.00 when we exercise our June 100 call, for a total profit of 8.00. If we ignore interest considerations, this is identical to the profit we would have realized had we instead bought the underlying contract at 102.00. This is shown in Figure 11-1.
假设标的合约的价格为102.00，想要做多。可以直接在市场上以102.00的价格买入合约，也可以通过买入6月看涨期权并卖出6月看跌期权构建合成多头头寸。哪种策略更优？与其他期权策略一样，这取决于市场上期权的价格。如果6月100看涨期权交易在5.00，看跌期权交易在3.00，则买入6月100看涨期权并卖出6月100看跌期权会产生2.00的成本。在到期时，标的合约价格为110.00，则行权看涨期权获得10.00的收入，总利润为8.00。忽略利息情况，这与直接以102.00买入合约实现的利润相同，如图11-1所示。
Suppose the underlying contract is still trading at 102.00, but now the June 100 call is trading at 4.90 and the June 100 put is trading at 3.05. If we take a synthetic long underlying position by purchasing the June 100 call and selling the June 100 put, we will show a debit of 1.85. Now if the underlying contract is at 110.00 at expiration we will show a total profit of 8.15 (the debit of 1.85 from the option trades, plus the 10-point credit when we exercise the 100 call). This is.15 better than we would do by taking a long position in the underlying contract at 102.00.
若标的合约仍交易在102.00，而6月100看涨期权价格为4.90，看跌期权价格为3.05，则合成多头头寸的成本为1.85。在到期时标的合约为110.00，总利润将为8.15，比直接买入合约多0.15。
As long as the price of the June 100 call is exactly two points greater than the price of the June 100 put, the profit or loss resulting from a synthetic position will be identical to an actual position taken in the underlying contract at a price of 102.00. The difference between the call and put price is often referred to as the synthetic market. In the absence of any interest or dividend considerations, the value of the synthetic market can be expressed as:
call price - put price = underlying price - exercise price
只要6月100看涨期权的价格比6月100看跌期权高出恰好两点，合成头寸的盈亏将等于以102.00的价格买入标的合约。买入看涨期权并卖出看跌期权时，这个差值称为“合成市场价”。在没有利息或股息影响下，合成市场价的价值可表示为：
看涨期权价格 - 看跌期权价格 = 标的合约价格 - 执行价格
If this equality holds, there is no difference between taking a position in the underlying market, or taking an equivalent synthetic position in the option market. With the June 100 call at 5.00 and the June 100 put at 3.00, we can write:
5.00 - 3.00 = 102.00 - 100.00
2.00 = 2.00
There is no difference between the synthetic and its real equivalent.
如果满足这一等式，实际持仓与合成持仓没有差别。当6月100看涨期权为5.00，看跌期权为3.00时，可表示为：
5.00 - 3.00 = 102.00 - 100.00
2.00 = 2.00
此时，合成头寸与实际头寸没有差别。
With the June 100 call at 4.90 and the June 100 put at 3.05, we can write:
4.90 - 3.05 ≠ 102.00 - 100.00
1.85 ≠ 2.00
Here the synthetic side is cheaper, and we therefore prefer to take a long underlying position synthetically by purchasing the call and selling the put.
若6月100看涨期权价格为4.90，而看跌期权价格为3.05，则可表示为：
4.90 - 3.05 ≠ 102.00 - 100.00
1.85 ≠ 2.00
在这种情况下，合成头寸成本较低，因此优先通过买入看涨期权并卖出看跌期权来构建多头头寸。
Now suppose the price of the June 100 call is 5.15 and the price of the June 100 put is 2.90, with the underlying contract still trading at 102.00. We have:
5.15 - 2.90 ≠ 102.00 - 100.00
2.25 ≠ 2.00
Now the real side is cheaper, so we prefer to take a long position by purchasing the actual underlying contract. On the other hand, if we want to take a short underlying position, we will prefer to sell the underlying contract synthetically by selling the call and purchasing the put. If we do this, we are selling a contract which is only worth 2.00 for 2.25.
如果6月100看涨期权的价格为5.15，看跌期权为2.90，则可表示为：
5.15 - 2.90 ≠ 102.00 - 100.00
2.25 ≠ 2.00
此时，实际持仓成本更低，因此选择直接买入标的合约做多。相反，如果要做空，卖出看涨期权并买入看跌期权可以以2.25的价格构建合成空头头寸，这高于合约本身2.00的价值。
The reader may have noted that the synthetic and real markets will be the same when the time value of the call and the time value of the put are identical. In our example, the synthetic and real markets are identical when the June 100 call and the June 100 put both have the same amount of time value. If this is not true, there is always a synthetic position which is either too cheap or too expensive with respect to its real equivalent.
当看涨期权和看跌期权的时间价值相等时，合成市场与实际市场是相同的。例子中，当6月100看涨期权和6月100看跌期权的时间价值相同时，合成市场和实际市场是相等的。如果不满足此条件，总有一种合成头寸相对于其实际持仓要么过低要么过高。
The three-sided relationship between a call, a put, and its underlying contract means that we can always express the value of any one of these contracts in terms of the other two:
underlying price = call price - put price + exercise price
call price = underlying price + put price - exercise price
put price = call price - underlying price + exercise price
This three-sided relationship is sometimes referred to as put-call parity.
看涨期权、看跌期权与其标的合约之间的三重关系意味着我们总能通过其他两个合约来表达任意一个合约的价值：
标的合约价格 = 看涨期权价格 - 看跌期权价格 + 执行价
看涨期权价格 = 标的合约价格 + 看跌期权价格 - 执行价
看跌期权价格 = 看涨期权价格 - 标的合约价格 + 执行价
这种三重关系通常称为“看跌-看涨平价”。
CONVERSIONS AND REVERSALS
转换与反向转换
If we take a synthetic long or short underlying position, our primary concern, as with an actual underlying position, is the direction of the market. If the market moves in our favor, we expect to show a profit; if it moves against us, we expect to show a loss. If we execute the synthetic at favorable prices we may gain more or lose less, but it is still primarily the direction of the market in which we are interested.
当我们创建一个合成多头或空头标的头寸时，主要关注的是市场的方向性。若市场走势有利，我们预期会获利；若不利，可能会亏损。如果以有利的价格建仓合成头寸，则可能赚多赔少，但仍然主要取决于市场方向。
Suppose, as before, the underlying contract is trading at 102.00, the June 100 call is trading at 5.10, and the June 100 put is trading at 2.85. The synthetic market should be 2.00 (underlying price less the exercise price), but is actually 2.25. If we wanted to take a short position in the underlying market, we would certainly prefer to do so synthetically (sell the call, buy the put). Suppose we are not interested in taking a directional position. Is there any way we can profit from the difference in price between the real and synthetic markets?
假设标的合约当前价格为102.00，6月100看涨期权的价格为5.10，看跌期权为2.85。理论上合成市场应为2.00（标的价格减行权价），实际价格却为2.25。如果我们希望做空标的市场，显然会优先选择以合成方式建立空头（卖出看涨期权，买入看跌期权）。假如我们不想建立方向性头寸，是否可以通过实际市场和合成市场的价差获利？
When a trader identifies two contracts which are essentially the same but which are trading at different prices, the natural course is to execute an arbitrage by purchasing the cheaper contract and selling the more expensive. Since the synthetic position and its real equivalent are essentially the same, and since they are trading at different prices (2.25 vs. 2.00), an option trader might try to purchase the cheaper contract (the underlying contract) and sell the more expensive (the synthetic equivalent). That is, he might try to purchase the underlying contract for 102.00, and simultaneously sell the call for 5.10 and buy the put for 2.85. The cash flows from these transactions are:
| Transaction | Cash flow |
| underlying purchase | -102.00 |
| call sale | +5.10 |
| put purchase | -2.85 |
| exercise or assignment (at 100) at expiration | +100.00 |
| total | +.25 |
No matter what happens in the underlying market, the underlying position will do exactly .25 better than the synthetic position. The entire position will therefore show a profit of .25, regardless of movement in the underlying market. This is shown in Figure 11-2.
当交易员发现两个价格不同但本质相同的合约时，通常会通过买入较便宜的合约、卖出较贵的合约进行套利。既然合成头寸与实际标的等价，而价格却不同（2.25与2.00），期权交易员可以尝试以102.00的价格买入标的，同时卖出看涨期权（5.10）并买入看跌期权（2.85）。交易现金流如下：
| 交易 | 现金流 |
| 买入标的 | -102.00 |
| 卖出看涨期权 | +5.10 |
| 买入看跌期权 | -2.85 |
| 到期时行权或被指派（按100价格） | +100.00 |
| 总计 | +0.25 |
无论标的市场如何变动，标的头寸总会比合成头寸多赚0.25，因此整个头寸将锁定0.25的利润，这在图11-2中展示。
The foregoing position, where the purchase of an underlying contract is offset by the sale of a synthetic position, is known as a conversion. The opposite position, where the sale of an underlying contract is offset by the purchase of a synthetic position, is known as a reverse conversion or, more commonly, a reversal. A reversal would be profitable if the underlying contract were trading at 102.00, and the difference between the prices of the June 100 call and June 100 put were less than 2.00. For example, with the June 100 call at 4.90 and the June 100 put at 3.05, the synthetic price is 1.85. By purchasing the synthetic at 1.85 (buy the call at 4.90, sell the put at 3.05), and selling the underlying contract at 102.00, the reverse conversion would lock in a profit of .15. This is shown in Figure 11-3.
上述买入标的同时卖出合成头寸的策略称为转换。相反，卖出标的并买入合成头寸的策略称为反向转换或反转。如果标的交易价格为102.00，6月100看涨期权与看跌期权的价差小于2.00，则反转有利可图。例如，6月100看涨期权价格为4.90，看跌期权价格为3.05，合成价格为1.85。以1.85买入合成（4.90买入看涨，3.05卖出看跌），以102.00卖出标的，反向转换锁定0.15利润，见图11-3。
Summarizing:
| conversion | = long underlying + synthetic short underlying |
| = long underlying + short call + long put | |
| reversal | = short underlying + synthetic long underlying |
| = short underlying + long call + short put |
As before, we assume that the call and the put have the same exercise price and expiration date.
总结如下：
| 转换 | = 多头标的 + 合成空头标的 |
| = 多头标的 + 空头看涨期权 + 多头看跌期权 | |
| 反转 | = 空头标的 + 合成多头标的 |
| = 空头标的 + 多头看涨期权 + 空头看跌期权 |
在此假设看涨和看跌期权具有相同的行权价和到期日。
Conversions and reversals are classified as arbitrage strategies because of their similarity to traditional arbitrage. Typically, an arbitrageur will attempt to simultaneously buy and sell the same items in different markets to take advantage of price discrepancies between the two markets. He might, for example, buy gold in New York for $389 per ounce, and sell it in London for $392 per ounce. The profit margin of $3, while small, is secure since the risk of a trade in one market is almost immediately offset by an identical but opposing trade in another market. Unlike a speculator, who usually hopes to make a large profit on a small number of trades, an arbitrageur hopes to make a small profit on a large number of trades. An arbitrageur is willing to do much greater size than the speculator because the risk is much smaller.
转换和反转被归类为套利策略，因为其与传统套利类似。一般来说，套利者会在不同市场中同时买入和卖出相同的资产，以利用两者间的价格差异。例如，套利者可能在纽约以每盎司389美元买入黄金，并在伦敦以392美元卖出。虽然利润微小，但由于两个市场中几乎同时完成对冲交易，风险较低。与希望在少量交易中获大利的投机者不同，套利者希望通过大量交易获得微小的利润，因此其交易规模通常远大于投机者。
Like traditional arbitrages, conversions and reversals involve buying and selling the same thing in different markets. A conversion involves buying the underlying instrument, whether a stock or commodity, in the underlying market, and selling the underlying instrument, synthetically, in the option market. A reversal involves selling the underlying instrument in the underlying market, and buying the underlying instrument, synthetically, in the option market. The profitability of these strategies is determined by the relationship between the synthetic price and the actual underlying price. Synthetic positions are often used to execute conversions and reversals, so traders sometimes refer to the synthetic market (the difference between the call price and put price) as the conversion/ reversal market.
如同传统套利，转换和反转涉及在不同市场中买入和卖出相同资产。转换策略中，交易员在标的市场买入资产，并在期权市场通过合成方式卖出该资产；反转则在标的市场卖出资产，并在期权市场通过合成方式买入资产。其盈利取决于合成价格和实际标的价格的关系。合成头寸通常用于执行转换和反转，因此交易员有时会将合成市场（即看涨和看跌期权价差）称为转换/反转市场。
All experienced traders are familiar with the price relationship between a synthetic position and its underlying contract, so that any imbalance in the conversion/reversal market is likely to be short-lived. If the synthetic is overpriced, all traders will want to execute a conversion (buy the underlying, sell the call, buy the put). If the synthetic is underpriced, all traders will want to execute a reversal (sell the underlying, buy the call, sell the put). Such activity, where everyone is attempting to do the same thing, will quickly force the synthetic market back to equilibrium. Indeed, imbalances in the conversion/reversal market are usually small and rarely last for more than a few seconds. When imbalances do exist, an option trader is usually willing to execute conversions and reversals in very large size because of the low risk associated with such strategies.
所有经验丰富的交易员都熟悉合成头寸和标的合约之间的价格关系，因此转换/反转市场的失衡通常只会持续很短时间。当合成被高估时，所有交易员都会进行转换操作（买入标的，卖出看涨期权，买入看跌期权）。当合成被低估时，所有交易员都会进行反转操作（卖出标的，买入看涨期权，卖出看跌期权）。这种大量的同向操作会迅速将合成市场恢复至平衡。转换/反转市场失衡通常较小，持续时间很短，通常不超过几秒。当失衡出现时，期权交易员通常愿意执行大规模的转换和反转操作，因为这种策略风险较低。
We initially broke down a synthetic position into its component cash flows, enabling us to identify the basic relationship between a synthetic and its underlying instrument:
call price - put price = underlying price - exercise price
However, when and how the component cash flows occur can alter this relationship. Since there can be different settlement procedures associated with different underlying instruments as well as with options, in order to calculate the exact value of a synthetic position we need to consider how these settlement procedures affect the basic synthetic relationship.
我们最初通过拆分合成头寸的现金流来识别其与标的之间的基本关系：
看涨期权价格 - 看跌期权价格 = 标的价格 - 行权价
然而，现金流的发生时间和方式可能会改变这种关系。由于不同标的和期权的结算方式不同，为准确计算合成头寸的价值，我们需考虑这些结算方式如何影响基本的合成关系。
Futures Option Markets
期货期权市场
If the cash flow resulting from an option trade and a trade in the underlying instrument is identical, the synthetic relationship is simply:
call price - put price = underlying price - exercise price
如果期权交易和标的合约交易的现金流相同，那么合成关系为：
看涨期权价格 - 看跌期权价格 = 标的价格 - 行权价
This will be true if interest rates are zero, or in futures markets where both the underlying contract and options on that contract are subject to futures-type settlement. In the latter case there is no interest component because no cash changes hands when either the futures contract or the options on that contract are traded. This is currently the procedure on many non-U.S. exchanges.
在利率为零的情况下，或在期货市场中，当标的合约及其期权均采用期货类型结算时，此关系成立。在这种情况下，因为交易期货合约或其期权时没有现金交割，因此不存在利息因素。目前，许多非美交易所采用这种做法。
In futures markets, where options are subject to stock-type (cash) settlement, while no cash will change hands when the futures contract is traded, there will be an exchange of cash when options are traded. The interest considerations on the credit or debit will therefore have to be taken into consideration when calculating the value of a synthetic position.
在期货市场中，如果期权采用股票类型（现金）结算，虽然期货合约交易时没有现金交割，但期权交易时会有现金交割。因此，在计算合成头寸价值时，需要考虑借贷或付款的利息成本。
Using our earlier example, with the June futures contract at 102.00, the June 100 call at 5.00, and the June 100 put at 3.00, we have:
call price - put price = underlying price - exercise price
5.00 - 3.00 = 102.00 - 100.00
2.00 = 2.00
At these prices there does not appear to be any profit opportunity from either a conversion or reversal. If we do either of these strategies, the credit and debits from the two sides of the equation exactly offset each other.
根据之前的例子，假设6月期货合约为102.00，6月100看涨期权为5.00，6月100看跌期权为3.00，我们有：
看涨期权价格 - 看跌期权价格 = 标的价格 - 行权价
5.00 - 3.00 = 102.00 - 100.00
2.00 = 2.00
按此价格，执行转换或反向转换不会产生任何利润，两侧的收入和指出完全抵消。
Suppose, however, that we decide to do a reversal by selling the underlying futures contract for two points (underlying price minus exercise price) and buying the synthetic for two points (call price minus put price). Suppose also that the options are subject to stock-type (cash) settlement, as is generally true in U.S. futures markets. We will lay out 2.00 on the option trades, and we will not get this money back until expiration when we will buy the underlying futures contract, which we originally sold at 102, for 100. If interest rates are currently 8% per annum, and there are three months remaining to expiration, there will be a 2% carrying cost on the two-point debit resulting from the option trades. Based on the simple synthetic relationship, we expected to break even on the conversion. In fact we will end up losing 2% x 2.00, or .04, because of the cost of carrying the debit to expiration. If we really want to break even, we will have to find some way to offset this interest cost of .04. We might, for example, purchase the call for .04 less (4.96), or sell the put for 04 more (3.04), or sell the futures contract for .04 more (102.04). Or we might use a combination of these three transactions, for example by purchasing the call for 4.98, selling the put for 3.01, and selling the futures contract for 102.01. The result is a savings of 04, an amount equal to the carrying cost on the debit resulting from the option trades.
假设我们进行反向转换操作，以2点的价格（标的价格减去行权价）卖出标的期货合约，同时以2点的价格（看涨期权价格减去看跌期权价格）买入合成头寸。假设期权采用股票类型（现金）结算（通常在美国期货市场中如此）。我们在期权交易中需支付2.00，并在到期时才能通过以100的价格回购我们最初以102卖出的期货合约来收回这笔钱。如果当前年利率为8%，到期还有三个月，则此2点的借方将产生2%的持有成本。根据简单的合成关系，我们预期可保本，然而因持有成本，实际会亏损0.04（2% x 2.00）。要实现保本，我们需要补偿这0.04的利息成本。比如，将看涨期权价格降低0.04（至4.96）买入，或将看跌期权价格提高0.04（至3.04）卖出，或以更高的价格（102.04）卖出期货合约。或将三种操作结合，例如以4.98购买看涨期权、以3.01卖出看跌期权，并以102.01卖出期货合约。最终节省了0.04，恰好等于持有成本。
Assuming that all options are European (no early exercise permitted), we can now express the synthetic relationship in futures markets where the options are settle in cash as follows:
call price - put price = futures price - exercise price - carrying costs
假设所有期权为欧式期权（不可提前行权），此时在期货市场中采用现金结算的期权的合成关系可表示为：
看涨期权价格 - 看跌期权价格 = 期货价格 - 行权价 - 持有成本
where the carrying costs are calculated on either the difference between the futures price and the exercise price, or the difference between the call price and put price, both of which will be approximately the same. In our example, if the call is trading for 4.98, the put for 3.01, and the futures contract for 102.01, the synthetic market is exactly balanced since:
4.98 - 3.01 = 102.01 - 100.00 - .04
1.97 = 1.97
其中，持有成本可以基于期货价格和行权价的差异或看涨与看跌期权价格的差异计算，两者大致相同。在我们的例子中，如果看涨期权交易价格为4.98，看跌期权为3.01，期货合约为102.01，则合成市场完全平衡：
4.98 - 3.01 = 102.01 - 100.00 - 0.04
1.97 = 1.97
Using put-call parity, we can always calculate a call, put, or underlying futures price when we know the prices of any two of the other contracts. For example, with a June futures contract at 102.00, and the June 100 put at 2.75, we have:
| call price | = futures price - exercise price + put price - carrying costs |
| = 102.00 - 100.00 + 2.75 - .04 | |
| = 4.71 |
根据看跌-看涨平价原理，只要知道其中两者的价格，我们便可计算看涨期权、看跌期权或标的期货的价格。例如，假设6月期货合约为102.00，6月100看跌期权价格为2.75，我们有：
| 看涨期权价格 | = 期货价格 - 行权价 + 看跌期权价格 - 持有成本 |
| = 102.00 - 100.00 + 2.75 - 0.04 | |
| = 4.71 |
With the June 100 call at 5.35 and the June futures contract at 101.90, we have:
| put price | = call price + exercise price - futures price + carrying costs |
| = 5.35 + 100.00 - 101.90 + .04 | |
| = 3.49 |
假设6月100看涨期权价格为5.35，6月期货合约为101.90，则我们有：
| 看跌期权价格 | = 看涨期权价格 + 行权价 - 期货价格 + 持有成本 |
| = 5.35 + 100.00 - 101.90 + 0.04 | |
| = 3.49 |
Finally, with the June 100 call at 3.25 and the June 100 put at 1.25, we have:
| futures price | = call price- putprice + exercise price + carrying costs |
| = 3.25 - 1.25 + 100.00 + .04 | |
| = 102.04 |
最后，假设6月100看涨期权价格为3.25，6月100看跌期权价格为1.25，则我们有：
| 期货价格 | = 看涨期权价格 - 看跌期权价格 + 行权价 + 持有成本 |
| = 3.25 - 1.25 + 100.00 + 0.04 | |
| = 102.04 |
Stock Option Markets
股票期权市场
Consider a stock trading at 103, with a June 100 call trading at six and a June 100 put trading at three. If there are no interest or dividend considerations, the synthetic market seems to be balanced since:
call price - put price = stock price - exercise price
6 - 3 = 103 - 100
3 = 3
假设一只股票交易价格为103，6月100看涨期权价格为6，看跌期权价格为3。如果不考虑利息或股息因素，合成市场似乎是平衡的，因为：
看涨期权价格 - 看跌期权价格 = 股票价格 - 行权价
6 - 3 = 103 - 100
3 = 3
Suppose we decide to do a conversion by purchasing the stock for 103 and selling the synthetic for three points (call price minus put price). We will lay out 103 for the stock, take in six for the call, and lay out three for the put. These trades will create a total debit of 100, a debit which we will have to carry to expiration, at which time we will sell the stock for 100, either through exercise of the put or assignment of the call. Since interest rates in the real world are not zero, there will be a cost associated with carrying this debit. If interest rates are currently 8% per annum, and there are three months remaining to expiration, there will be a 2% carrying cost on the 100-point debit resulting from the trades. Based on the simple synthetic relationship, we expected to break even on the conversion. But in fact we will end up losing 2% x 100, or two points, because of the cost of carrying the debit to expiration. If we really want to break even, we will have to find some way to offset this two-point interest cost. We might do this by selling the call for two points more (8), or by purchasing the put for two points less (1), or by purchasing the stock for two points less (101). Or we might use a combination of these three transactions, for example by selling the call for 7, purchasing the put for 2/2, and purchasing the stock for 102/2. The result is a savings of two points, an amount equal to the carrying cost on the debit resulting from the trades.
假设我们决定通过以103的价格购买股票并以3点（看涨期权价格减去看跌期权价格）卖出合成市场来进行转换。我们将以103购买股票，收取6的看涨期权价格，并支付3的看跌期权价格。这些交易将产生总计100的借方余额，需持有至到期，此时我们将以100的价格卖出股票（通过行使看跌期权或被指派看涨期权）。由于实际利率非零，持有该借方余额将产生成本。如果当前年利率为8%，到期时间为三个月，则100点借方余额将产生2%的持有成本。基于简单的合成关系，我们预期可保本，但实际上我们将因持有成本亏损2% x 100，即2点。若要保本，我们需补偿这2点的利息成本，可通过以8的价格卖出看涨期权，或以1的价格买入看跌期权，或以101的价格买入股票来实现，或者结合以上交易，例如以7的价格卖出看涨期权、2½的价格买入看跌期权、102½的价格买入股票。这样可以节省2点，正好等于交易产生的持有成本。
Taking into consideration the interest rate component, we can express the synthetic relationship as:
call price - put price = stock price - exercise price + carrying costs
where the carrying costs are calculated on the exercise price. (footnote 1: The exact value depends on the discounted price, or present value, of the exercise price. For a more detailed explanation, sec Appendix B.) In our example, if the call is trading for 7, the put for 2½, and the stock for 102½, the synthetic market is exactly balanced since:
7 - 2½ = 102½ - 100 + 2
4½ = 4½
考虑利息因素，我们可以将合成关系表示为：
看涨期权价格 - 看跌期权价格 = 股票价格 - 行权价 + 持有成本
其中持有成本按行权价计算。(脚注1: 具体值取决于行权价的贴现值或现值。详细说明见附录B。) 在我们的例子中，若看涨期权交易价格为7，看跌期权为2½，股票价格为102½，则合成市场完全平衡：
7 - 2½ = 102½ - 100 + 2
4½ = 4½
Now suppose that prior to expiration the stock will pay a dividend of 1½. When we did our conversion we purchased stock, so we will receive an extra 1½ points when the dividend is paid. Given this, we can afford to pay up to 1½ points more for the conversion and still do no worse than break even. We could, for example, sell the call for 5½, or buy the put for four, or buy the stock for 104. Or, as before, we might combine these three trades by selling the call for six, buying the put for 3½, and buying the stock for 103½
现在假设到期前股票将支付1½的股息。进行转换时我们购买了股票，因此我们将获得额外的1½点股息。因此，我们可以多支付1½点的费用而仍然能保本。例如，卖出看涨期权价格为5½，或以4的价格买入看跌期权，或以104的价格购买股票。或者，可以组合这三种操作，如以6的价格卖出看涨期权、3½的价格买入看跌期权和103½的价格购买股票。
Assuming that all options are European (no early exercise permitted), and taking into consideration both interest rates and dividends, we can now express the full value of the synthetic relationship in the stock option market as:
call price - put price = stock price - exercise price + carrying costs - dividends
where the carrying costs are calculated on the exercise price and the dividends are those expected prior to expiration.
假设所有期权为欧式期权（不可提前行权），并考虑利息和股息因素，我们现在可以在股票期权市场中将合成关系表示为：
看涨期权价格 - 看跌期权价格 = 股票价格 - 行权价 + 持有成本 - 股息
其中持有成本按行权价计算，股息为到期前预计的股息。
Using put-call parity, we can always calculate a call, put, or underlying stock price when we know the prices of any two of the other contracts. For example, with the stock at 102 and the June 100 put at 3¼, we have:
| call price | = stock price - exercise price + put price + carrying costs - dividends |
| = 102. - 100 + 3¼ + 2 - 1½ | |
| = 5¾ |
利用看跌-看涨平价原理，只要知道其中两个合约的价格，我们可以计算看涨期权、看跌期权或标的股票的价格。例如，假设股票价格为102，6月100看跌期权为3¼，我们有：
| 看涨期权价格 | = 股票价格 - 行权价 + 看跌期权价格 + 持有成本 - 股息 |
| = 102 - 100 + 3¼ + 2 - 1½ | |
| = 5¾ |
With the June 100 call at 6½ and the stock at 101, we have:
| put price | = call price + exercise price - stock price - carrying costs + dividends |
| = 6½ + 100 - 101¼- 2 + 1½ | |
| = 4¾ |
当6月100看涨期权价格为6½，股票价格为101时，我们有：
| 看跌期权价格 | = 看涨期权价格 + 行权价 - 股票价格 - 持有成本 + 股息 |
| = 6½ + 100 - 101¼ - 2 + 1½ | |
| = 4¾ |
Finally, with the June 100 call at 7 and the June 100 put at 1¾, we have:
| stock price | = call price - put price + exercise price - carrying costs + dividends |
| = 7 - 1¾ + 100 - 2 + 1½ | |
| = 104¾ |
最后，当6月100看涨期权价格为7，6月100看跌期权价格为1¾时，我们有：
| 股票价格 | = 看涨期权价格 - 看跌期权价格 + 行权价 - 持有成本 + 股息 |
| = 7 - 1¾ + 100 - 2 + 1½ | |
| = 104¾ |
ARBITRAGE RISK
套利风险
New option traders are often instructed to go into the market and concentrate on executing conversions and reversals because these strategies are riskless. Beware: There are no riskless strategies. There are only strategies with greater or lesser risk. The risks of doing conversions or reversals may not be immediately apparent, but they exist nonetheless.
新手期权交易者通常被建议进入市场，专注于执行转换和反向转换，因为这些策略被认为是无风险的。请注意：没有无风险的策略，只有风险大小不同的策略。转换或反向转换的风险可能并不明显，但它们确实存在。
Interest Rate Risk
利率风险
Whenever there is a cash flow associated with a trade, the value of the trade depends on the interest which can be earned on a credit or which must be paid to carry a debit over the life of the trade. This is a function of interest rates, and since interest rates may not be constant, the interest considerations can change over time. If part of the expected profit from a trade depends on our earning 8% on a credit balance, a reduction in interest rates to 6% will certainly reduce our profit, and might even result in a loss. If part of the expected profit from a trade depends on our paying 7% on a debit balance, an increase in interest rates to 10% will likewise reduce our profit. In practice, large changes in interest rates over the limited life of most options (footnote 2: We are referring here primarily to exchange traded options with expirations up to approximately nine months. Long-term options with expirations of up to several years will obviously be much more sensitive to changes in interest rates.) are the exception rather than the rule. For this reason the interest rate risk associated with conversions and reversals is relatively small.
任何涉及现金流的交易，其价值都依赖于交易期间可以赚取的利息或需要支付的持仓成本。这取决于利率，而利率可能会变动，因此利息因素也可能随时间变化。如果部分预期利润基于8%的信用余额利息，而利率降至6%，那么利润会减少，甚至可能导致亏损。同样地，如果部分预期利润取决于7%的借方余额利率，而利率升至10%，也会降低利润。实际上，在大多数期权的有限到期期间内（脚注2: 这里主要指的是期限约九个月的交易所期权。对于期限长达数年的长期期权，对利率变动的敏感性显然会更高。）出现大幅利率变动的情况较少，因此转换和反向转换的利率风险相对较小。
Note also that synthetic strategies in the stock option market are much more sensitive to interest rates than the same strategies in the futures option market. A stock option conversion or reversal includes the cash flow from the underlying stock, while a similar trade in the futures option market includes no such cash flow. The cash flow from a stock trade is always greater than the cash flow from an option trade since the price of stock is always greater than the price of the options.
需要注意的是，股票期权市场中的合成策略对利率更为敏感，而期货期权市场中的相同策略则不然。股票期权的转换或反向转换包含来自股票的现金流，而期货期权市场中的类似交易则没有这种现金流。股票交易的现金流始终大于期权交易的现金流，因为股票价格通常高于期权价格。
Execution Risk
执行风险
Since no market participant wants to give away money, a trader is unlikely to be offered a profitable conversion or reversal all at one time. Consequently, he will have to execute one or two legs of the strategy, and hope to be able to execute the final leg(s) at a later time. He may, for example, initially purchase the underlying contract and puts, and later hope to complete the conversion by selling calls. However, if call prices begin to fall, he may never be able to profitably complete the conversion. Even a professional trader on an exchange, who would seem to be in a good position to know the prices of all three contracts, can be fooled. He may purchase a call and sell a put (synthetic long underlying) at what he thinks are good prices. However, when he tries to sell the underlying contract to complete the reversal, he may find that the price is much lower than what he thought it was. Anytime a strategy is executed one leg at a time, there is always the risk of an adverse change in prices before the strategy can be completed.
由于市场参与者都不愿亏钱，交易者不太可能一次性获得一个有利可图的转换或反向转换。因此，他需要先执行策略中的一到两部分，再期望稍后完成剩余的部分。例如，他可以先买入标的合约和认沽期权，稍后通过卖出认购期权来完成转换。然而，如果认购期权价格开始下跌，他可能无法有利地完成转换。即使是在交易所的专业交易者，也可能会出错。他可能会以为自己在较好价格买入了认购期权并卖出了认沽期权（合成标的多头），但在试图卖出标的合约以完成反向转换时，发现价格远低于预期。任何逐步完成的策略都有可能在完成之前因价格不利变动而带来风险。
Pin Risk
钉住风险
When we introduced the concept of a synthetic position, we assumed that at expiration either the underlying market would be above the exercise price, in which case any call would be exercised, or below the exercise price, in which case any put would be exercised. But what will happen if the underlying market is exactly equal, or pinned, to the exercise price at the moment of expiration?
在引入合成头寸的概念时，我们假设到期时标的市场在执行价格之上，这样认购期权将被行权；或者标的市场在执行价格之下，这样认沽期权将被行权。但如果到期时标的市场恰好与执行价格一致，出现“钉住”现象，会发生什么情况呢？
Suppose a trader has executed a June 100 conversion: he is short a June 100 call, long a June 100 put, and long the underlying contract. If the underlying market is above or below 100 at expiration, there is no problem. He will either be assigned on the call or will exercise the put. In either case he will offset his long underlying position so that he will have no market position on the day following expiration.
假设一位交易者执行了一个6月100的转换：他持有6月100的认购空头、认沽多头，同时持有标的合约多头。如果到期时市场价格高于或低于100，没有问题；他将被指派执行认购或行使认沽，这样他的标的多头头寸将在到期日次日被对冲掉。
Now suppose that at the moment of expiration the underlying market is right at 100. The trader would like to be rid of his underlying position. If he doesn't get assigned on the call, he plans to exercise his put; if he does get assigned on the call, he will let the put expire worthless. In order to make a decision, he must know whether the call will be exercised. Unfortunately, he won't know this until the day after expiration, when he either does or does not receive an assignment notice. But then it will be too late because the call will have expired.
但假设到期时市场价格正好是100。交易者希望抛售他的标的头寸。如果没有被指派行权认购期权，他计划行使认沽期权；如果被指派行权认购期权，他则让认沽期权到期作废。然而，他必须知道认购期权是否会被行权才能做出决定。不幸的是，直到到期日次日收到或不收到行权通知时，他才知道，但那时已为时已晚，因为认购期权已过期。
It may seem that an option which is exactly at-the-money at expiration will never be exercised since, in theory, it has no value. In fact many at-the-money options do get exercised. Even though such an option has no theoretical value, it does have some practical value. For example, suppose the owner of a call which is exactly at-the-money at expiration wants to take a long position in the underlying contract. He has two choices. He can either exercise the call or buy the underlying contract. Since most exchanges where options are traded include the right of exercise in the original transaction cost, it is almost always cheaper to exercise the call. Even if there is a charge for exercise, it will be less than the cost of trading the underlying contract. Anyone owning an at-the-money option, and choosing to take a long or short position at expiration, will find that it is cheaper to exercise the option than buy or sell the underlying contract.
看似到期正好处于平价的期权不会被行权，因为理论上它没有价值。但实际上，许多平价期权确实会被行权。即使这种期权没有理论价值，它仍然具有一些实用价值。例如，假设某人持有一份到期时处于平价的认购期权，并希望建立一个标的多头头寸。他可以选择行使认购期权或直接买入标的合约。由于大多数期权交易所包括行权成本在原始交易费用中，因此行使认购期权几乎总是更便宜。即使需要支付行权费用，也比直接交易标的合约成本低。因此，持有平价期权并在到期时选择多头或空头头寸的交易者会发现行使期权比直接买卖标的更便宜。
Clearly, the trader who is short an at-the-money option at expiration has a problem. What can he do?
显然，到期时持有平价期权空头的交易者会遇到麻烦。那么他该怎么办呢？
One course might be to guess whether the trader will be assigned. If the market appears to be strong on the last trading day, the trader might assume that it will continue higher on the day following expiration. If the person who holds the call sees the situation similarly, it is logical to assume that the call will be exercised. Hence, the trader will choose not to exercise the put. Unfortunately, if the trader is wrong, and he does not get assigned on the call, he will find himself with a long underlying position which he would rather not have. Conversely, if the market appears to be weak on the last trading day, the trader might make the assumption that he will not be assigned on the call. He will therefore choose to exercise the put. But again, if he is wrong and does get an assignment notice, he will find himself with an unwanted short underlying position on the day following expiration.
一种做法是猜测他是否会被指派行权。如果在最后交易日市场看涨，交易者可能会认为次日市场会继续上涨。因此，假设认购期权持有者也如此认为，可以假设认购期权将被行权。因此，交易者会选择不行使认沽期权。但如果他猜错了，且未被指派认购行权，他将持有一个不想要的标的多头。同样，如果市场在最后交易日看跌，交易者可能认为不会被指派认购行权，因此他选择行使认沽期权。然而，如果他猜错了且被指派认购行权，则次日他将持有不想要的标的空头。
The risk of a wrong guess can be further compounded by the fact that conversions and reversals, because of their low risk, are usually done in large size. If the trader guesses wrong, he may find himself naked long or short not one, but several hundred underlying contracts.
错误猜测的风险还可能因转换和反向转换的低风险特点而进一步放大，因为这些策略通常会大规模进行。如果交易者猜错了，他可能会裸多或裸空几百张标的合约。
There can be no certain solution to the problem of pin risk. With many, perhaps thousands, of open contracts outstanding, some at-the-money options will be exercised and some won't. If the trader lets the position go to expiration and relies on luck, he is at the mercy of the fates, and that is a position which an intelligent option trader prefers to avoid. The practical solution is to avoid carrying conversions and reversals to expiration when there is a real possibility of expiration right at the exercise price. If the trader has a large number of June 100 conversions or reversals, and expiration is approaching with the underlying market close to 100, the sensible course is to reduce the pin risk by reducing the size of the position. If the trader doesn't reduce the size, he may find that he is under considerable pressure to get out of a large number of risky positions at the last moment.
对于钉住风险，没有明确的解决方案。由于存在大量未平仓合约，有些平价期权会被行权，有些则不会。如果交易者让头寸到期并依赖运气，他将受制于命运，而这是一个聪明的期权交易者宁愿避免的情境。实际的解决办法是在到期价接近执行价时避免将转换和反向转换持仓到期。如果交易者持有大量6月100转换或反向转换，且临近到期市场价格接近100，合理的做法是通过减少头寸规模来降低钉住风险。如果交易者不缩减规模，他可能会发现自己在最后一刻不得不紧急退出大量风险头寸。
Sometimes even a careful trader will find that he has some at-the-money conversions or reversals outstanding as expiration approaches. One way to eliminate the pin risk which still exists is to liquidate the position at the prevailing market prices. Unfortunately, this is likely to be a losing proposition since the trader will be forced to trade each contract at an unfavorable price, either buying at the offer or selling at the bid. Fortunately, it is often possible to trade out of such a position all at once at a fair price.
即便是谨慎的交易者，有时也会在接近到期时持有一些平价转换或反向转换头寸。消除剩余钉住风险的一种方法是以当前市场价格平仓。然而，这通常会导致损失，因为交易者会被迫以不利的价格买卖合约——要么买在卖价，要么卖在买价。幸运的是，通常可以以公平价格一次性平仓这样的头寸。
Since conversions and reversals are common strategies in all option markets, a trader who has an at-the-money conversion and is worried about pin risk can be fairly certain that there are also traders in the market who have at-the-money reversals and are also worried about pin risk. If the trader with the conversion could find a trader with a reversal and cross positions with him, both traders would eliminate the pin risk associated with their positions. This is why, on option exchanges, one often finds traders looking for other traders who want to trade conversions or reversals at even money. This simply means that a trader wants to trade out of his position at a price which is fair to everyone involved, so that everyone can avoid the problem of pin risk. Whatever profit a trader expected to make from the conversion or reversal presumably resulted from the initial trade, not from the closing trade.
由于转换和反向转换在所有期权市场中都是常见策略，因此持有平价转换的交易者担心钉住风险时，可以确定市场中也有持有平价反向转换的交易者同样担心钉住风险。如果持有转换的交易者找到一个持有反向转换的交易者并与他交换头寸，则双方都能消除与钉住风险相关的头寸。这就是为什么在期权交易所中，经常可以看到交易者寻找其他交易者，希望在平价时进行转换或反向转换交易。这意味着交易者希望以对双方都公平的价格退出头寸，从而避免钉住风险的问题。假设转换或反向转换的预期收益来自最初的交易，而非平仓交易。
Certain options, such as stock indices and Eurodollars, are settled at expiration in cash rather than with the delivery of an actual underlying contract. When such an option expires, the amount of cash which flows into or out of a trader's account is simply the amount by which the option is in-the-money, i.e., the difference between the underlying price and the option's exercise price. There is no pin risk associated with options which settle in this manner because no underlying position results from exercise or assignment.
某些期权，如股票指数和欧洲美元，采用现金交割方式到期，而不是实际标的合约的交割。当这些期权到期时，进入或离开交易者账户的现金金额仅为期权的实值部分，即标的价格与期权执行价格之差。此类期权的行权或指派不会带来任何标的头寸，因此不存在钉住风险。
Interest rate risk, execution risk, and pin risk are common to all markets, regardless of the underlying contract. There are, however, certain risks which are unique to a market depending on the characteristics of the underlying market.
利率风险、执行风险和钉住风险在所有市场中都存在，而与标的合约无关。然而，有些风险是特定于市场的，取决于标的市场的特性。
Settlement Risk in the Futures Market
期货市场的结算风险
Let's go back to our original conversion where a trader is short a June 100 call, long a June 100 put, and long an underlying contract. Suppose the underlying contract is a June futures contract which is trading at 102.00. If there are three months remaining to June expiration, interest rates are 8%, and all options are subject to stock-type (cash) settlement, the value of the June 100 synthetic market (the difference between the June 100 call and the June 100 put) is:
futures price - exercise price - carry on 2 points for 3 months =
102 - 100 - (2 × 3/12 × 8% = 1.96
回到之前的转换案例，假设一位交易者持有6月100认购期权空头、6月100认沽期权多头和标的合约多头。假设标的合约为6月期货，当前交易价为102.00，距离6月到期还有三个月，利率为8%，所有期权都采用股票型（现金）结算方式。此时，6月100合成市场（即6月100认购期权和认沽期权的价差）的价值为：
期货价格 - 行权价格 - 2点的三个月持有成本
102 - 100 - (2 × 3/12 × 8%) = 1.96
Suppose a trader is able to sell a June 100 call for 5.00, buy a June 100 put for 3.00, and sell a June futures contract for 102.00. If interest rates do not change, and assuming there will not be a problem with pin risk, at expiration the trader should realize a profit of .04, since he has done the June 100 conversion at .04 better than its value.
假设交易者能够以5.00卖出6月100认购期权，以3.00买入6月100认沽期权，并以102.00卖出6月期货。若利率保持不变，且没有钉住风险问题，到期时该交易者应获得0.04的利润，因为他的6月100转换优于理论值0.04。
Suppose that shortly after the trader initiates this conversion, the underlying June futures contract falls to 98.00. The synthetic (short) side of the position will now show a profit of four points: the short call and long put together will appreciate by four points. But because the options are settled like stock, the profit on the synthetic side will only be a paper profit, and will not be fully realized until expiration. On the other hand, the trader is also long a June futures contract, and this contract, since it is subject to futures-type settlement, will result in an immediate four point debit when the market drops four points. To cover this four point debit, the trader will either have to borrow the money or take the money out of an existing, interest-bearing account. In either case there will be a loss in interest, and this Interest loss will not be offset by the paper profit from the option position. If the loss in interest is great enough, it may more than offset the profit of .04 which the trader originally expected from the position. And in the most extreme case, where the trader does not have access to the funds required to feed the futures position, the trader may be forced to liquidate the position. Needless to say, forced liquidations are never profitable.
假设交易刚建立后不久，6月期货价格下跌至98.00，合成头寸的空头部分将显示4点的账面利润：认购空头和认沽多头共计上涨了4点。但由于期权是股票型结算，这一账面利润无法完全实现，需等到到期。而交易者的6月期货多头会因期货型结算产生4点的即时账面亏损。为了弥补这4点亏损，交易者必须借款或从现有的利息账户中提取资金。在这种情况下，产生的利息损失不会被期权头寸的账面利润抵消。如果利息损失过大，可能会超出交易者原本预期的0.04利润。极端情况下，若交易者无法获得足够资金弥补期货头寸，他可能被迫平仓，而被迫平仓通常不利可图。
Of course, this works both ways. A rise in the price of the underlying futures contract to 106.00 will result in a four point loss to the synthetic side: the short call and long put together will decline by four points. But the loss on the synthetic side will only be a paper loss, and will not be fully realized until expiration. On the other hand, the rise in the futures contract will result in an immediate cash credit, a credit on which the trader can earn interest. This interest will increase his potential profit beyond the original expected profit of .04.
当然，这种情况也有相反的可能。如果期货价格上涨至106.00，合成头寸将显示4点的账面亏损：认购空头和认沽多头一共下跌了4点。但合成头寸的账面亏损直到到期才会实现，而期货头寸的上涨则会带来即时的现金收益，并可赚取利息。这部分利息会使交易者的潜在利润超过最初预期的0.04。
Most futures option traders think of conversions and reversals as being delta neutral, but in fact this is not necessarily true. With the underlying futures contract at 102, the deltas in our example might be:
| Option Position | Delta Position |
|---|---|
| short June 100 call | -60 |
| long June 100 put | -38 |
| long June futures contract | +100 |
| total | +2 |
多数期权交易者认为转换和反向转换是无风险的，但事实上并非如此。在102价位的期货合约下，本例中的delta可能如下：
| 期权头寸 | Delta头寸 |
|---|---|
| 6月100认购空头 | -60 |
| 6月100认沽多头 | -38 |
| 6月期货多头 | +100 |
| 合计 | +2 |
The two extra deltas reflect the fact that the trader would prefer the market to rise rather than fall, so that cash will flow into his account from the futures position. The interest from this cash flow can represent an unexpected profit or loss, in the case of a decline in the futures price.
这多出的2个delta表明交易者希望市场上涨，从而获得期货头寸的现金收益，而现金利息的变动会带来意外的盈利或亏损。
Two deltas are really no risk, except when we remember that conversions and reversals, because of their low risk, are often done in very large size. A trader who executes 500 conversions has a risk delta equal to 500 x +2 = +1000. This is the same as being naked long 10 futures contracts. The risk comes from the interest that can be earned on any cash credit, or which must be paid on any cash debit, resulting from movement in the underlying futures contract.
尽管2个delta的风险很小，但考虑到转换和反向转换交易往往大规模执行，这种风险可能被放大。若交易者执行500笔转换，他的风险delta相当于+1000，相当于裸多10份期货合约。风险源于标的合约波动带来的现金信用或现金借贷的利息收益或成本。
The amount by which the delta of a synthetic futures position will differ from 100 depends on the interest risk associated with the position. This in turn depends on two factors: the general level of interest rates and the amount of time remaining to expiration. The higher the interest rate and the more time remaining to expiration, the greater the risk. The lower the interest rate and the less time remaining to expiration, the less the risk. A 10% interest rate with nine months to expiration represents a much greater risk than a 4% interest rate with one month remaining to expiration. In the former case, the deltas of a synthetic position may add up to 94, while in the latter case the deltas may add up to 99.
合成期货头寸的delta与100的偏差取决于利息风险，受利率水平和到期时间的影响。利率越高、到期时间越长，风险越大；利率越低、到期时间越短，风险越小。比如，10%利率和9个月到期的风险远大于4%利率和1个月到期的情况。前者可能导致合成头寸的delta合计为94，而后者则可能接近99。
Note that there is no settlement risk when both contracts are subject to the same settlement procedure. If all contracts are subject to stock-type settlement, no cash flow results from fluctuations in the prices of the contracts prior to expiration. If all contracts are subject to futures-type settlement, any credit or debit resulting from changes in the price of the underlying futures contract will be offset by an equal but opposite cash flow from changes in prices of the option contracts.
如果所有合约采用相同的结算方式，则不会存在结算风险。若所有合约为股票型结算，价格波动不会带来现金流动；若所有合约为期货型结算，期货价格变动带来的现金流动会被期权价格变动产生的相反现金流抵消。
Dividend Risk in the Stock Market
股票市场的股息风险
Consider a stock trading at 102½, with three months to June expiration, interest rates at 8%, and a dividend of 1½ expected prior to expiration. The value of the June 100 synthetic (the difference between the 100 call and the June 100 put) is:
stock price - exercise price + carry on 100 to expiration - expected dividends =
102½ - 100 + (100 × 3/12 × 8%) - 1½ = 3
假设一只股票价格为102½，距6月到期有三个月，利率为8%，预计到期前将派发1½的股息。6月100合成市场（100认购和100认沽价差）的价值为：
股票价格 - 行权价格 + 100的持有成本 - 预期股息 =
102½ - 100 + (100 × 3/12 × 8%) - 1½ = 3
Suppose a trader is able to sell a June 100 call for 7¾ buy a June 100 put for 4½, and buy stock for 102½. If interest rates do not change, and assuming there will not be a problem with pin risk, at expiration the trader should realize a profit of ¼, since he has done the June 100 conversion at ¼ better than its value.
假设交易者以7¾卖出6月100认购，以4½买入6月100认沽，并以102½买入股票。若利率不变且无钉住风险，到期时交易者应获得¼的利润，因为其转换优于理论值0.25。
Since the trader owns the stock, part of his profit comes from the 1½ point dividend which he expects to receive when the stock goes ex-dividend. If the dividend changes unexpectedly, it could affect the trader's eventual profit. If, for example, the company is doing poorly and decides to cut its dividend in half, to ¾, the trader's conversion will be worth ¾ less, and his profit of ¼ will turn into a loss of ½. Of course, if the company is doing well and decides to increase its dividend to two points, the conversion will be worth ½ more, and the trader's profit will increase from ¼ to ¾. Clearly, the possibility of a change in the expected dividend represents a risk to a conversion or reversal. Moreover, if multiple dividends are expected over the life of the strategy, the impact of a change in the company's dividend policy can be greatly magnified.
由于交易者持有股票，利润的一部分来自1½点的预期股息。如果股息意外变化，将影响最终利润。例如，如果公司经营不佳并将股息减半至¾，交易者的转换价值将减少¾，原本的¼利润将转为½的亏损。当然，如果公司经营良好并将股息增加至2点，转换价值将增加½，利润将从¼增加至¾。显然，股息预期变化是转换或反向转换的风险因素，若在策略周期内有多次股息派发，股息政策变化的影响可能会被放大。
BOXES
盒式策略
As we have seen, a conversion or reversal entails risk because these strategies combine a synthetic position in options with a position in the underlying contract. The risk arises because a synthetic position in options and an actual position in the underlying contract can have different characteristics, either in terms of settlement procedure, as in the futures option market, or in terms of the dividend payout, as in the stock option market. How might we eliminate this risk?
正如我们所见，转换或反向转换策略具有一定风险，因为这些策略结合了期权的合成头寸和标的合约的头寸。风险源于期权的合成头寸和标的合约在结算方式上（如期货市场）或股息分配上（如股票期权市场）可能存在不同特性。那么如何消除这种风险呢？
One way to eliminate this risk is to eliminate the position in the underlying contract. Consider a conversion:
short a call
long a put
long an underlying contract
一种方法是消除标的合约的头寸。例如，在一个转换策略中：
卖出认购期权
买入认沽期权
买入标的合约
If we want to maintain this position, but would also like to eliminate the risk that goes with holding an underlying position, we might replace the long underlying position with something that acts like an underlying contract, but which isn't an underlying contract. For example, we might replace the long underlying position with a deeply in-the-money call. Now our position is:
short a call
long a put
long a deeply in-the-money call
如果想保留该头寸，又希望消除持有标的合约的风险，可以用其他类似于标的合约的头寸来替代它，比如换成深度实值认购期权。此时头寸变为：
卖出认购期权
买入认沽期权
买入深度实值认购期权
If the deeply in-the-money call has a delta of 100, and therefore acts like a long underlying contract, the position will have the same characteristics as the conversion.
若该深度实值认购期权的delta为100，表现如同持有标的合约，则该头寸的特性将与原转换头寸相同。
In the same way, instead of replacing the underlying position with a deeply in-the-money call, we can sell a deeply in-the-money put:
short a call
long a put
short a deeply in-the-money put
If the deeply in-the-money put has a delta of -100, and therefore acts like an underlying contract, the position will again have the same characteristics as the conversion.
同样，我们可以选择卖出深度实值认沽期权替代标的合约：
卖出认购期权
买入认沽期权
卖出深度实值认沽期权
如果深度实值认沽期权的delta为-100，即表现如同标的合约，则该头寸仍保持与转换相同的特性。
This type of position, where the underlying instrument in a conversion or reversal is replaced with a deeply in-the-money option, is known as a three-way. While eliminating some risks, a three-way is not without its own problems. If a trader sells a deeply in-the-money option to complete his three-way, he still has the risk of the market going through the exercise price. Indeed, as the underlying market moves closer and closer to the exercise price of the deeply in the money option, the option will act less and less like a position in the underlying contract. The combination of contracts will then act less and less like a true conversion or reversal.
这种用深度实值期权替代转换或反向转换策略中标的头寸的方式称为三方策略。虽然它减少了部分风险，但也带来一些问题。若交易者卖出深度实值期权以完成“三方策略”，仍面临价格穿越行权价的风险。随着市场接近该行权价，期权将逐渐不像标的合约，组合也会逐渐失去真实的转换或反向转换效果。
What else acts like an underlying contract but isn't an underlying contract? A synthetic underlying position obviously fills the bill. In addition to a three-way, we might also consider replacing an underlying position in a conversion or reversal with a synthetic equivalent. For example, our original position might be a 100 conversion:
short a June 100 call
long a June 100 put
long an underlying contract
是否还有其他类似标的合约的替代品？一个合成标的头寸显然满足这一需求。除了三方策略，我们还可以用合成等价头寸替代转换或反向转换中的标的头寸。例如，原始头寸可以是一个100转换：
卖出6月100认购期权
买入6月100认沽期权
买入标的合约
Suppose we also execute a reversal at 90:
| long a June 90 call | short a June 100 call |
| short a June 90 put | long a June 100 put |
| short an underlying contract | long an underlying contract |
假设我们还在90点位执行了一个反向转换：
| 买入6月90认购期权 | 卖出6月100认购期权 |
| 卖出6月90认沽期权 | 买入6月100认沽期权 |
| 卖出标的合约 | 买入标的合约 |
The long and short underlying contracts cancel out, leaving:
| long a June 90 call | short a June 100 call |
| short a June 90 put | long a June 100 put |
买入和卖出的标的合约互相抵消，剩下：
| 买入6月90认购期权 | 卖出6月100认购期权 |
| 卖出6月90认沽期权 | 买入6月100认沽期权 |
We are left with a synthetic long position at the 90 exercise price, and a synthetic short position at the 100 exercise price. Or we can think of the position as a reversal at one exercise price, and a conversion at a different exercise price.
这样形成了一个90行权价的合成多头头寸和100行权价的合成空头头寸，或看作是在不同行权价上建立的反向转换和转换。
This position, known as a box, is similar to a conversion or reversal, except that any risk associated with holding a position in the underlying contract has been eliminated because the underlying position has been replaced with a synthetic underlying position at a different exercise price. A trader is long (short) the box when he is synthetically long (short) the lower exercise price and synthetically short (long) the higher exercise price. The example position is long a June 90/100 box.
这种头寸称为盒式策略，与转换或反向转换类似，但因替换了标的合约，因此消除了持仓风险。例如该头寸为6月90/100盒式多头。
At expiration, regardless of the price of the underlying contract, a trader who has a box will simultaneously buy the underlying contract at one exercise price and sell the underlying contract at the other exercise price. The value of the box at expiration will therefore be exactly the amount between exercise prices. In our example, at expiration the 90/100 box will be worth exactly 10 points because the trader will simultaneously buy the underlying contract at 90 and sell it at 100. If it is worth 10 points at expiration, how much is it worth today? If the options are subject to stock-type settlement, the value of the box today will be the value of the box at expiration less carrying costs. If our 10-point box expires in three months with interest rates at 8%, the value today is:
10 - (10 × 3/12 × 8%) = 10 - .20 = 9.80
到期时，无论标的合约价格如何，持有盒式策略的交易者将同时以一个行权价买入标的合约，以另一个行权价卖出，最终该盒式策略的价值正好等于行权价之间的差额。上述90/100盒式策略到期时将价值10点，因为交易者会同时以90点买入标的合约、以100点卖出。若到期价值为10点，则今日价值为到期价值减去持有成本。若10点的盒式策略在三个月后到期、利率为8%，其当前价值为：
10 - (10 × 3/12 × 8%) = 10 - .20 = 9.80
Since a box eliminates the risk associated with carrying a position in the underlying contract, boxes are even less risky than conversions and reversals, which are themselves low-risk strategies. When all options are European (there is no risk of early exercise), and the options are settled in cash rather than through delivery of the underlying contract (there is no pin risk), the purchase or sale of a box is identical to lending or borrowing funds over the life of the options. In our example, a trader who sells a 10-point box for 9.80 has essentially borrowed funds from the buyer of the box for three months at an interest rate of 8%. If a trader were willing to pay a higher rate to borrow funds, he could sell the box at a lower price, for example 9.70. This corresponds to a rate over three months of 12%. When no other method is available, a trading firm might be able to raise needed short-term cash by selling boxes. Since the firm will probably have to sell the boxes at a price lower than the theoretical value, this will increase the firm's borrowing costs. Moreover, if the options can be exercised early, or if there is a danger of pin risk at expiration, this method of borrowing may not be without its own risks.
由于盒式策略消除了持仓风险，其风险比转换和反向转换更低。当所有期权为欧式（无提前行权风险）且期权为现金结算（无钉住风险），盒式策略等同于在期权生命周期内的借贷资金。例如，若交易者以9.80卖出一个10点的盒式策略，实际上是在8%利率下向买方借款三个月。若交易者愿意以更高利率借款，可以更低价格卖出盒式策略，例如9.70，相当于三个月利率为12%。当没有其他融资途径时，交易公司可以通过卖出盒式策略筹集短期现金。不过，由于通常需低于理论价格出售盒式策略，这会增加借款成本。此外，若期权可提前行权，或到期时有钉住风险，这种借款方式可能也会面临一定风险。
We originally introduced a box as a conversion at one exercise price and a reversal at a different exercise price. Since the long and short underlying positions cancel out, we are left with two synthetic underlying positions. The June 90/100 box was written:
| long a June 90 call | short a June 100 call |
| short a June 90 put | long a June 100 put |
我们最初介绍盒式策略是作为不同行权价上的转换和反向转换。由于多头和空头标的头寸抵消，只剩下两个合成标的头寸。6月90/100盒式策略写作如下：
| 买入6月90认购期权 | 卖出6月100认购期权 |
| 卖出6月90认沽期权 | 买入6月100认沽期权 |
The left side of the box is a synthetic long at 90 and the right side is a synthetic short at 100. Instead of dividing the box into right and left sides, we can also divide it into top and bottom sides. Notice that the top side is a bull vertical call spread (long June 90 call/short June 100 call, while the bottom side is a bear vertical put spread (long June 100 put/short June 90 put). Since a box is a combination of two vertical spreads, the combined prices of the vertical spreads must add up to the value of the box.
盒式策略的左侧是90的合成多头，右侧是100的合成空头。除了按左右分开，也可以按上下分解。注意到上侧是看涨垂直价差（买入6月90认购/卖出6月100认购），下侧是看跌垂直价差（买入6月100认沽/卖出6月90认沽）。盒式策略是两个垂直价差的组合，垂直价差的总价必须等于盒式策略的价值。
For example, with three months remaining to expiration and interest rates at 8%, the value of our June 90/100 box is 9.80. Suppose a trader knows that the June 90/100 vertical call spread is trading for six points. Without any additional information, the trader can estimate the fair market price of for the June 90/100 vertical put spread. He knows 90/100 box is worth 9.80, and that the value of a call and put vertical spread must add up to the value of the box. The price of the put spread must therefore be:
9.80 - 6 = 3.80
例如，在三个月到期、8%利率下，6月90/100盒式策略的价值为9.80。若交易者知道6月90/100认购垂直价差交易价为6点，则他可以估算6月90/100认沽垂直价差的公平市场价格。盒式策略价值为9.80，而认购和认沽垂直价差的总和等于盒式策略价值，因此认沽垂直价差的价格为：
9.80 - 6 = 3.80
If the trader believes he can either buy or sell the call vertical spread at six points, and he is asked for a market in the put spread, he will probably make his market around an assumed value of 3.80. He might, for example, make a market of 3.50 bid /4.10 ask. If he is able to buy the put vertical for 3.50, he can then try to buy the call vertical for 6.00. If he is successful, he will have paid a total of 9.50 for a box with a theoretical value of 9.80. Conversely, if he is able to sell the put vertical for 4.10, he can then try to sell the call vertical for 6.00. If he is successful, he will have sold a box with a theoretical value of 9.80 at a price of 10.10.
若交易者可以以6点买入或卖出认购垂直价差，同时被要求对认沽垂直价差报价，则可能会以3.80为基础。例如，他可以报价3.50买入/4.10卖出。若能以3.50买入认沽垂直价差，再以6.00买入认购垂直价差，则其总成本为9.50，而盒式策略理论值为9.80。反之，若能以4.10卖出认沽垂直价差，再以6.00卖出认购垂直价差，则以10.10卖出一个理论价值为9.80的盒式策略。
JELLY ROLLS
果冻卷策略
Another method of eliminating a position in the underlying contract is to take a synthetic position in a different expiration month, rather than at a different exercise price as with a box. For example, suppose we have executed the following reversal:
long a June 100 call
short a June 100 put
short an underlying contract
Suppose we now execute the following conversion:
short a September 100 call
long a September 100 put
long an underlying contract
If the underlying contract for both June and September is identical, they will cancel out, leaving us with:
| long a June 100 call | short a September 100 call |
| short a June 100 put | long a September 100 put |
These combined long and short synthetic positions taken at the same exercise prices but in different expiration months is known as a jelly roll, (footnote 3: This very unscientific sounding term seems to have originated among traders on the Chicago Board Options Exchange.) or simply a roll. The strategy is most common in the stock option market where the underlying for both months is the same.
另一种消除标的头寸的方法是选择不同到期月份的合成头寸，而非像盒式策略那样使用不同的行权价。例如，假设我们已执行以下反向转换：
买入6月100看涨期权
卖出6月100看跌期权
卖出标的合约
然后我们再执行以下转换：
卖出9月100看涨期权
买入9月100看跌期权
买入标的合约
如果6月和9月的标的合约相同，它们将相互抵消，留下：
| 买入6月100看涨期权 | 卖出9月100看涨期权 |
| 卖出6月100看跌期权 | 买入9月100看跌期权 |
这种在相同行权价但不同到期月份采取的合成多空头寸组合称为果冻卷，或简称“卷”。这一策略在标的相同的股票期权市场中最为常见。
What should be the value of a roll in the stock option market? In our example, the sale of the June/September 100 roll, the trader will buy stock at 100 at June expiration and sell the same stock at 100 at September expiration. The value of the roll is the cost of holding the stock for the three month period from June to September. If interest rates are 8%, the holding costs are:
100 × 3/12 x 8% = 2.00
Therefore, the roll should have a value of two points. Expressed another way, the difference between the June 100 synthetic (the June 100 call less the June 100 put) and the September 100 synthetic (the September 100 call less the September 100 put) should be exactly two points.
在股票期权市场中，卷的价值应为多少？在我们的例子中，出售6月/9月100卷，交易者将在6月到期时以100买入股票，在9月到期时以100卖出股票。卷的价值即为从6月到9月持有股票的成本。如果利率为8%，持有成本为：
100 × 3/12 x 8% = 2.00
因此，卷的价值应为两点。换句话说，6月100合成（6月100看涨期权减去6月100看跌期权）和9月100合成（9月100看涨期权减去9月100看跌期权）的差异应恰好为两点。
Now suppose that the stock pays a quarterly dividend of ¾ point. Since the trader will own the stock for the period from June to September, he will receive the ¾ point dividend. The value of the roll is now the two-point carrying cost less the ¾ point dividend, or 1¼. The full value of the roll is:
| jelly roll | = long term synthetic - short-term synthetic |
| = cost of carry - expected dividends |
where the cost of carry is calculated on the exercise price over the period between expiration months.
假设股票每季度支付0.75点的股息。由于交易者在6月至9月期间持有股票，他将收到0.75点的股息。卷的价值现在为2点持有成本减去0.75点股息，即1.25点。卷的完整价值为：
| 果冻卷 | = 长期合成 - 短期合成 |
| = 持有成本 - 预期股息 |
其中持有成本基于行权价并按照到期月之间的时间计算。
In the same way that we evaluated a box as the combination of two vertical spreads, we can also evaluate a jelly roll as the difference between two time spreads:
| long a June 100 call | short a September 100 call |
| short a June 100 put | long a September 100 put |
与我们将盒式策略视为两个垂直价差的组合相似，果冻卷也可以看作是两个时间价差的差额：
| 买入6月100看涨期权 | 卖出9月100看涨期权 |
| 卖出6月100看跌期权 | 买入9月100看跌期权 |
This is simply a short June/September 100 call time spread and a long June/September put time spread. Since the roll is a combination of the sale of a call (put) time spread and the purchase of put (call) time spread, the values of the call and put times spreads should differ by exactly the value of the roll. If the roll is worth 1, the difference between the call and put time spreads should also be 1/4. If the call time spread is trading for 3/, the put time spread should be trading for 2. This is logical when we realize that we can rewrite the jelly roll in a slightly different form:
| jelly roll | = long-term synthetic - short-term synthetic |
| = (long-term call - long-term put) - (short-term call - short-term put) | |
| = (long-term call - short-term call) - (long term put - short-term put) | |
| = carrying costs - expected dividends |
这只是一个短6月/9月100看涨期权时间价差和一个长6月/9月看跌期权时间价差。由于果冻卷是卖出看涨（看跌）时间价差并买入看跌（看涨）时间价差的组合，看涨和看跌时间价差的差值应正好等于果冻卷的价值。如果果冻卷价值为1，那么看涨和看跌时间价差的差异也应为1/4。如果看涨时间价差的交易价为3/4，那么看跌时间价差的交易价应为2。这可以理解为，我们可以用另一种形式来表达果冻卷：
| 果冻卷 | = 长期合成 - 短期合成 |
| = (长期看涨 - 长期看跌) - (短期看涨 - 短期看跌) | |
| = (长期看涨 - 短期看涨) - (长期看跌 - 短期看跌) | |
| = 持有成本 - 预期股息 |
The difference between the call and put time spreads should therefore be equal to the cost of carry on the exercise price less the expected dividends.
因此，看涨和看跌时间价差之间的差额应等于行权价的持有成本减去预期股息。
Since jelly rolls involve holding stock positions for some period of time, they have risks similar to those affecting conversions and reversals. If interest rates rise, or the dividend is cut, a jelly roll will become more valuable. If interest rates fall, or the dividend is increased, a jelly roll will become less valuable.
由于果冻卷涉及一段时间的持仓，它的风险类似于转换和反向转换。如果利率上升或股息减少，果冻卷将更有价值；反之，如果利率下降或股息增加，果冻卷的价值将减少。
Finally, we can create a time box by taking opposing synthetic positions at different exercise prices and expiration dates:
| long a June 90 call | short a September 100 call |
| short a June 90 put | long a September 100 put |
The value of this position is the combined value of purchasing the June 90/100 box and selling the June/September 100 jelly roll. If the June 90/100 box is worth 9¾, and the June/September 100 jelly roll is worth 1½, the time box should be worth 8½.
最后，我们可以通过在不同的行权价和到期日采取对立的合成头寸来创建时间盒：
| 买入6月90看涨期权 | 卖出9月100看涨期权 |
| 卖出6月90看跌期权 | 买入9月100看跌期权 |
该头寸的价值等于购买6月90/100盒与卖出6月/9月100果冻卷的组合价值。如果6月90/100盒的价值为9¾，而6月/9月100果冻卷的价值为1½，则时间盒的价值应为8½。
USING SYNTHETICS IN VOLATILITY SPREADS
利用合成头寸进行波动性价差
A trader need not restrict himself to using synthetics only in arbitrage strategies. He can also execute a volatility or directional strategy using a synthetic. Consider the following situation. Dealing in a futures option market where the underlying March futures contract is trading very actively at 100.00, a trader wants to execute the following backspread for a one-point credit:
buy 20 March 105 calls
sell 10 March 100 calls
交易者不必将合成头寸仅限于套利策略，也可以利用合成头寸执行波动性或方向性策略。假设在一个期货期权市场中，标的3月期货合约的交易价格非常活跃，为100.00。此时交易者希望以1点的收益进行以下反向价差：
买入20张3月105看涨期权
卖出10张3月100看涨期权
That is, the amount he expects to receive for the sale of one March 100 call less the amount he expects to pay for two March 105 calls should equal one point. Suppose the current market in these options is:
| Option | Bid/Ask |
| March 100 call | 2.70/2.90 |
| March 105 call | .85/.95 |
也就是说，他预期从卖出一张3月100看涨期权中获得的收益减去买入两张3月105看涨期权的成本应等于1点。假设这些期权当前的市场报价为：
| 期权 | 买入/卖出价 |
| 3月100看涨期权 | 2.70/2.90 |
| 3月105看涨期权 | .85/.95 |
If the trader were able sell the March 100 calls at the ask price of 2.90, he would be willing to pay the ask price of .95 for the March 105 calls, since 2.90 - (2 x.95) = 1.00. In the same way, if he were able to buy the March 105 calls at the bid price of .85, he would be willing to sell the March 100 calls at the bid price of 2.70, since 2.70 - 12 x .85) = 1.00.
如果交易者能够以2.90的卖出价卖出3月100看涨期权，则他愿意以0.95的买入价买入3月105看涨期权，因为2.90 - (2 x .95) = 1.00。同样，如果他能以0.85的买入价买入3月105看涨期权，他也愿意以2.70的买入价卖出3月100看涨期权，因为2.70 - (2 x .85) = 1.00。
Suppose that an offer comes into the marketplace to sell March 105 cails at .85. The trader immediately buys these calls and is about to offer the March 100 calls at 2.70, when suddenly a bid of 2.80 is made for March 100 puts. Does this affect what the trader might do?
假设市场出现一个卖出报价，以0.85出售3月105看涨期权，交易者立即买入这些看涨期权，并计划以2.70出售3月100看涨期权。这时，突然出现2.80的3月100看跌期权买入价。这会影响他的策略吗？
Since the new bid is for the March 100 put, and the trader is interested in selling the March 100 call, it may seem that the put bid will have no affect on his strategy. But recall the synthetic relationship:
synthetic short call = short put + short underlying
In the futures market, the value of the call is:
call price = put price + futures price - exercise price - carrying costs
If the underlying March futures contract is trading at 100, there is no carrying cost associated with a March 100 synthetic because the difference between the futures price and exercise price is zero. Therefore:
March 100 call = March 100 put + 100 - 100 - 0
March 100 call = March 100 put
由于新的买入报价是针对3月100看跌期权，而交易者关心的是卖出3月100看涨期权，看跌期权的买入报价似乎与他的策略无关。但请回想一下合成关系：
合成卖出看涨期权 = 卖出看跌期权 + 卖出标的合约
在期货市场中，看涨期权的价值为：
看涨期权价格 = 看跌期权价格 + 期货价格 - 行权价格 - 持有成本
如果标的3月期货合约的交易价格为100，由于期货价格和行权价格之间的差额为零，因此3月100合成头寸没有持有成本。因此：
3月100看涨期权 = 3月100看跌期权
In other words, if the trader believes he can trade the March futures contract at 100, then the prices of the March 100 call and put ought to be identical. He originally intended to sell the March 100 call at 2.70 to complete his backspread. He is now given the chance to sell the March 100 call, synthetically, at 2.80. If he sells the March 100 put at 2.80, and simultaneously sells the March futures contract at 100.00, he has effectively sold the March 100 call at 2.80, which is .10 better than he originally intended
换句话说，如果交易者认为他可以以100的价格交易3月期货合约，那么3月100看涨期权和看跌期权的价格应相同。他原本打算以2.70的价格卖出3月100看涨期权来完成他的反向价差，现在他有机会通过合成卖出3月100看涨期权，以2.80的价格卖出，这比他预期的要多0.10。
Whenever a trader is considering a strategy, he ought to always ask himself whether he can do better by executing some part of his strategy synthetically. Usually he won't be able to because synthetic relationships tend to be very efficient. But every now and then the trader will find that the synthetic position is slightly more favorable. Over a career of trading even small savings can add up.
每当交易者考虑一个策略时，他都应当问自己是否可以通过合成执行某部分策略来获得更好的收益。通常他可能无法做到，因为合成关系往往非常高效。但偶尔会发现合成头寸稍微更有利。经过长时间交易，小的节省也会累积成显著收益。
Using synthetic equivalents, we can also rewrite some volatility spreads in more familiar terms. For example, we said that call and put butterflies with the same exercise prices are identical. Suppose we take a typical call butterfly and rewrite it in terms of its synthetic equivalents:
| Contract | Synthetic Equivalent |
|---|---|
| long 1 March 95 call | long 1 March 95 put/long an underlying contract |
| short 2 March 100 calls | short 2 March 100 puts/short 2 underlying contracts |
| long 1 March 105 call | long 1 March 105 put/long an underlying contract |
利用合成等价物，我们还可以将一些波动性价差重新表达成更熟悉的形式。例如，我们曾提到具有相同行权价的看涨和看跌蝶式期权是相同的。假设我们取一个典型的看涨蝶式期权，用其合成等价物重新表达：
| 合约 | 合成等价物 |
|---|---|
| 买入1张3月95看涨期权 | 买入1张3月95看跌期权/买入一个标的合约 |
| 卖出2张3月100看涨期权 | 卖出2张3月100看跌期权/卖出2个标的合约 |
| 买入1张3月105看涨期权 | 买入1张3月105看跌期权/买入一个标的合约 |
Since the long and short underlying contracts in our synthetic equivalent cancel out, we are left with a put butterfly:
long 1 March 95 put
short 2 March 100 puts
long 1 March 105 put
由于合成等价物中的标的合约相互抵消，我们剩下一个看跌蝶式期权：
买入1张3月95看跌期权
卖出2张3月100看跌期权
买入1张3月105看跌期权
Using synthetic equivalents, we can also express the following strategy in a more familiar form:
long 2 March 100 calls
short an underlying contract
利用合成等价物，我们还可以将以下策略表达成更熟悉的形式：
买入2张3月100看涨期权
卖出一个标的合约
This is simply a long straddle, since we can take one of the March 100 calls and rewrite it synthetically:
long March 100 call = long March 100 put + long underlying contract
leaving:
long 1 March 100 call
short an underlying contract
long 1 March 100 put
long an underlying contract
这实际上是一个长跨式期权，因为我们可以将其中一个3月100看涨期权重新合成：
买入3月100看涨期权 = 买入3月100看跌期权 + 买入标的合约
得到：
买入1张3月100看涨期权
卖出一个标的合约
买入1张3月100看跌期权
买入一个标的合约
Since the long and short underlying contracts cancel out, we are left with a typical long straddle:
long 1 March 100 call
long 1 March 100 put
由于标的合约的多头和空头相互抵消，我们剩下一个典型的长跨式期权：
买入1张3月100看涨期权
买入1张3月100看跌期权
TRADING WITHOUT THEORETICAL VALUES
无需理论价值的交易
Synthetic relationships can often enable a trader to make logical trading decisions without the aid of values generated by a theoretical pricing model. Usually all that is required is an ability to calculate the carrying cost on a position and, in the case of stocks, to determine the dividend payout.
合成关系可以让交易者在没有理论定价模型的情况下做出逻辑性的交易决策。通常只需计算头寸的持有成本，并在涉及股票时确定股息支付即可。
While a trader can't always be as certain of a profit as he is with arbitrage strategies such as conversions, reversals, boxes, and jelly rolls, there are many logical relationships between options and underlying contracts, or between options and other options, which enable a trader to identify potentially profitable trading strategies without the use of a theoretical pricing model.
虽然交易者无法像在转换、反转、箱形或果冻卷套利策略中那样确定获利机会，但选项与标的合约之间或选项与其他选项之间存在许多逻辑关系，这些关系可帮助交易者识别潜在的获利机会，而无需理论定价模型的支持。
Consider the following option prices with the underlying price at 101.50:
| 95 call | 100 call | 105 call |
| 8.00 | 4.80 | 1.60 |
Is there anything wrong with these prices?
假设当前标的价格为101.50，期权价格如下：
| 95看涨期权 | 100看涨期权 | 105看涨期权 |
| 8.00 | 4.80 | 1.60 |
这些价格是否存在问题？
If the trader purchases one 95 call for 8.00, sells two 100 calls at 4.80, and purchases one 105 call for 1.60, he has bought the 95/100/105 butterfly for zero since:
(2 × 4.80) - (8.00 + 1.60) = 9.60 - 9.60 = 0
Since a butterfly can never be worth less than zero at expiration, and in this case might be worth as much as five points, the purchase of this butterfly represents a riskless trading opportunity.
如果交易者购买一张95看涨期权（价格8.00）、卖出两张100看涨期权（价格4.80），并购买一张105看涨期权（价格1.60），他会获得95/100/105蝶式价差的净成本为零，因为：
(2 × 4.80) - (8.00 + 1.60) = 9.60 - 9.60 = 0
由于蝶式价差在到期时不可能低于零，且最高可达5点，因此这种买入策略构成了一个无风险的交易机会。
Another way of reaching the same conclusion is to realize that a butterfly is made up of two consecutive vertical spreads. In our example, the purchase of the 95/100/105 butterfly consists or purchasing the 95/100 vertical call spread and selling the 100/105 vertical call spread. We know that as the market rises, a vertical call spread goes more deeply into-the-money and becomes more valuable. Therefore, as long as there is an equal amount between exercise prices, a vertical call spread with lower exercise prices will always be more valuable than a vertical call spread with higher exercise prices. The opposite is true of vertical put spreads. Those with higher exercise prices are always more valuable than those with lower exercise prices. If two vertical spreads violate this relationship, a trader can take advantage of the situation by purchasing the spread which ought to be more valuable, and selling the spread which ought to be less valuable. In our example, the vertical call spreads are:
| 95/100 spread | = | 8.00 - 4.80 | = | 3.20 |
| 100/105 spread | = | 4.80 - 1.60 | = | 3.20 |
另一种得出相同结论的方法是理解蝶式价差由两个连续的垂直价差构成。在本例中，买入95/100/105蝶式价差即意味着买入95/100垂直看涨价差并卖出100/105垂直看涨价差。我们知道，随着市场上涨，垂直看涨价差会更深入价内并变得更有价值。因此，只要行权价间隔相同，较低行权价的垂直看涨价差总是比较高行权价的更有价值。而看跌价差则相反，较高行权价的比较低行权价的更有价值。如果两个垂直价差不符合此关系，交易者可以利用这个差异，买入应更有价值的价差并卖出应较便宜的价差。在本例中，垂直看涨价差为：
| 95/100价差 | = | 8.00 - 4.80 | = | 3.20 |
| 100/105价差 | = | 4.80 - 1.60 | = | 3.20 |
While both spreads are trading at the same price, the 95/100 spread is intuitively more valuable than the 100/105 spread. Consequently, a trader will try to purchase the 95/100 spread and sell the 100/105 spread. If he succeeds, he will have purchased the butterfly for zero. While there is no guarantee that this will be profitable (the underlying market could finish below 95 or above 105), the strategy still makes sense because there is no risk associated with it.
虽然两个价差的价格相同，但95/100价差直观上比100/105价差更有价值。因此，交易者会尝试买入95/100价差并卖出100/105价差。如果成功，他将以零成本获得蝶式价差。虽然无法保证一定获利（标的市场可能最终低于95或高于105），但由于无风险，这一策略仍然合理。
Consider a different type of relationship:
Underlying price = 99.75 Interest rate = 0
| 95 | 100 | 105 | |
|---|---|---|---|
| calls: | 6.85 | 3.70 | 1.10 |
| puts: | 2.10 | 3.95 | 6.35 |
考虑另一种关系：
标的价格 = 99.75 利率 = 0
| 95 | 100 | 105 | |
|---|---|---|---|
| 认购期权: | 6.85 | 3.70 | 1.10 |
| 认沽期权: | 2.10 | 3.95 | 6.35 |
First, we might check to make sure that all the synthetic relationships balance. Since they do, there is no profit opportunity there. Next, we might check the butterflies. Both the call and put butterflies are trading for .55. This might or might not be a reasonable price; without a theoretical pricing model it would be difficult to say. But at least the butterflies are trading for some positive number, so it is not immediately apparent that they represent a good trading opportunity. Is there any other logical relationship that might be violated?
首先，我们可以检查所有合成关系是否平衡。既然平衡，就没有套利机会。接下来，可以查看蝶式价差。认购和认沽蝶式价差均为0.55，这个价格是否合理暂且不论；没有定价模型难以判断。但至少蝶式价差的价格是正的，因此还不能确定它们是一个好的交易机会。是否还有其他逻辑关系可能被违反？
Look at the prices of the straddles:
| 95 | 100 | 105 | |
|---|---|---|---|
| call + put: | 8.95 | 7.65 | 7.45 |
再看看跨式价差的价格：
| 95 | 100 | 105 | |
|---|---|---|---|
| 认购期权+认沽期权: | 8.95 | 7.65 | 7.45 |
We know that a straddle becomes more valuable as the underlying price moves away from the exercise price, resulting in one side of the straddle, either the call or put, going more deeply into-the money. With the underlying price at 99.75, the 95 straddle should be more valuable than the 100 straddle, and this is reflected in the respective prices of 8.95 and 7.65. The 105 straddle, being further away from 99.75, should also be more valuable than the 100 straddle. But here the prices do not reflect the additional value. The 105 straddle is .30 cheaper than the 100 straddle. Even though we don't know the theoretical value of either the 100 or 105 straddle, we do know that the 105 straddle is too cheap with respect to the 100 straddle. Therefore, if given the opportu-nity, a trader would attempt to purchase the 105 straddle at 7.45 and sell the 100 straddle at 7.65. There is no guarantee that this strategy will be profitable. The underlying contract could finish above 105, in which case the 100 straddle will be worth five points more than the 105 straddle. But if we assume that movement is random, then the sale of the 100 straddle and the purchase of the 105 straddle puts the laws of probability heavily on the trader's side.
我们知道，随着标的价格远离行权价，跨式价差的价值会增加，因为跨式价差的某一边（认购或认沽）将更深度价内。当前标的价格为99.75，因此95跨式价差应比100跨式价差更有价值，这在价格上有所体现，分别为8.95和7.65。同理，105跨式价差距离99.75更远，理论上应比100跨式价差更有价值。但这里的价格并未反映这一点——105跨式价差比100跨式价差便宜0.30。即使我们不知道100或105跨式价差的理论价格，也可以确定105跨式价差相对100跨式价差过于便宜。因此，如果有机会，交易者可能会选择以7.45买入105跨式价差并以7.65卖出100跨式价差。这一策略并不能保证盈利，因为标的价格可能会收于105以上，此时100跨式价差将比105跨式价差高出五点。然而，若假设价格波动是随机的，那么卖出100跨式价差并买入105跨式价差将显著提高交易者获利的概率。
Finally, consider this scenario:
Underlying price = 100.75 Interest rate = 0
| 95 | 100 | 105 | |
|---|---|---|---|
| March calls: | 7.50 | 3.85 | 2.35 |
| June calls: | 9.65 | 5.70 | 3.30 |
Is there any apparent mispricing here?
最后，考虑以下情形：
标的价格 = 100.75 利率 = 0
| 95 | 100 | 105 | |
|---|---|---|---|
| 三月认购: | 7.50 | 3.85 | 2.35 |
| 六月认购: | 9.65 | 5.70 | 3.30 |
这里有明显的错价吗？
First, we might look at the butterflies. The March butterfly is trading for (7.50 + 2.35) - (2 × 3.85) = 2.15. The June butterfly is trading for (9.65 + 3.30) - (2 × 5.70) - 1.55. Since both butterflies are trading for some positive amount, it isn't immediately clear that we should buy or sell elther one? What about the prices of the butterflies with respect to each other? Should the June butterfly, at 1.55, be less expensive than the March butterfly, at 2.15? It may seem that the June butterfly, with more time to expiration, ought to be more expensive. In option evaluation we usually associate more time with greater value. But if the market remains at 100.75, the price of the butterfly will expand towards 4.25 as time passes. Hence, it is logical that the March butterfly is more valuable than the June butterfly.
首先，我们可以观察蝶式价差。三月蝶式价差的价格为(7.50 + 2.35) - (2 × 3.85) = 2.15。六月蝶式价差的价格为(9.65 + 3.30) - (2 × 5.70) = 1.55。由于两个蝶式价差都有正值，目前不清楚是否应该买入或卖出任何一个。接下来比较它们的价格，六月蝶式价差（1.55）应比三月蝶式价差（2.15）便宜吗？从理论上看，剩余时间越长，期权价值通常越高。然而，如果市场保持在100.75，蝶式价差价格会随着时间接近4.25。因此，三月蝶式价差更贵是合理的。
Is there any other relationship we might consider? When we see options with different expiration dates, time spreads certainly come to mind.
| 95 | 100 | 105 | |
|---|---|---|---|
| June call less March call: | 2.15 | 1.85 | .95 |
Is there anything wrong with these prices?
是否还有其他值得考虑的关系？当我们看到具有不同到期日的期权时，时间价差自然浮现。
| 95 | 100 | 105 | |
|---|---|---|---|
| 六月认购 - 三月认购： | 2.15 | 1.85 | 0.95 |
这些价格有问题吗？
We know that a time spread maintains its greatest value with the underlying contract at the exercise price. The 100 time spread, being more at-the-money, is trading for more than the 105 time spread, as one would expect. The 100 time spread should also be more valuable than the 95 time spread. Here the 95 time spread is trading at a greater price, and, again, if we assume random movement in the underlying contract, this is clearly not logical. Without a theoretical pricing model, we can't be sure what each of these spreads is worth. But we do know that the 95 call time spread is too expensive with respect to the 100 call time spread. If we sell the former and buy the latter we can't be certain of a profit, but we do have the laws of probability on our side.
我们知道，当标的价格接近行权价时，时间价差的价值最大。100时间价差更接近平值，价格高于105时间价差，符合预期。理论上，100时间价差也应高于95时间价差。然而，95时间价差价格反而更高。如果我们假设标的价格随机波动，这显然不合理。虽然我们无法确定这些价差的理论价值，但可以断定95认购时间价差相对过高，而100认购时间价差相对便宜。卖出95认购时间价差并买入100认购时间价差，尽管不能确保盈利，但概率对交易者有利。
Note that in each of these examples we made an important assumption:
Regardless of the exact theoretical value, there ought to be a uniform progression of both individual option prices and spread prices in the marketplace. If this uniform progression is violated, a trader can take advantage of the situation by purchasing the option or spread which is relatively cheap and selling the option or spread which is relatively expensive.
请注意，在以上例子中，我们作了一个重要假设：
无论确切的理论值如何，市场上个别期权和价差价格应呈现一定的规律。如果这种规律被打破，交易者可以通过买入相对便宜的期权或价差，卖出相对昂贵的，来利用这种情况。
While the great majority of option traders work with theoretical values, a good habit for the new trader is to quickly check the relattonship of option prices and spreads in the market to ensure that there are no obvious mispricings. The trader can start with conversions and reversals, then look at vertical spreads and butterflies, and finally consider straddles and time spreads. Usually there will not be any obvious mispricing. But if there is, the trader can exploit the situation by purchasing the cheap side of the relationship and selling the expensive.
大多数期权交易者依赖理论值，但新手交易者可以养成一个好习惯，快速检查市场中的期权和价差关系，确保没有明显的错价。交易者可以从转换和反向策略开始，接着观察垂直价差和蝶式价差，最后考虑跨式和时间价差。通常不会发现明显错价，但如果有，交易者可以通过买入便宜的一方、卖出昂贵的一方来获利。
Two evaluation tables, demonstrating typical progressions of option prices and spreads in a stock option and futures option market, are shown in Figures 11-4 and 11-5.
图11-4和图11-5展示了股票期权和期货期权市场中期权和价差价格的典型变化进程。
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