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
How Options Are Priced?
The value of an option has two components:
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.
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.
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.
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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