Black & Scholes article : option pricing

Question

I am currently reading the famous article by Fischer Black and Myron Scholes called 'The pricing of Options and Corporate liabilities'.

Just at the beginning of the article, they go on and explain what an option is on the example of a 'call' option. This means, the option one takes is the right to buy a stock at 'strike price' on expiration date. Thus, to earn something, the strike-price must be lower than the stock price at maturity to make money. What puzzles me is the following sentence :

In general, it seems clear that the higher the price of the stock, the greater the value of the option. When the stock price is much greater than the exercise price, the option is almost sure to be exercised. [...] On the other hand, if the price of the stock is much less than the exercise price, the option i sure to expire without being exercised, so its value will be near zero.

If the expiration date is very far in the future, then the price of a bond that pays the exercise price on the maturiy date will be very low, and the value of the option will be approximately equal to the price of the stock. On the other hand, if the expiration date is very near, the alue of the option will be approximately equal to the stock price minus the exercise price, or zero, if the stock price is less than the exercise price. Normally, the value of an option declines as its maturity date approaches, if the value of the stock does not change.

http://www3.nccu.edu.tw/~cclu/FinTheory/Papers/Black-Scholes73.pdf

Could someone explain to me the part in bold : in particular 'the price of the a bond that pays..' and why the value of the option should decline in time.

Cheers

Answer 1

Consider the black-scholes-merton result. Notice that the expected value of the bond is its present value, discounted from the expiration date.

The same is not applied to the price of the stock.

The further in the future you go, the less value the bond carries because it's being discounted into oblivion.

Now, looking at d1, as time tends towards infinity, so does d1.

N(d1) is a probability. The higher d1, the higher the probability and vice versa, so as time increases, the probability for S trends to 100% while K is discounted away.

Note that the math doesn't yet fully model reality, as extremely long dated options such as the European puts Buffett wrote were traded at ~1/2 the value the model said he should've.

He still had to take a GAAP loss: http://www.berkshirehathaway.com/letters/2008ltr.pdf

Answer 2

Theta is a variable in options pricing. Theta aka time decay decreases the price of the option over time.

The reason of this is that you have to think of the option as insurance. It is a hedge against actual holdings in an asset. Would you pay more or less for insurance that covers you for a year's time, would you pay more or less for insurance that covers you for a week? The answer is that the market will pay less for insurance that covers them for a lower period of time.

This is one of several ways of thinking about it.

There is also the probability that the option will be profitable at all, the further out in the future, the more likely it will be profitable and people will pay a premium for it.

There are other variables in the black-scholes formula and it is the most widely used options pricing formula. But keep in mind, the geniuses that made up the formula blew up their hedge fund thinking they could sell the options at an inflated premium from their own formula to everyone. Ironic really.