Let us look at the different types of options.
There are two parties in any option:
Types of options based on purpose:
Types of options based on when they can be exercised:
Value of European Option at Expiration
Call option
Put option
Relationship between the value of the option and the value of the underlying
Effect of the Exercise Price
Consider two call options with similar attributes (time to expiration, underlying) but different exercise prices. Assume the exercise price for call option 1 is 25 and that for call option 2 is 30. The right to buy at a lower price of 25 will be more valuable than the right to buy at a higher price of 30.
Similarly, in the case of a put option, the right to sell for a higher price is more valuable than the right to sell for a lower price.
Relationship between the value of the option and the exercise price
Moneyness of an option: Indicates whether an option is in, at, or out of the money. The table below shows the moneyness of an option for various relative values of S and X.
Call Option | Put Option | |
In the money | S > X | S < X |
At the money | S = X | S = X |
Out of the money | S < X | S > X |
Effect of Time to Expiration
Longer-term options should be worth more than shorter-term options. For instance, you have a bought a call option on a stock at an exercise price of $25. Which one would be more valuable to you; an option that expires in a week or an option that expires a year later? Without doubt, the option that expires a year later as it gives enough time for the stock price to increase and make the option valuable.
But, is it true for a put option? Partly, yes. The more the time to expiration, the more the opportunity for the stock to fall below its exercise price. What does the put option holder get in return when the underlying falls below the exercise price? Just the exercise price. The longer the holder waits to exercise the option, the lower the present value of the payoff. If the discount rate is high, then a longer time to expiration results in a lower present value for the payoff.
Relationship between the value of the option and the time to expiration
Effect of the Risk-free Rate of Interest
If the risk-free rate is high, then the call price is higher. For example, you want to invest in a stock whose price is $100. You can either buy the stock for $100 or buy the call option on the stock for $5. Both these actions give you exposure to the stock. If you buy the call option, then you can invest the remaining $95. If the interest rates are high, then buying the call option is more valuable because of the interest earned.
In the case of puts, the longer time to expiration with a higher risk-free rate lowers the present value of the exercise price when the option is exercised. That is, when the time to expiration is longer you get the money later and, if the risk-free rate is high, it is discounted by a higher number which lowers the value of the payoff.
Relationship between the value of the option and the risk-free interest rate
Effect of Volatility of the Underlying
Volatility is good for both call and put options. If the underlying stock becomes more volatile, then the probability of the options expiring deep in the money becomes greater.
Relationship between the value of the option and the volatility of the underlying
Effect of Payments on the Underlying and the Cost of Carry
Call option | Put option | ||
Benefits (dividends on stocks, interest on bonds, convenience yield on commodities) | Stocks and bonds fall in value when dividends and interest are paid. | Decreases because call holders do not get these benefits. | Increases |
Carrying costs | Increases as call holders do not incur these costs. | Decreases |
Relationship between the value of the option and benefits/costs of the underlying
Lowest Price of Calls and Puts
In this section, we look at the least price one would be interested in paying for a call option. Let us consider a call option on a stock with a strike price of X that expires at time T. The initial price of the underlying at time t = 0 is _{, }the underlying price at expiration is _{, }and the risk-free rate is r%. If the option is in the money, the payoff at expiration will be S_{T }– X. Lowest price of the call option is given by:
Interpreting the above equation:
Similarly, a put option is equivalent to selling short an asset and investing the proceeds (present value of X) in a risk-free bond at r% for T periods. It will pay X at expiration. A put can never be worth less than zero as the holder cannot be forced to exercise it. The lowest price for a put option is given by: