Chapter 12 Early Exercise of American Options
第十二章 提前行使美式期权
Thus far we have assumed that all option strategies involve holding a position to expiration. Since the great majority of exchange traded options are American, carrying with them the right of early exercise, it will be worthwhile to consider some of the characteristics of American options. Specifically, we will want to answer two questions:
- Given the opportunity, under what circumstances might a trader consider exercising an American option prior to expiration?
- How much more should a trader be willing to pay for an American option over an equivalent European option?
到目前为止，我们假设所有的期权策略都涉及持仓到期。由于大部分交易所交易的期权为美式期权，拥有提前行权的权利，因此我们有必要探讨美式期权的一些特性。特别是我们需要回答两个问题：
- 在什么情况下，交易者可能会考虑提前行权美式期权？
- 交易者愿意为美式期权多支付多少溢价来超过等值的欧式期权？
In order for early exercise to be desirable, there must be some positive cash flow which will result from early exercise, and the value of this cash must be greater than the insurance value of the option. Because the cash flow which results from the exercise of a stock option is significantly different from the cash flow which results from the exercise of a futures option, the conditions under which early exercise will be desirable are different. For this reason, we will look at these two situations separately.
提前行权是否有利，取决于提前行权带来的正向现金流是否大于期权的保险价值。由于股票期权和期货期权行权带来的现金流不同，提前行权的条件也不同，因此我们将分别探讨这两种情况。
FUTURES OPTIONS
期货期权
From previous discussions (see Chapter 6) we know that in evaluating an option there are some factors which make the option more valuable and there are some factors which make the option less valuable. Consider a futures option. We might list the factors which affect its value as follows:
option value = intrinsic value + volatility value - interest rate value
根据之前的讨论（见第六章），在评估期权时，有些因素会增加期权价值，而有些因素会减少期权价值。考虑期货期权，其价值影响因素如下：
期权价值 = 内在价值 + 波动率价值 - 利率价值
Since the intrinsic value and volatility components can never be less than zero, these factors always enhance the option's value. As either rises, the option value rises. Only the interest rate component might affect the option's value negatively. As interest considerations rise, the option's value falls. If the negative effects of interest rates are greater than the positive effects of volatility, it might be possible for an option, if it is European, to be worth less than parity. (Footnote 1: Note that a European option with a theoretical value less than parity will have a positive theta. Its value will gradually rise to parity as expiration approaches.) In such a case, if the option were an American option it would become an early exercise candidate.
由于内在价值和波动率部分不会为负，因此这些因素总是增加期权价值。只有利率因素可能对期权价值产生负面影响。随着利率上升，期权价值下降。如果利率的负面影响大于波动率的正面影响，那么欧式期权的理论价值可能低于平价。（脚注1：理论价值低于平价的欧式期权将有正theta值，其价值会在到期日临近时逐渐升至平价。）在这种情况下，如果该期权是美式期权，就会成为提前行权的候选。
For example, suppose a certain futures contract is trading at 100. Suppose also that we own an 80 call which will expire in two months, and that this option is subject to stock-type settlement. When we evaluate the option, we find that it has a theoretical value of 20 and a delta of 100. This means that the option has essentially the same characteristics as the futures contract. If the option is American, and we want to maintain the same delta position, we have three possible choices:
- Hold the option
- Exercise the option
- Sell the option and buy a futures contract
Which of these choices is best?
例如，假设某期货合约的价格为100，假设我们持有一张将在两个月后到期的80看涨期权，并且该期权采用股票型结算方式。经评估，期权的理论价值为20，delta为100。这意味着该期权与期货合约特性相同。如果期权是美式期权，并且我们希望维持相同的delta头寸，有三种选择：
- 持有期权
- 行权期权
- 卖出期权并购买期货合约
哪种选择更好？
If we choose the first alternative and hold the position, there is no change to our account. The position we take home is identical to the one with which we begin the next trading day.
如果选择第一种，持有期权头寸，账户不会发生变化。我们的持仓状况在第二天交易开始时保持不变。
If we choose the second alternative and exercise the 80 call, it is as if we had gone into the futures market and purchased a futures contract at 80. Since we now have a futures position, we are subject to futures-type settlement. If the futures contract which we purchased at 80 settles at 100, 20 points will be credited to our account—20 points on which we can earn interest. How much interest will we earn? That depends on the level of interest rates as well as the amount of time remaining to expiration. If interest rates are 6%, with two months remaining to expiration, we will earn:
20 x 2/12 × 6% = .20
如果选择第二种，行权80看涨期权，相当于在期货市场以80买入期货合约。此时，我们持有期货头寸，受到期货型结算的影响。如果以80买入的期货合约在100结算，账户将获得20点的收益，并能赚取相应的利息。利息收益取决于利率水平和剩余到期时间。如果利率为6%，剩余两个月，利息收益为：
20 × 2/12 × 6% = 0.20
Through early exercise of the option we will earn an additional .20 in interest over the time remaining to expiration. Since no trader wants to throw away an additional profit, exercising the option early makes better sense than simply holding the position.
通过提前行权，我们可以在到期前额外赚取0.20的利息收益，因此提前行权比单纯持仓更有利。
What if the option is trading for more than parity, say 20.50? If we sell the 80 call for 20.50 and buy the futures contract for 100, we will have the same position we would have had through early exercise. But now, instead 20 points, we will have 20.50 points credited to our account. We will realize an additional profit of .50 through the sale of the option, as well as the slight additional interest earnings which go with it.
如果期权的交易价格超过平价，比如为20.50，我们可以选择第三种方式：以20.50卖出80看涨期权，再以100买入期货合约，这样我们得到的头寸与提前行权相同，但账户会获得20.50点，比提前行权多出0.50点的收益。
A trader who exercises a futures option early does so to capture the interest on the option's intrinsic value. This intrinsic value will be credited to his account only if the option is subject to stock-type settlement. When he abandons the option through exercise, he receives the variation resulting from settlement of the futures contract. If, however, futures options are subject to futures-type settlement, as they are on many non-U.S. exchanges, no cash flow results from the early exercise of an option. With an 80 call trading at 20 and the underlying futures contract trading at 100, if the option is subject to futures-type settlement when exercised, 20 points will be credited to the trader's account from the settlement of the futures contract. But 20 points will simultaneously be debited from his account when the 20-point value of the option disappears. Since the cash credits and debits exactly offset each other, no additional interest earnings can accrue.
提前行权期货期权的主要目的是获得期权内在价值的利息收益。这一收益仅在期权为股票型结算时有效。当通过行权放弃期权时，交易者会收到期货合约的结算收益。然而，如果期货期权采用期货型结算（如非美国市场的许多交易所），提前行权不会产生现金流。在这种情况下，当80看涨期权交易价格为20、标的期货价格为100时，行权后账户会因期货合约结算获得20点的收益，但期权20点的价值会同时从账户扣除。由于现金收付完全抵消，因此无法获得额外的利息收益。
In futures markets where the options are subject to futures type settlement there is never any economic justification for early exercise, and early exercise can never be an optimal choice. A trader will always be better off either holding the option or selling it, rather than exercising it early.
在采用期货型结算的期货市场中，提前行权没有经济上的合理性，也不可能成为最佳选择。交易者始终会发现持有或卖出期权比提前行权更有利。
STOCK OPTIONS
股票期权
Early Exercise of Calls for the Dividends
为了股息提前行权
We can separate a stock option call into its components, as we did with a futures option. Now we have the additional consideration of dividends:
call value - intrinsic value + interest rate value + volatility value - dividend value
我们可以将股票期权的看涨期权分解成几个组成部分，与期货期权类似，但需要考虑股息：
看涨期权价值 = 内在价值 + 利率价值 + 波动率价值 - 股息价值
Since the intrinsic value, interest rate, and volatility components can never be less than zero, these factors always enhance the call's value. As any one of them rises, the call value rises. Only the dividend component might affect the option's value negatively. As the dividend rises, the call option's value falls. If the underlying stock pays no dividend, or no dividend is expected prior to expiration of the option, a call option can never have a value less than parity (intrinsic value). If, however, the negative effects of the dividend are greater than the positive effects of interest rates and volatility, it might be possible for a call, if it is European, to be worth less than parity.
由于内在价值、利率和波动率的影响不会为负，因此这些因素总是增加看涨期权的价值。只有股息可能对期权价值产生负面影响。股息增加时，看涨期权的价值下降。如果标的股票不支付股息，或在期权到期前没有预期股息，看涨期权的价值不会低于平价（内在价值）。但如果股息的负面影响大于利率和波动率的正面作用，则欧式看涨期权可能低于平价。
For example, suppose a certain stock is trading at 100 and that the stock will go ex-dividend two points tomorrow. Suppose also that there is a 90 call available which will expire in two weeks. When we evaluate the option, we find that it has a theoretical value of 10 and a delta of 100. This means that the option has essentially the same characteristics as the stock. If the option is American, and we want to maintain the same delta position, we have three possible choices:
- Hold the option
- Exercise the option
- Sell the option and buy stock
Which of these choices is best?
例如，假设某股票交易价格为100，明天将除息两点。同时有一个两周后到期的90看涨期权，经评估其理论价值为10，delta为100。这意味着该期权与股票特性相同。如果期权为美式期权，且我们希望维持相同的delta头寸，有三种选择：
- 持有期权
- 行权
- 卖出期权并买入股票
哪种选择最佳？
Suppose we simply hold the option. Certainly we will maintain our delta position. But what will happen tomorrow when the stock gives up its dividend? If the stock opens unchanged, it will open ex-dividend at 98, (Footnote 2: On the day a stock gives up its dividend, prices are typically disseminated with the dividend already deducted from the stock's price. Hence, a 100-polnt stock which gives up a 2-point dividend overnight and opens the next day at 98 would be deemed to have opened unchanged since the 2-point drop in Its price results from the dividend payout, not from any change in investor demand.) since the two-point dividend will be deducted from its price. Since the option has a value of parity, it will open not at 10, the previous day's parity price, but at 8, today's parity price. In other words, if we hold the option we can be certain of losing two points on our position.
假设我们选择持有期权，虽然可以保持delta头寸，但明天股票除息时会发生什么？如果股票开盘价不变，将以98除息开盘（100减去2点股息），此时期权的平价价值为8，而不是前一天的平价10。这意味着我们持有期权将损失两点。
Can we do any better with the second choice, exercising the option? If we exercise the option, we will pay the exercise price of 90 for the stock, and we will discard the 10-point value of the option, effectively purchasing the stock at 100. When the stock goes ex-dividend we will lose two points when it opens two points lower the next day, but we will also receive the dividend since we now own the stock. We will break even because the two-point loss on the stock price will be offset by the two-point dividend we receive. Clearly, we are better off exercising the option than holding it, not because we will show some additional profit, but because we will avoid a two-point loss. We must exercise the option to ensure that we break even.
第二种选择，即行权，会更有利吗？如果行权，我们以90的行权价购入股票，期权的10点价值将被放弃，相当于以100买入股票。当股票除息两点后，股价下跌两点，但我们作为股东将获得股息，从而抵消了股价的下跌。因此，通过行权，我们避免了两点的损失，从而实现盈亏平衡。
What about the third choice, selling the option and buying stock ourselves? This seems to be very similar to early exercise. In both cases we are replacing the option with the stock. If the option is trading at parity, in this case 10, there is no difference between exercising the option, or selling the option and buying the stock. In each case the cash flow will be identical and we will own the stock when it goes ex-dividend. But suppose the 90 call is trading at a price other than 10. Suppose, for example, it is trading at 9½. Clearly, this price is not possible in a market where the options are American and can be exercised at any time. Purchasing a 90 call for 9½ is equivalent to purchasing the stock for 991/2. If the stock is trading at 100, this means the call is trading at less than parity. If this were indeed true, all traders would buy the call, sell the stock, and immediately exercise the call. This would result in a riskless profit of 1/ point, the amount by which the option was below parity. In a market where options are subject to early exercise, no option ought to be trading at less than parity.
第三种选择是卖出期权并自行买入股票。这看似与提前行权相似，都是用股票替代期权。如果期权按平价交易，比如10，那么行权与卖出期权买入股票的现金流是相同的。但如果90看涨期权的交易价格低于平价，比如9½，在美式市场中这将不可能出现，因为买入90看涨期权并以100的价格买入股票可以实现无风险套利，获取1/2点的利润。因此，在美式期权市场中，期权价格不应低于平价。
What if the 90 call is trading for more than parity, say 10½? Now if we sell the option and purchase the stock we will still receive the dividend, since we will own the stock. But we will end up with an additional / point which we would not have collected through exercise of the call. Therefore, our third choice, selling the call and buying the stock, is the optimum choice.
如果90看涨期权的交易价格高于平价，比如10½，此时卖出期权并买入股票是最佳选择。因为此操作不仅让我们获得股息，还能额外获得1/2点收益，这是通过行权无法实现的。
Since the only reason a trader would ever consider exercising a stock option call early is to receive the dividend, if a stock pays no dividend there is no reason to exercise a call early. If the stock does pay a dividend, the only time a trader ought to consider early exercise is the day before the stock goes ex-dividend. At no other time in its life is a stock option call an early exercise candidate.
因为交易者考虑提前行权看涨期权的唯一理由是获得股息，如果股票不支付股息，则没有提前行权的理由。如果股票支付股息，交易者应考虑在除息日前一天提前行权。除此之外的时间，看涨期权不适合提前行权。
We noted that in an American (exercise) market an option should never trade at less than parity. Is the same true in a European (exercise) market? In our example of the 90 call and stock at 100, with an expected two-point dividend, the call price will clearly drop when the stock goes ex-dividend. If the option cannot be turned into stock through exercise prior to the ex-dividend date, the option is clearly going to lose value. If it has very little time value, the amount it will lose when the dividend is given up may make the option worth less than parity. In our example, if the option expires in two weeks with interest rates at 8% and a volatility of 20%, the European value of the 90 call is approximately 8/4. This is 1¾ under parity, due to the fact that a call option, no matter how deeply in-the-money, does not carry with it the right to the dividend. Only ownership of the stock carries that right.
在美式行权市场中，期权不应低于平价。在欧式行权市场中也是如此吗？在我们90看涨期权和100股票的例子中，预期两点股息会导致股票除息后看涨期权价格下降。如果期权无法在除息前通过行权转为股票，期权必然会失去价值。如果其时间价值很低，那么除息时的下跌可能会使期权低于平价。在我们例子中，若期权到期剩两周，利率为8%，波动率为20%，则欧式90看涨期权的理论价值约为8¼，比平价低1¾，因为无论看涨期权多么深度实值，只有持有股票才享有股息权。
Early Exercise of Puts for Interest
为了利息提前行权看跌期权
As we did with a stock option call, we can express a stock option put's value in terms of its components:
put value = intrinsic value - interest rate value + volatility value + dividend value
与股票期权的看涨期权类似，我们可以将股票期权的看跌期权价值分解为几个组成部分：
看跌期权价值 = 内在价值 - 利率价值 + 波动率价值 + 股息价值
In the case of a put, the only component that affects its value negatively is the interest rate component. If the negative effects of interest rates are greater than the positive effects of volatility and dividends, it might be possible for a put, if it is European, to be worth less than parity. Consider this situation:
Stock price = 100, Time to expiration = 8 weeks, Volatility = 20%
Interest rate = 8.00%, Dividend = 0
在看跌期权中，唯一可能对其价值产生负面影响的是利率。如果利率的负面影响超过波动率和股息的正面影响，则欧式看跌期权的价值可能低于平价。假设以下情况：
股票价格 = 100, 距到期时间 = 8周, 波动率 = 20%
利率 = 8.00%, 股息 = 0
With these assumptions, the value of a 110 call is approximately .70. Using the put/call parity relationship, we can calculate the value of the 110 put:
put value = call value + exercise price - stock price - carrying costs
The carrying costs on the exercise price of 110 are:
110 × 56/365 × 8% = 1.35
The European put value is therefore:
70 + 110 - 100 - 1.35 = 9.35
根据这些假设，110看涨期权的价值约为0.70。利用看跌/看涨平价关系，可以计算出110看跌期权的价值：
看跌期权价值 = 看涨期权价值 + 行权价 - 股票价格 - 持有成本
行权价为110的持有成本为：
110 × 56/365 × 8% = 1.35
因此欧式看跌期权的价值为：
0.70 + 110 - 100 - 1.35 = 9.35
Since the 110 put is only worth 9.35 if we hold it to expiration, but is worth 10 points if we exercise it today, the option is apparently worth more dead (exercised) than alive (unexercised). When we exercise the 100 put we get to sell stock at 110, and we can earn interest on this 110-point credit to expiration.
由于110看跌期权持有至到期仅值9.35，但若今天行权则值10点，因此该期权行权后价值高于未行权。当行权100看跌期权时，我们可以以110的行权价卖出股票，并在到期前赚取该110点的利息。
Whereas a stock option call can only be an early exercise candidate on the day prior to the stock's ex-dividend date, a stock option put can become an early exercise candidate anytime the interest which can be earned through the sale of the stock at the exercise price is sufficiently large. Determining exactly when this happens is a difficult problem, but if the stock pays a dividend it is most likely to occur on the day after the stock goes ex-dividend. Since a put is a substitute for a short stock position, one of the advantages of holding a put is to avoid paying the dividend. Hence, a trader will almost always want to hold the put through the ex-dividend date. Then, if the interest considerations are sufficient, the trader will exercise his put.
与股票期权看涨期权只有在股票除息前一天可能成为提前行权候选不同，股票期权看跌期权可以在任何时候因行权后卖出股票所得的利息足够大而成为提前行权的候选。确定何时适合提前行权是个复杂问题，但如果股票支付股息，最可能在股票除息日后的那天出现这种情况。由于看跌期权是空头股票的替代方式，持有看跌期权的优势之一是避免支付股息。因此，交易者通常会在除息日之后持有看跌期权，然后在利息收益足够高时行权。
In an American (early exercise) market no put option can be trading at less than parity. Otherwise there is an immediate arbitrage opportunity available by purchasing the stock, purchasing the put, and immediately exercising the put. The profit will be equal to the amount by which the put is trading at less than parity. This is not necessarily true in a European (no early exercise) market, where the option must be carried to expiration. In our example of the 110 put where the theoretical value derived from the call price was 9.35 (.65 less than parity), the market in the European 110 put might be 914-9½. A market maker is willing to sell the option at 9%/ (/2 less than parity) because he can hedge himself by selling stock, and he will be confident in the knowledge that he will not be required to buy back the stock until expiration. The interest he earns on the stock to expiration will more than offset the amount he will lose by selling the put below parity.
在美式（可提前行权）市场中，没有看跌期权的交易价格会低于平价，否则就会出现立即套利的机会，通过购买股票、购买看跌期权并立即行权即可获得利润，等于看跌期权低于平价的差额。在欧式（不可提前行权）市场中情况不同，期权必须持有至到期。在我们110看跌期权的例子中，根据看涨期权计算出的理论价值为9.35（低于平价0.65），欧式110看跌期权的市场价格可能为9¼-9½。做市商愿意以9½（低于平价½点）的价格卖出期权，因为他可以通过卖出股票进行对冲，并且知道不必在到期前回购股票，到期前的利息收益足以弥补以低于平价出售看跌期权所带来的损失。
Conditions for Early Exercise
提前行权的条件
If given the chance, there are clearly reasons why a trader might want to exercise an American option early. In the case of futures options, when the option is subject to stock-type settlement, the trader is trying to capture the interest on the option's intrinsic value. In the case of stock option calls, he is trying to capture the dividend which the stock pays. In the case of stock option puts, he is trying to capture the interest on the proceeds from the sale of the stock at the exercise price. From the foregoing discussion we can infer two conditions which are necessary before a trader considers exercising an option early to capture this additional profit:
- The option must be trading at parity.
- The option must have a delta close to 100.
在有提前行权机会的情况下，交易者可能会出于一些原因选择提前行权。对于期货期权，提前行权可以捕获期权内在价值的利息；对于股票看涨期权，则是为了获取股票支付的股息；而对于股票看跌期权，则是获取行权价卖出股票后获得的利息。从以上讨论中我们可以推断出交易者考虑提前行权以获取额外收益的两个必要条件：
- 期权价格应处于平价。
- 期权的delta接近100。
If the option is trading at more than parity, the trader will always do better selling the option and taking a position in the underlying market himself. Determining whether an option is trading at parity is simply a matter of finding out what the market in that option is. In the great majority of cases, if an option is deeply enough in-the money to be an early exercise candidate, the bid/ask spread is probably so wide that for practical purposes the option is trading at parity.
如果期权价格高于平价，交易者卖出期权并自行建立标的头寸会更有利。确定期权是否在平价交易，只需查看该期权的市场价。大多数情况下，如果期权深度实值以致成为提前行权候选，买卖价差可能已经足够宽，使其价格实际等同于平价。
Why should the delta be close to 100? If we exercise an option early, we are trading in the option for a position in the underlying contract. We therefore want to be reasonably certain that the option and the underlying contract have the same characteristics. In other words, we want to be certain that the option has no additional insurance value left, so that when we exercise the option we are not throwing away this insurance value.
为什么delta要接近100？因为提前行权相当于将期权转化为标的合约的头寸，因此我们希望期权和标的合约的特性一致。换句话说，我们不希望期权仍有额外的保险价值，行权时不会损失该保险价值。
For example, suppose there is an 80 call available with the underlying contract trading at 100. If we want to take a long position we can purchase either the 80 call or the underlying contract. If we feel there is no chance that the market can fall below 80 prior to expiration, there is no difference between holding the 80 call or holding the underlying contract. But suppose we feel that there is some chance that the market could fall below 80. In that case we will prefer to hold the 80 call. If the market were to fall below 80, holding the call would limit our potential loss to the option's premium. If, however, we hold a long underlying position and the market drops below 80, our potential loss is unlimited.
例如，假设有一个80的看涨期权，标的合约价格为100。如果我们希望建立多头，可以购买80看涨期权或直接持有标的合约。如果我们认为市场在到期前不可能跌破80，那么持有80看涨期权或标的合约都没有差异。但如果市场可能跌破80，我们会倾向持有看涨期权，因为市场跌破80时，持有期权的损失仅限于期权价格，而标的多头则面临无限损失。
Recall that one interpretation of the delta is the probability that the option will finish in-the-money. A delta close to 100 means that the underlying market has almost no chance of going through the exercise price, and the option therefore has almost no chance of going out-of-the-money.
记住，delta的一个解释是期权到期时仍为实值的概率。delta接近100表示标的市场几乎不可能穿越行权价，期权几乎不可能变成虚值。
What degree of certainty justifies early exercise? With a delta of 100 one certainly would consider early exercise. What about 99? 96? 90? With a delta above 95, many traders will at least consider early exercise. How much interest will be earned, or what dividend will be collected, through early exercise? With a delta of 95 or lower, the risk is probably too great to consider early exercise. There is at least a 5% chance that the underlying market will go through the exercise price. If that happens, a trader will regret having exercised an option early, regardless of the additional benefits to be gained through early exercise.
什么程度的确定性可以支撑提前行权？如果delta为100，可以考虑提前行权；但如果是99、96、90呢？delta高于95时，许多交易者会考虑提前行权。然而如果delta低于95，提前行权的风险较大，因为标的市场可能有5%的概率会穿越行权价，损失保险价值的风险不容忽视。
By ensuring that the delta is close to 100, we are also ensuring that we do not throw away any insurance value that the option might still retain. As an option goes more deeply into-the-money, its time value (insurance value) begins to disappear. With a delta of 100, in theory the option has no time value left. However, as the delta declines, the time value increases. If the delta were significantly less than 100, the option would still retain a significant amount of time value. If we were to exercise such an option, we would be throwing away this insurance value.
确保delta接近100可以避免损失期权可能保留的保险价值。期权越深度实值，其时间价值（保险价值）越少。delta为100理论上表示期权不再有时间价值。然而，delta越小，时间价值越大。如果在delta显著低于100时行权，会浪费保险价值。
Since the desirability of early exercise is dependent, at least in part, on the option's delta, the accuracy of the delta is important. But there are many factors which affect the delta, and we might incorrectly assess any one of these factors. If we feel that the correct volatility is 15%, an option might have a delta of 98. But if we decide to raise our volatility to 17%, the delta of the option will decline. Under the new conditions its delta might only be 93. At a 15% volatility a trader might consider early exercise, while at a 17% volatility he might not. For this reason, options in low-volatility markets are exercised early much more often than options in high-volatility markets.
由于提前行权的可取性部分取决于期权的delta，因此delta的准确性非常重要。然而，影响delta的因素很多，任何因素的误判都可能影响判断。如果认为波动率为15%，期权的delta可能为98；而若波动率提升至17%，delta可能降至93。在低波动率市场中，期权更容易被提前行权，而高波动率市场则不然。
In the same way, an option with three months remaining to expiration might have a delta of 92, and would therefore not be an early exercise candidate. But if two months pass, and the price of the underlying contract is unchanged, the delta will rise, and the new delta might be 99. With three months remaining the option was not an early exercise candidate, but with one month remaining the option might be early exercise candidate.
类似地，剩余到期时间三个月的期权delta可能为92，尚不适合提前行权；但若再过两个月，标的价格保持不变，delta可能升至99。三个月时该期权不适合提前行权，但一个月时则可能成为行权候选。
Of course, we ought to ask why we were considering early exercise in the first place. In many cases, we exercise to earn interest over the life of the option. We can earn more interest over three months than over one month, so from this point of view a three-month option is more likely to be an early exercise candidate than a one month option. We can see that many factors affect the desirability of early exercise, and these factors can pull in different directions, with some factors dictating for early exercise and some dictating against.
需要问的是，为什么要考虑提前行权？许多情况下，提前行权是为了赚取期权的利息。三个月期权的利息收益大于一个月期权，因此从这一点来看，三个月期权比一个月期权更适合提前行权。影响提前行权的因素很多，这些因素可能相互作用，一些支持提前行权，一些则反对。
Thus far we have only addressed the question of when a trader might find it desirable to exercise an option early. There should be an additional value associated with an American option over an identical European option because the American option carries with it additional rights. Even an out-of-the-money option, which no one would consider exercising today, might someday go deeply enough into-the-money so that it could become an early exercise candidate. This possibility should always make an American option more valuable than a European option.
迄今为止我们只讨论了何时适合提前行权。美式期权相较于欧式期权应该具备额外价值，因为美式期权带有更多权利。即使是虚值期权，将来也可能变得深度实值，从而成为提前行权候选。这种可能性总是使美式期权比欧式期权更有价值。
How much more should a trader be willing to pay for an American option over an Identical European option? The Black-Scholes model makes no attempt to answer this question since it is a European pricing model. In spite of the fact that Black-Scholes generated values are likely to show some inaccuracies in an American market, for many years traders continued to use the model because no model with equal ease of use existed to evaluate American options. The problem of early exercise was dealt with through intuition, or through slight adjustments to Black-Scholes generated values. For example, when a stock is expected to pay a dividend, an American call value can be approximated by comparing the Black-Scholes value of the call option under two circumstances:
- The call expires the day before the stock goes ex-dividend.
- The call expires on its customary date, but the underlying stock price used to evaluate the call is the current price less the expected dividend.
Whichever value is greater is the pseudo-American call value.
交易者愿意为美式期权多支付多少溢价？Black-Scholes模型并未回答这一问题，因为它是欧式期权定价模型。尽管Black-Scholes模型在美式市场中会出现一定偏差，长期以来交易者仍然使用它，因为没有同样简便的美式期权定价模型。早期行权的问题通常通过直觉或对Black-Scholes模型略微调整来解决。例如，股票预期支付股息时，可以通过比较Black-Scholes模型下两种情形的看涨期权价值来近似美式看涨期权：
- 在股票除息日前一天到期的看涨期权。
- 到期日为常规时间的看涨期权，但标的股价减去预期股息。
较大的价值即为伪美式看涨期权价值。
In the case of options on futures, or of put options on stock, traders used Black-Scholes generated values, but raised any option with a theoretical value less than parity to exactly parity.
对于期货期权或股票看跌期权，交易者使用Black-Scholes模型计算值，但将任何低于平价的理论值提升至平价。
Models eventually were developed to more accurately evaluate American options. The most widely used of these are the Cox-Ross-Rubenstein model, developed by John Cox, Stephen Ross, and Mark Rubenstein; and the Whaley model, developed by Giovanni Barone-Adesi and Robert Whaley. Unlike the Black-Scholes model, neither of these models is closed form. One can't simply sit down and add up all the numbers to get the correct value. Rather, both models are algorithms, or loops, and each time the user makes a pass through the loop, the closer he comes to the true American value of the option. While the Cox-Ross-Rubenstein model is quite easy to understand, both intuitively and mathematically, it may require numerous passes through the loop to generate an acceptable value. On the other hand, the Whaley model is more complex mathematically, but converges to an acceptable value much more quickly. The Cox-Ross-Rubenstein model may require 40 or 50 passes through the loop to reach the same degree of accuracy as the Whaley model achieves with four or five passes through the loop. In addition to evaluating American options, both models also determine when an option should be exercised early. We were somewhat vague on this point in our earlier discussion, stating that the delta of an option should be close to 100 to justify early exercise. When using a true American model, an option is optimally exercised early when its theoretical value is exactly parity and its delta is exactly 100.
While the Cox-Ross-Rubenstein and Whaley models in most cases generate similar values, more and more computer services are using the Whaley model to evaluate American options because of the speed with which it can achieve results. However, the Cox-Ross-Rubenstein model, while slower, is still very widely used because it offers some advantages which are not available in the Whaley model. For example, the Whaley model handles the dividend payout of a stock as if it were a continuous interest payment over the life of the option. In reality dividends are paid all at once, and the Cox-Ross-Rubenstein model more accurately reflects the impact of this one-time payment on a call option on the stock. Unlike the Whaley model, the Cox-Ross-Rubenstein model can also be used to evaluate some of the recently introduced, exotic options. Such options are path dependent since their value depends not only on the likely price distribution of the underlying contract at expiration, but also on the various paths which the underlying price might take to achieve that distribution. Both these models are discussed further in Appendix B.
之后开发了更精确的美式期权定价模型，最常用的是Cox-Ross-Rubenstein模型和Whaley模型。与Black-Scholes模型不同，这两个模型都不是闭式模型。Cox-Ross-Rubenstein模型理解简单，但需要多次循环才能达到可接受的精度；Whaley模型更复杂，但收敛速度更快。大多数情况下，这两个模型生成的值相似，但越来越多的计算服务使用Whaley模型，因为它收敛快。然而，Cox-Ross-Rubenstein模型在处理一次性股息支付和评价新型复杂期权方面具有优势。这两个模型在附录B中有详细讨论。
Regardless of the model a trader chooses, the accuracy of model-generated values will depend at least as much on the inputs into the model as the theoretical accuracy of the model itself. If a trader evaluates an American option using an incorrect volatility, or an incorrect interest rate, or an incorrect underlying price, the fact that he derives his values from an American rather than a European model is likely to make little difference. Both models will generate incorrect values because the inputs are incorrect. The American model may produce less error, but that will be small consolation if the incorrect inputs lead to a large trading loss.
无论使用哪种模型，计算结果的准确性不仅取决于模型本身，还取决于输入参数。若波动率、利率或标的价格输入不准确，即使用美式模型也可能出错。美式模型可能会减少误差，但如果错误的输入导致重大交易损失，这点小优势也无济于事。
American option values are most important when there is a significant difference between the carrying cost associated with the option and the carrying cost associated with the underlying contract. The greater the difference between the two carrying costs, the greater the value of early exercise. When the underlying contract is a futures contract and the option on the contract is subject to futures-type settlement, the cost of carrying either position is, in theory, zero. This is the same as assuming an effective interest rate of zero. If we use a zero interest rate, there is no difference between European and American option values.
当期权与标的合约的持仓成本差异显著时，美式期权值的重要性最大。差异越大，提前行权的价值越大。若标的合约为期货且期权采用期货式结算，持仓成本为零，欧式和美式期权无差异。
When the options on a futures contract are subject to stock-type settlement, there is a small difference between the carrying cost on the option and the carrying cost on the underlying contract. Although the option is subject to stock-type settlement, the price of the option is only a small fraction of the price of the underlying contract. The additional value for early exercise is therefore small, and is only likely to show up in deeply in-the-money options. Even in this case, the difference between European and American values is often less than the minimum price increment. In such a market, a trader is unlikely to profit significantly because he is using an American pricing model while other traders are using a European pricing model. Practical considerations, such as the accuracy of the trader's volatility estimate, his ability to anticipate directional trends in the underlying market, and his ability to control risk through effective spreading strategies, will far outweigh any small advantage gained by using an American rather than a European model.
当期货合约的期权采用类似股票的结算方式时，期权持有成本与标的合约持有成本之间的差异很小。尽管期权采用股票型结算，但其价格只是标的合约价格的一小部分。因此，提前行权的额外价值较小，通常只会体现在深度实值期权中。即便如此，欧式和美式期权之间的价值差异通常也低于最小价格增量。在这种市场中，交易者因为使用美式定价模型而获得显著利润的可能性不大。实际因素，如波动率估计的准确性、预测标的市场趋势的能力，以及通过有效的价差策略控制风险的能力，远比使用美式模型带来的微小优势更为重要。
The importance of early exercise is greatest when the underlying contract is a stock or physical commodity. (Footnote 3: Early exercise can also be an important consideration in the foreign exchange market if the interest rates associated with the domestic currency (the currency in which the option is settled) and foreign currency (the currency to be delivered in the event of exercise) are significantly different.) In such a case there is a significant difference between the carrying cost on an option and the carrying cost on an underlying position. This difference will especially affect the difference between European and American put values, since early exercise will allow the trader to earn interest on the proceeds from the sale at the exercise price. An option trader in either the stock or physical commodity market will find that the additional accuracy offered by an American model, such as the Cox-Ross-Rubenstein or Whaley models, will indeed be worthwhile. The significance of this difference is shown in Figure 12-1.
当标的是股票或实物商品时，提前行权的意义最大。（脚注3：在外汇市场中，如果结算货币（本国货币）和交割货币（外国货币）的利率差异显著，提前行权也值得考虑。）在这种情况下，期权的持仓成本与标的持仓成本之间有显著差异，尤其影响欧式和美式看跌期权的价值差异，因为提前行权使交易者能够获得按行权价卖出标的所得的利息。在股票或实物商品市场中，期权交易者会发现像Cox-Ross-Rubenstein或Whaley模型这样的美式定价模型所带来的精确性确实有价值。图 12-1 展示了这一差异的重要性。
THE EFFECT OF EARLY EXERCISE ON TRADING STRATEGIES
提前行权对交易策略的影响
The delta of an American option will always be greater than the delta of an equivalent European option. The extent to which it is greater will depend on how deeply in-the-money the option is, as well as the interest considerations resulting from early exercise. In most cases the delta of an American option will differ only slightly from a European option, and the possibility of early exercise is therefore unlikely to have a significant impact on volatility or directional strategies. In the former case, a trader may have to adjust the ratio of a strategy slightly if he wants to remain close to delta neutral. In the latter case, a trader may find that he is a few deltas longer or shorter than he would be if employing the same strategy with European options.
美式期权的delta值总是高于等值的欧式期权，其差异大小取决于期权的实值程度以及提前行权的利息因素。大多数情况下，美式期权的delta值与欧式期权的差异很小，因此提前行权对波动率或方向性策略的影响不大。在波动率策略中，交易者可能需要略微调整策略的比率以保持接近delta中性；在方向性策略中，交易者可能会发现持仓略微偏长或偏短。
Because early exercise of an option is not automatic, there are strategies which depend on someone making an error and not exercising an option early when he ought to do so. For example, a stock option trader might try to execute a dividend play. This strategy consists of buying stock and selling deeply in-the-money calls as the ex-dividend date for the stock approaches. If the trader is not assigned on the calls, he will break even on the stock (the stock price will fall, but he will collect the dividend). At the same time he will profit when the deeply in-the-money calls which he has sold fall by the amount of the dividend. Of course, if he is assigned on the calls, as he ought to be, he will only break even. But for each call which goes unexercised the trader will profit by the amount of the dividend. Dividend plays were much more common in the early days of option trading when the market was less sophisticated and many options which should have been exercised were not. As markets have become more efficient, only a professional trader, with very low transaction costs, can afford to take advantage of such a possibility. Even then, he may find that he is assigned on the great majority of the calls he has sold.
由于提前行权并非自动触发，有些策略依赖于其他交易者未在适当时候提前行权的错误。例如，股票期权交易者可能会尝试进行分红策略，即在股票接近除息日期时，买入股票并卖出深度实值的看涨期权。如果卖出的看涨期权未被指派执行，交易者将在股票价格下跌的同时获得分红，且从深度实值期权随分红金额下跌中获利。当然，如果期权被指派执行，交易者将只能打平，但每一个未被行权的期权都会带来等于分红金额的利润。早期市场不够成熟时，分红策略更为常见，许多应被行权的期权未被执行。如今市场更加高效，只有低交易成本的专业交易者才可能从中获利，即便如此，大部分卖出的期权也可能会被指派执行。
A trader can execute a similar type of interest play by selling stock and simultaneously selling deeply in-the-money puts. Now, instead of profiting by the amount of the dividend, the trader will profit by the amount of the interest he can earn on the exercise price (the proceeds of the stock sale and the put sale combined). This profit will continue to accrue as long as the puts remain unexercised. If the puts are exercised, the trader does no worse than break even. Again, only a professional trader, with his low transaction costs, is likely to employ such a strategy.
交易者可以进行类似的利息策略，通过卖出股票并同时卖出深度实值的看跌期权来获利。此时，交易者的利润不是来自分红，而是来自行权价的利息收益（股票出售和期权出售的总收益）。只要看跌期权未被行权，这一利润会持续累积。如果看跌期权被行权，交易者最多只能持平。通常，只有低交易成本的专业交易者才会采用这种策略。
If options are subject to stock-type settlement, an interest play can also be done in the futures option market by either purchasing a futures contract and simultaneously selling a deeply in the money call, or selling a futures contract and simultaneously selling a deeply in-the-money put. If the option is deeply enough in-the money, it ought to be exercised early. If, however, the option remains unexercised, the trader will continue to earn interest on the proceeds from the option sale. Since the amount on which the trader will earn interest is the difference between the exercise price and futures price, this will not be as profitable as a similar strategy in the stock option market, where the trader will earn interest on the exercise price. Still, if the transaction costs are low enough, it may be worth doing.
如果期权采用股票类型的结算方式，在期货期权市场也可以通过类似的利息策略操作，即买入期货合约并卖出深度实值的看涨期权，或者卖出期货合约并同时卖出深度实值的看跌期权。如果期权足够深度实值，理论上应被提前行权。但如果未被行权，交易者仍然可以从期权出售的收益中赚取利息。由于交易者赚取利息的金额是行权价与期货价格的差额，这种策略的收益不如股票期权市场上赚取行权价利息的策略高。但若交易成本足够低，仍然具有一定的可操作性。
A variation on dividend and interest plays can also be executed using deeply In-the-money vertical spreads. For example, suppose a stock is trading at 100 with the ex-dividend date approaching If both the 80 and 85 calls are deeply enough in-the-money to be early exercise candidates, a trader might try to purchase the 80/85 call vertical spread for five points. If he does so, he will exercise his 80 call in order to collect the dividend, and at the same time hope not to be assigned on the 85 call. Oddly, if both the 80 and 85 calls are early exercise candidates, the trader should also be willing to sell the 80/85 call vertical for five points. If he does so, he will exercise his 85 call in order to collect the dividend, and at the same time hope not to be assigned on the 80 call. In other words, a professional trader might logically make a market under these circumstances of 5 bid/5 ask. He is willing to buy or sell the 80/85 call vertical at five points. He intends to immediately exercise whichever call he buys, while hoping not be assigned on the call which he sells.
分红和利息策略也可以通过深度实值的垂直价差来实现。例如，假设一只股票当前价格为100，且即将分红。如果80和85的看涨期权足够深度实值，成为提前行权的候选，交易者可以尝试以五点的价格买入80/85看涨垂直价差。此时，他可以行使80看涨期权来获得分红，同时希望不被分配到85看涨期权的行权。奇怪的是，如果80和85看涨期权都符合提前行权的条件，交易者也可以接受以五点卖出80/85看涨垂直价差。在这种情况下，他会行使85看涨期权来收取分红，同时希望不被分配到80看涨期权的行权。也就是说，在这种情况下，专业交易者可能会合理地在5点报价/5点要价，愿意以五点价格买入或卖出80/85看涨垂直价差，并打算立即行使买入的期权，而希望卖出的期权不被行权。
The possibility of early exercise can also have an impact on arbitrage strategies. For example, suppose a stock option trader executes a reverse conversion:
buy a call
sell a put
sell stock
提前行权的可能性也会影响套利策略。例如，假设一位股票期权交易者执行了反向转换：
买入看涨期权
卖出看跌期权
卖出股票
If he executes this strategy at what he believes to be profitable prices, part of his profit will come from the interest he expects to earn on the sale of the stock. But what will happen if the stock price begins to fall, and in fact falls so far that the trader is assigned on the put? This will eliminate the interest earnings, since he will be required to buy back the stock. Of course, he can still sell the call and take in some cash. But if the value of the call is insufficient to offset the interest loss, the trader may find that his profitable reversal has in fact become unprofitable.
如果他以认为有利可图的价格执行该策略，部分利润将来自他预计在卖出股票上获得的利息。但如果股价开始下跌，甚至跌至被要求执行看跌期权的程度会怎样？这将终止利息收益，因为他必须回购股票。当然，他仍然可以卖出看涨期权来获得一些现金收入，但如果看涨期权的价值不足以抵消利息损失，这位交易者可能发现自己原本盈利的反向转换策略实际上变成了亏损。
The trader in our example is worried about being assigned on his put. If the market drops there will be a greater likelihood of assignment; if the market rises there will be a lesser likelihood of assignment. Since the trader prefers the market to rise, he must be delta long. This is confirmed by the following delta values:
Stock price = 100; Time to expiration = 3 months; Volatility = 25%
Interest rate = 8.00%; Dividend = 0
| Option | European Value |
European Delta |
American Value |
American Delta |
|---|---|---|---|---|
| 100 call | 5.97 | 58.8 | 5.97 | 58.8 |
| 100 put | 4.00 | -41.2 | 4.20 | -44.0 |
示例中的交易者担心被分配执行看跌期权。如果市场下跌，被分配的可能性会增加；若市场上涨，则被分配的可能性会降低。由于交易者希望市场上涨，因此他的delta应该为正。这从以下delta值得到确认：
股票价格 = 100；到期时间 = 3个月；波动率 = 25%
利率 = 8.00%；股息 = 0
| 期权 | 欧洲期权 价值 | 欧洲期权 Delta | 美式期权 价值 | 美式期权 Delta |
|---|---|---|---|---|
| 100 看涨期权 | 5.97 | 58.8 | 5.97 | 58.8 |
| 100 看跌期权 | 4.00 | -41.2 | 4.20 | -44.0 |
If these are European options, the total delta of the reversal is:
+58.8 +41.2 -100 = 0
If, however, the options are American, the total delta is:
+58.8 +44.0 -100 = +2.8
The positive delta of 2.8 reflects a slight preference for the market to rise so that the trader will avoid being assigned on the put.
For a similar reason, if the trader executes a conversion (sell call, buy put, buy stock), he is 2.8 deltas short. He wants the stock to fall so that he can exercise his put early and thereby avoid the interest costs of carrying a long stock position.
如果这些是欧洲期权，则反向转换的总delta为：
+58.8 +41.2 -100 = 0
但如果这些是美式期权，总delta为：
+58.8 +44.0 -100 = +2.8
2.8的正delta反映了交易者希望市场上涨，以避免看跌期权被分配。
同样，如果交易者执行转换（卖出看涨期权，买入看跌期权，买入股票），他的delta为-2.8，表示他希望股票下跌，以便提前行使看跌期权，从而避免持有多头股票的利息成本。
Because the desirability of early exercise, and therefore the likelihood of early exercise, can increase or decrease as the underlying market rises or falls, conversions and reversals using American options are not delta neutral. While these strategies may be unbalanced by only two or three deltas, the fact that they are often done in large size can result in an additional risk that the trader cannot afford to ignore. This also applies to boxes and jelly rolls, which are simply combinations of conversions and reversals. Consider these option values:
Stock price = 100; Volatility = 25%; Interest rate = 8.00%; Dividend = 0
Time to expiration: March = 3 months, June = 6 months
| Option | European Value |
European Delta |
American Value |
American Delta |
|---|---|---|---|---|
| March 95 call | 9.00 | 73.7 | 9.00 | 73.7 |
| March 95 put | 2.13 | -26.3 | 2.24 | -27.8 |
| March 100 call | 5.97 | 58.8 | 5.97 | 58.8 |
| March 100 put | 4.00 | -41.2 | 4.20 | -44.0 |
| June 95 call | 11.96 | 72.7 | 11.96 | 72.7 |
| June 95 put | 3.24 | -27.3 | 3.49 | -29.9 |
| June 100 call | 9.03 | 62.3 | 9.03 | 62.3 |
| June 100 put | 5.11 | -37.7 | 5.55 | -42.1 |
由于提早行权的需求可能随标的市场的涨跌而变化，因此使用美式期权的转换和反向转换策略并非完全Delta中性。尽管这些策略的Delta可能只有两到三个的偏差，但因为通常涉及大额交易，这会带来额外风险，不容忽视。对于盒式套利和果冻卷等策略，情况也是如此，这些策略实际上是转换和反向转换的组合。考虑以下期权价值：
股票价格=100；波动率=25%；利率=8.00%；股息=0
到期时间：3月=3个月，6月=6个月
| 期权 | 欧洲期权 价值 | 欧洲期权 Delta | 美式期权 价值 | 美式期权 Delta |
|---|---|---|---|---|
| 3月95看涨期权 | 9.00 | 73.7 | 9.00 | 73.7 |
| 3月95看跌期权 | 2.13 | -26.3 | 2.24 | -27.8 |
| 3月100看涨期权 | 5.97 | 58.8 | 5.97 | 58.8 |
| 3月100看跌期权 | 4.00 | -41.2 | 4.20 | -44.0 |
| 6月95看涨期权 | 11.96 | 72.7 | 11.96 | 72.7 |
| 6月95看跌期权 | 3.24 | -27.3 | 3.49 | -29.9 |
| 6月100看涨期权 | 9.03 | 62.3 | 9.03 | 62.3 |
| 6月100看跌期权 | 5.11 | -37.7 | 5.55 | -42.1 |
We can see how the possibility of early exercise affects the values and deltas of various arbitrages:
| Strategy | European Value |
European Delta |
American Value |
American Delta |
|---|---|---|---|---|
| March 95/100 box | 4.90 | 0 | 4.99 | -1.3 |
| June 95/100 box | 4.80 | 0 | 4.99 | -1.8 |
| March/June 95 jelly roll | 1.85 | 0 | 1.71 | +1.1 |
| March/June 100 jelly roll | 1.95 | 0 | 1.71 | +1.6 |
我们可以看到提前行权的可能性如何影响各种套利策略的价值和Delta：
| 策略 | 欧洲 价值 | 欧洲 Delta | 美式 价值 | 美式 Delta |
|---|---|---|---|---|
| 3月95/100盒式套利 | 4.90 | 0 | 4.99 | -1.3 |
| 6月95/100盒式套利 | 4.80 | 0 | 4.99 | -1.8 |
| 3月/6月95果冻卷套利 | 1.85 | 0 | 1.71 | +1.1 |
| 3月/6月100果冻卷套利 | 1.95 | 0 | 1.71 | +1.6 |
Boxes become more valuable using American options because a trader who is long the box owns a put with a higher exercise price. If a trader buys the 95/100 box, he is long a 95 call and a 100 put, and short a 95 put and 100 call. The 100 put will become an early exercise candidate before the 95 put, making the American box more valuable than the same European box. The delta of the box is negative with American options because a trader who owns the box would like the market to decline so that he will be able to exercise the 100 put as quickly as possible.
使用美式期权时，盒式套利更有价值，因为持有盒式套利的交易者拥有行权价较高的看跌期权。例如，购买95/100盒式套利的交易者持有一个95看涨期权和一个100看跌期权，同时卖出一个95看跌期权和一个100看涨期权。在这种情况下，100看跌期权比95看跌期权更有可能成为提前行权的候选，从而使美式盒式套利比同样的欧洲盒式套利更有价值。由于美式盒式套利的Delta为负数，因此持有盒式套利的交易者希望市场下跌，以便能尽早行权100看跌期权。
Jelly rolls become less valuable using American options because a trader who owns the jelly roll has sold a long-term put. If a trader buys the March/June 100 jelly roll, he is long a March 100 put and June 100 call, and is short a March 100 call and June 100 put. The June 100 put, being a longer term option, has more potential for early exercise than the March 100 put. The delta of the jelly roll is therefore positive because there is a greater danger of being assigned on the June 100 put. Hence, the trader who owns the jelly roll would like the market to rise so that he is less likely to be assigned on the June 100 put.
使用美式期权时，果冻卷套利的价值会降低，因为持有果冻卷的交易者卖出的是长期看跌期权。若交易者购买3月/6月100果冻卷，他持有一个3月100看跌期权和一个6月100看涨期权，同时卖出一个3月100看涨期权和一个6月100看跌期权。由于6月100看跌期权期限更长，更有可能提前行权，因此果冻卷的Delta为正。持有果冻卷的交易者希望市场上涨，这样被分配到6月100看跌期权的可能性会降低。
A unique, early exercise situation affecting the value of boxes can also arise in a stock option market if there is a tender offer to buy a portion of the stock in a company. For example, with a stock trading at 100, one would expect to see the 100/105 box trading at close to five points. But suppose that a tender offer is made to buy half the outstanding shares at a price of 110. A trader who owns the 100/105 box will exercise his 100 calls in order to be able to tender stock. If he tenders 1000 shares, he might expect half, or 500 shares, to be accepted at the tender price of 110. But after the tender is completed, because it was for only half the outstanding shares, the stock price will return to its pre-tender price of 100. The trader's remaining 500 shares will still be trading at 100. In other words, prior to the tender offer the 100 calls are worth 10, but after the tender the 105 puts are worth a minimum of five. Of course, one cannot say that the box is worth 15, because only half the outstanding shares are worth 110. The other half are worth 100. Still, the 100/105 box is likely to trade for significantly more than its usual price of five.
在股票期权市场中，如果公司发布部分股票的要约收购，提前行权的特殊情况还会影响盒式套利的价值。例如，假设股票价格为100，通常会期望100/105盒式套利价格接近5点。但如果出现要约收购，以110的价格收购一半在外流通的股票，持有100/105盒式套利的交易者可以行权100看涨期权并提交股票。如果他提交1000股，预计会有一半，即500股，以110的要约价被接受。然而，由于只收购了一半股份，收购完成后股价会回到收购前的100。交易者剩余的500股仍会以100交易。换言之，要约收购前100看涨期权价值10，但收购后105看跌期权至少值5。当然，不能说盒式套利价值15，因为只有一半股份值110，另一半值100。然而，100/105盒式套利的交易价格可能会显著高于平时的5点。
The possibility of early exercise can also affect arbitrage strategies in the futures option market, but the difference between American versus European values is much less than in the stock option market because there is less interest consideration associated with futures options. For practical purposes, unless the options are very deeply in-the-money or are very long-term, the difference between American and European values for futures options is negligible. Some sample option and arbitrage values are shown below (we assure that all options are subject to stock-type settlement):
Futures price = 100; Volatility = 25%; Interest rate = 8.00%;
Time to expiration: March = 3 months, June = 6 months
| Option | European Value |
European Delta |
American Value |
American Delta |
|---|---|---|---|---|
| March 95 call | 7.60 | 66.9 | 7.67 | 67.5 |
| March 95 put | 2.70 | -31.2 | 2.73 | -31.3 |
| March 100 call | 4.88 | 51.5 | 4.88 | 51.8 |
| March 100 put | 4.88 | -46.6 | 4.88 | -46.9 |
| June 95 call | 9.27 | 62.2 | 9.34 | 63.5 |
| June 95 put | 4.46 | -33.9 | 4.48 | -34.2 |
| June 100 call | 6.76 | 51.4 | 6.80 | 52.2 |
| June 100 put | 6.76 | -44.7 | 6.80 | -45.4 |
| Strategy | European Value |
European Delta |
American Value |
American Delta |
|---|---|---|---|---|
| March 95 conversion | 4.90 | +1.9 | 4.94 | +1.2 |
| June 95 conversion | 4.81 | +3.9 | 4.86 | +2.3 |
| March 100 conversion | 0 | +1.9 | 0 | +1.3 |
| June 100 conversion | 0 | +3.9 | 0 | +2.4 |
| March 95/100 box | 4.90 | 0 | 4.94 | +0.1 |
| June 95/100 box | 4.81 | 0 | 4.86 | +0.1 |
| March/June 95 jelly roll | 1.85 | 0 | 1.71 | +1.1 |
| March/June 100 jelly roll | 1.95 | 0 | 1.71 | +1.6 |
提前行权的可能性也会影响期货期权市场中的套利策略，但由于期货期权涉及的利息考虑较少，美式和欧式期权的价值差异比股票期权市场小得多。通常情况下，除非期权处于深度实值或期限很长，美式和欧式期货期权的价值差异可以忽略不计。以下展示了一些期权和套利价值示例（假设所有期权均采用股票型结算）：
期货价格 = 100；波动率 = 25%；利率 = 8.00%；
到期时间：3月 = 3个月，6月 = 6个月
| 期权 | 欧式 价值 | 欧式 Delta | 美式 价值 | 美式 Delta |
|---|---|---|---|---|
| 3月95看涨期权 | 7.60 | 66.9 | 7.67 | 67.5 |
| 3月95看跌期权 | 2.70 | -31.2 | 2.73 | -31.3 |
| 3月100看涨期权 | 4.88 | 51.5 | 4.88 | 51.8 |
| 3月100看跌期权 | 4.88 | -46.6 | 4.88 | -46.9 |
| 6月95看涨期权 | 9.27 | 62.2 | 9.34 | 63.5 |
| 6月95看跌期权 | 4.46 | -33.9 | 4.48 | -34.2 |
| 6月100看涨期权 | 6.76 | 51.4 | 6.80 | 52.2 |
| 6月100看跌期权 | 6.76 | -44.7 | 6.80 | -45.4 |
| 策略 | 欧式 价值 | 欧式 Delta | 美式 价值 | 美式 Delta |
|---|---|---|---|---|
| 3月95转换 | 4.90 | +1.9 | 4.94 | +1.2 |
| 6月95转换 | 4.81 | +3.9 | 4.86 | +2.3 |
| 3月100转换 | 0 | +1.9 | 0 | +1.3 |
| 6月100转换 | 0 | +3.9 | 0 | +2.4 |
| 3月95/100盒式套利 | 4.90 | 0 | 4.94 | +0.1 |
| 6月95/100盒式套利 | 4.81 | 0 | 4.86 | +0.1 |
| 3月/6月95果冻卷 | 1.85 | 0 | 1.71 | +1.1 |
| 3月/6月100果冻卷 | 1.95 | 0 | 1.71 | +1.6 |
Many new traders seem to worry unduly about the possibility of early exercise: "If I sell an option, what happens If I am suddenly assigned?" It is true that sometimes early assignment can cause a loss. But there are many factors which can cause a trader to lose money; early exercise is only one such factor. A trader should be prepared to deal with the possibility of early exercise just as he should be prepared to deal with the possibility of movement in the price of the underlying contract, or the possibility of changes in volatility. Margin requirements established by the clearing houses usually require a trader to keep sufficient funds in his account to cover the possibility of early assignment. But this is not always true. If he is short deeply in-the-money options, an early assignment notice may cause a cash squeeze. If this happens, he will need sufficient capital to cover the situation. Otherwise, he may be forced to liquidate some or all of the remaining position. And forced liquidations are invariably losing propositions.
许多新手交易员过于担心提前行权的问题：“如果我卖出一个期权，突然被指派怎么办？”确实，提前指派有时可能带来损失，但导致亏损的因素有很多，提前行权只是其中之一。交易员应准备好应对提前行权的可能性，就像应对标的合约价格波动或波动性变化的可能性一样。结算所通常会设定保证金要求，要求交易员账户中有足够的资金应对提前指派的可能性。但这并非总是如此。如果他持有的是深度实值的空头期权，提前指派可能导致资金紧张。这时，他需要有足够的资金应对，否则可能被迫清仓部分或全部头寸，而被迫清仓通常是亏损的。
An experienced trader should be able to foresee the likelihood of early exercise. He need only ask himself: "If I owned this option, would logically exercise it now?" If the answer is "yes," then the trader ought to be prepared for assignment. Early exercise rarely comes as a surprise. If it does, it is probably good for the trader who was assigned. If an option is exercised too early, someone has erroneously abandoned the time value or protective characteristics associated with the option. When that happens, the trader who is assigned will find that he has just received an unexpected gift.
有经验的交易员应能预见提前行权的可能性。他只需问自己：“如果我是这期权的持有者，现在会理性地行权吗？”如果答案是“是”，那他就应该做好被指派的准备。提前行权很少会令人意外。如果真的意外了，对被指派的交易员通常反而是有利的。如果有人过早行权，那就意味着他错误地放弃了期权的时间价值或保护性特点。此时，被指派的交易员就相当于收到了一个意外的“礼物”。
No comments:
Post a Comment