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  • Give Me Options - How Are Options Priced?


    Markets Overview

    For the US stock market, our proprietary system has identified two immediately buyable stocks: BEAT and LULU. These stocks are buyable right now according to our system.

    Disclosure: Dr. Zhu just took positions in BEAT and LULU for his fund shortly after US market had opened today.











    For the Chinese stock market, nothing is buyable right now according to our system.

    For the Hong Kong stock market, there is one stock that is immediately buyable according to our system: 3344.HK







    上证50ETF期权2月9日上市 A股迎来股票期权时代



    来源:第一财经日报

    1月9日盘后,证监会新闻发言人邓舸在证监会例行新闻发布会上表示,日前已经批准上证所开展股票期权交易试点,批复品种为上证50ETF期权,该期权将于2月9日正式上市。

    随着上证50ETF期权试点进入倒计时,A股也将迎来股票期权时代。在专业人士看来,相比于三年前谢幕的权证,股票期权时代的来临,期权的推出短期可能造成有限波动,长期而言是金融发展的必经之路,将使国内市场更加成熟。

    从权证到期权
    在多位市场人士看来,相比三年前的权证,股市已经硝烟弥漫,再给配权证这样既老式又不好使的武器,这仗可怎么打?权证为何不受待见,下述两个历史时刻可见一斑。

    2007年6月22日,是盐湖钾肥认沽权证的最后交易日,权证行权价格15.1元,纸面上看认沽权证在手,卖股无忧:持有人如果选择行权,发行人就必须以15.1元买入盐湖钾肥的正股,不愿买也得买。

    但是,盐湖钾肥前一交易日的收盘价为47元,由于涨跌幅的限制,盐湖钾肥不可能在一天之内跌到15.1元以下,权证实际上已是废纸一张。投资者如果买了权证,就需要把能卖40多元一股的股票以15.1元的价格卖给发行人,只能说是“有钱任性”了。

    不差钱的投资者也的确造就了“废纸逆袭”的历史奇观:当日该认沽权证开盘价高达0.42元,盘中甚至最高涨至1元,最终收盘也还能卖出1毛多钱。爆炒废纸背后固然有投资者不理性的因素在,但更凸显了权证设计上的缺陷:创设供给量有限。

    海通证券研究所高道德、朱剑涛表示:“权证的发行方只能是上市公司或者有资质的券商,发行上市规模有限,虽然券商可以后续创设权证,但是创设过程耗时,创设规模受券商资本金限制。”

    除了数量有限,权证的行权价格也极为有限。2007年7月招商银行的股价在20元以上,此时券商创设的认沽权证行权价5.65元。这与一年前该权证刚上市价格相同。

    在专业人士看来,并不是5.65元的行权价不合理,而是创设规则本身:“创设是指权证上市交易后,有资格的机构发行与原有权证条款完全一致的权证的行为。”受该规则限制,权证行权价无法与时俱进,而一个极度虚值的权证也失去了对冲的作用,更大程度上沦为对赌的工具。

    A股迎股票期权时代
    与权证相比反观期权,期权兼具权证的好处,两者均能赋予投资者特定时间以特定价格买卖标的的权利,均实行T+0交易,交易的印花税也都不用交,而且上述权证的两大缺陷在期权身上不复存在,变得更加有吸引力。

    首先,不仅是机构投资者,个人投资者也可以是期权的“供应商”。期权交易里投资者只要满足一定资产门槛,完成相关知识测试获取三级交易权限后就可以进行保证金卖出开仓。

    结合当年盐湖钾肥认沽权证的例子,如果还有土豪愿意行使花1块钱“买47元一股的股票卖15.1元”这种权利,投资者可以交纳权利金开出一张期权给这个土豪,随后面临两种情况:一是土豪不行权,投资者收获1块钱的权利金;二是土豪继续任性选择行权,投资者“必须”以15.1元买入土豪手上的股票,转手二级市场40多元卖出。

    当众多投资者加入卖出开仓行列时,市场竞争将使得虚高的期权价格归零,爆炒废纸的故事在期权时代将很难重演。

    其次,行权价也不再受限“与原有权证条款完全一致”,交易所会在股价附加加挂不同行权价的合约,价外期权(虚值)、价内期权(实值)均有,确保对冲功能。不同于当年招行认沽权证几乎没有对冲作用,期权的对冲功能令众多机构翘首以待。

    以海外对冲基金为例,在锁定现货利润时除了清仓外还可以选择买入看跌期权:持仓下跌时可以通过行权价卖出,持仓上涨的情况仍可以享有收益,损失的只是购买看跌期权的权利金。上周五,A股经历了半小时百点大跳水,进入期权时代后,内地投资者也可以选择通过看跌期权撤退,避免现货市场的踩踏事件。

    新配方、新味道、新工具带来的玩法也绝不止上述锁定利润一种而是花样繁多。要想参与其中,对期权还不熟悉的投资者当务之急是恶补知识,考取好分数:按上证所规定,“个人投资者申请的交易权限级别分为一级、二级、三级交易权限等,个人投资者申请各级别交易权限,应当在相应的知识测试中达到规定的合格分数,并具备相应的期权模拟交易经历”。


    How Options Are Priced?


    The value of an option has two components:

    Intrinsic Value. This measures any money that can be released by exercising an option. It is either zero or positive. An option that has positive intrinsic value is said to be in-the-money.

    Time Value. This measures the value of an option over-and-above any intrinsic value it has.

    Even if an unexpired option has no intrinsic value it will still have some time value. Time value reflects the chance that the option may move in-the-money before expiry. Generally speaking, this chance is greater:
    • the longer the time remaining to expiry;
    • the greater the volatility of the underlying asset (the more that returns on the asset fluctuate).

    Taken together these factors – time to expiry and volatility – represent opportunities for the buyer of an option and risks for the writer. Time value is also affected by the level of interest rates. For example, the buyer of a call can deposit the strike price until the contract is exercised. Higher interest rates provide a greater income advantage in buying the call compared to buying the underlying asset in the first instance.

    Calculating intrinsic value is easy, but time value is another matter. The problem is that, unlike (say) a Treasury bill, the eventual payout from an option is not fixed. It depends critically on what happens to the price of the underlying asset over the life of the contract. Option valuation requires a way of modelling the possible payouts resulting from buying or selling an option and the probabilities that these will occur.

    Black-Scholes model
    The standard model for pricing European stock options is commonly known as the Black-Scholes model. We introduced the model in the first article in this series introducing options. Myron S. Scholes and Robert C. Merton were awarded the Nobel Prize in Economics in 1997 for their work on options pricing model. Fischer Sheffey Black (January 11, 1938 – August 30, 1995) unfortunatley had died before a Nobel Prize would have been awarded to him.









    In the financial markets relatively few people work through all the mathematics underlying option pricing, especially the techniques used to price the more complex exotic options developed in recent years. Nevertheless many people in finance rely on pricing models in their day-to-day work and need to develop a reasonable understanding of the inputs and the outputs, the key assumptions and the practical limitations.

    At first glance it might seem that the obvious solution to pricing an option is to forecast what is likely to happen to the price of the underlying asset in the future.

    The problem with this approach is that it is based on subjective probability. Someone who is convinced that the price of a given share is certain to rise would be prepared to pay a high premium for an at-the-money call option on that share. Meantime, someone else who forecast a sharp fall in the share price would think that the call option was virtually worthless. There would be no ‘fair price’ for the option on which everyone could agree.

    The Black-Scholes model does not use subjective probabilities. It is based on the idea that a trader can write an option and eliminate the risks involved in doing so. This is the concept of a riskless hedge. In effect, the model says that the value of an option is determined by the cost of managing the hedge.

    The Black-Scholes model (adapted for a share that pays dividends) needs only five inputs to price a European-style option. The fair value of an option – the theoretical price that should be paid for the contract – is the expected payout at expiry discounted back to the day the option is purchased and the premium paid. The model inputs and outputs are pictured in the figure below:





    The first two inputs are the spot or cash price of the underlying asset and the strike price of the option. These establish whether or not the option has any intrinsic value. They also help to determine how likely or otherwise it is that the option will be exercised.

    For example, if an unexpired call has a strike of $100 and the spot share price is $100, then the option has zero intrinsic value. However there is a good chance – something like an even chance – that the share price will be above $100 at expiry and the option will expire in-the-money. However, if the spot price is $100 and the strike of a call is $200 it is far less likely that the call will ever be exercised. Assuming they share the same underlying and expiry date, the value of an out-of-the-money option is generally less than that of an at-the-money option.

    Obviously, the time to expiry is also important in valuing an option. There is a greater chance that the price of a share will change substantially over a year than during a day. Other things being equal, therefore, a longer-dated option tends to be more valuable because it provides more profit opportunities for the holder.

    Input number five to the model – the cost of carry – is also quite straightforward. It is the rate of interest that applies to the expiry of the option, less any dividends that will be paid out on the underlying over that time period. The binomial example showed that the writer of a call option can hedge the risk by buying shares in the underlying, partially funded by a loan. Therefore the cost of borrowing, less any dividends that are earned on the share while it is held in the hedge portfolio, affects the premium the writer has to charge for the option.

    Finally, the model requires an estimate of the volatility of the underlying share over the life of the option. The reason why the model requires this input is clear. Other things being equal, an option on a highly volatile share is more expensive than one on a share that trades in a narrow range. The chance of an extreme price movement is greater, and the option has a higher expected payout.

    If a share price is highly volatile this increases the chance that it will rise sharply, which increases the potential profits for the buyer of a call. But it also makes it more likely that the share price will fall. Don’t the two effects cancel out? The answer is ‘no’ because the situation is not symmetrical. If the share price rises to high levels the buyer of the call can exercise and make a substantial profit. However if the share price falls the buyer is not forced to exercise and can only lose the initial premium paid for the contract.

    The figure below shows that once we know the five inputs feeding into the pricing of an option, we can easily calcualte the "fair" value of the option. For example, assuming that our stock's price is currently at 100, our strike price is 105, the option will expire in 60 days, interest rate is 1%, and the stock's volatility is 30%, then according to the B-S model, an American-style call's fair value (this is an out of the money call with zero intrinsic value) is 2.9 dollars per share (290 dollars per call option as each option "controls" 100 shares by definition) and a put's fair value (this is an in the money put, which has 5 dollar of intrinsic value) is 7.8 dollars per share (780 dollars per put option).



    Of the five inputs to the model, only the volatility assumption is really problematical. The spot price is available on the stock market. Nowadays it is likely to be broadcast widely on electronic news services such as Reuters or Bloomberg. The strike of an option is a matter of agreement between the various parties, as is the time to expiry. It is not too difficult to forecast the dividend income on a share if the option expires in a few weeks or months (although with longer-dated contracts forecasting dividends becomes increasingly speculative).

    The problem is that the model requires an assumption about the volatility of the underlying asset over the life of the option. This will determine the expected payout on the contract. Unfortunately the future volatility of an asset cannot be directly observed, so it has to be estimated or forecast in some way. In other words, this critical input in determing the options price is more or less "guessed".

    A useful starting point is to look at the past price behaviour of the underlying share and calculate its historical volatility. This can be used as the basis for a forecast of the future. The greater the volatility of a share, the greater the chance of an extreme price movement. This increases the expected payout to the option buyer, and hence the initial premium charged by the writer of the contract.

    Conclusions:

    The Nobel-Prize winning Black-Scholes model requires five inputs to derive an option's "fair" value: the spot price of the underlying; the strike; time to expiry; the volatility of the underlying; and the net carry cost – the cost of borrowing money less any income earned on the underlying. The most problematical input is volatility. This cannot be directly observed and must be estimated. Historical volatility is based on past movements in the price of the underlying and may not reflect the future. Thus, options may not be always priced fairly. Therefore, profit is possible trading mis-priced options if you can identify them correctly.




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