In this article we are considering determination of the present values of options.

Boundaries: Before we identify the factors determining options values, it is helpful to specify the boundaries within which the value of an option falls.

The minimum value at which a call option sells before the expiration date, say, at time zero, is max (0, S0 – E). This means that C0, the value of a call options can never fall blow zero (this happens when S0 E). To see why this is so, consider a call option with E = 150, S0 = 250 and C0 = 75. In this case it pays an investor to buy the call option for 75 exercise it for 150, and finally sell the stock or 250. By doing so he earns a profit of:

S0 – (C0 + E) = 250 – (75 + 150) = 25

This profit, reflecting arbitrage profits, comes without incurring any risk or cost. Such a profit cannot occur in a well functioning financial market. Hence in such a market C0 cannot sell for less than S0 – E

What is the upper limit for the option price? A call option entitles the holder to buy the underlying stock on payment of a certain exercise price. Hence its value cannot be greater than that of the underlying stock. If it were so, the investor would be better off by buying the stock directly.

Key Factors

As indicated above, the price of call option must fall in shaded region. Put formally,

Max (S0 – E,0) ≤ C0 ≤ S0

Where exactly in the shaded region will the value of as call options be? The precise location of the option value depends on five key factors:

1. Exercise price

2. Expiration date

3. Stock price

4. Stock price variability

5. Interest rate

Exercise Price: BY now it is obvious that, other things being constant, the higher the exercise price the lower the value of the call option. Remember that the value of a call option can never be negative regardless of how high the exercise price is set. Further it has a positive value of there is some possibility that the stock price will be higher than the exercise price before the expiration date.

Expiration date: Other things being equal, the longer the time to expiration date the more valuable the call option. Consider two American calls with maturities of one year and two years. The two year call obviously is more valuable than the one year call because it gives its holder one more year within which it can be exercised.

Stock Price: The value of a call option, other things being constant, increases with the stock price. This point is obvious from the figures showing the relationship between the stock price and the value of call option.

Variability of the Stock Price: A call options has value when there is a possibility that the stock price exceeds the exercise price before the expiration date. Other things being equal, the higher the variability of the stock price, the greater is the likelihood that the stock price will exceed the exercise price.

So fundamental is this point that it calls for another illustration. Consider the probability distribution of the price of two stocks, P and Q just before the call option (with a exercise price of 80) on them expires.

P

Price Probability

60 0.5

80 0.5

Q

Price Probability

50 0.5

90 0.5

While the expected price of stock Q is same as that of stock P, the variance of Q is higher than that of P. The call option (exercise price: 80) on stock P is worthless as there is no likelihood that the price of stock P will exceed 80. However, the call option on stock Q is valuable because there is a distinct possibility that the stock price will exceed the exercise.

Remember that there is a basic difference between holding a stock and holding a call option on the stock. If you are risk averse investor try to avoid buying a high variance stock as it exposes you to the possibility of negative returns. However, you will like to buy option on that stock because you receive the profit from the right tail of the probability distribution, while avoiding the loss on the left tail. Thus, regardless of your risk disposition you will find a high variance in the underlying stock desirable.

Interest Rate: When you buy a call option you do not pay the exercise price until you decide to exercise the call option. Put differently the payment, if any, is made in future, the higher the interest rate, the greater the benefit will be from delayed payment and vice versa. So the value of a call option is positively related to the interest rate.

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