# Factors affecting value of an option

CFA level I / Derivatives / Basics of Derivative Pricing and Valuation / Factors affecting value of an option

There are primarily six factors that determine the value of an option. The factors are underlying price, exercise price, time to expiration, risk-free rate, volatility, and interim cash flows & costs.

Effect of the value of the underlying: The call option can be viewed as buying the underlying and the put option can be viewed as selling the underlying. So, the value of call option increases with an increase in the value of the underlying and the value of put option decreases with an increase in the value of the underlying.

Effect of the exercise price: The lower is the exercise price, the higher will be the value of the call option because we would be able to buy the underlying at a lower price. The opposite is true for the put option i.e. the higher is the exercise price; the higher will be the value of the put option as we would be able to sell the underlying at a higher price. Hence, the value of a European call option is inversely proportional to the exercise price and the value of a European put option is directly proportional to the exercise price.

Effect of time to expiration: The more is the time to expiration, the greater is the value of the option. The logic is that the underlying has more potential for movement and thus the option will have a higher value. With the same logic, even the put option will increase with an increase in the time to expiration. But there is some exception to the European put options. If the risk-free rate is high, the volatility is lower, and the European put option is deep-in-the-money, then the value of put option can decrease with increase in the time to expiration.

You can easily remember it with the example of a bankrupt company. Suppose you buy a European put option with one year to expiry for the exercise price of \$100. Just after the option purchase, the company gets caught in a scandal and goes bankrupt. The price of the stock falls to zero and is never going to recover and is going to remain at the price of zero. Since the option is European, we cannot exercise the option before the expiration. So, the value of the option will be simply the present value of \$100. The value will keep on increasing as the time to expiration decreases and we move closer to the expiry.

The long-dated European put option having 10-years, 15-years or 20-years to expiration almost always decrease in value with increase in the time to expiration because the negative impact of the discount factor of the risk-free rate dwarfs the positive impact of the movement of the underlying due to longer time to expiration. Because the lower movement is limited because the underlying price cannot fall below zero.

Effect of the risk-free rate of interest: The value of call option increases in the value with an increase in the risk-free rate and the value of put option decreases with an increase in the risk-free rate. It is easier to remember if we know the put-call parity for European options which will be discussed later in this chapter. As per put-call parity, c0 + X*(1+r)-T = p0 + S0. If we increase the risk-free rate, then the value of factor X*(1+r)-T falls and the value of call option has to increase for the parity of the equation.

Effect of volatility: Both call options and put options increase in value with an increase in volatility. The call option increases in value because the underlying price can increase to a higher price because of high volatility. Similarly, the put option increases in value because the underlying price can fall to a lower price due to higher volatility. The volatility factor and time to expiration factor are combined to get the time value of an option. The volatility can have more impact if the time to expiration is longer. The option prices generally decrease as the options approach expiration date and this is referred to as time value decay.

Effect of payments on the underlying and the cost of carry: The call option is equivalent to the long position in the underlying and the put option is equivalent to the short position in the underlying. The value of the underlying decreases with benefits and increases with the cost of carry. So, the value of European call option is inversely proportional to the benefits and directly proportional to the cost incurred in holding the underlying. The opposite is true for the European put option i.e. the value of European put option increases with more benefits and decreases with more cost of carry.

 Factors Value of European call option Value of European put option Value of underlying Directly proportional Inversely proportional Exercise price Inversely proportional Directly proportional Time to expiration Directly proportional Directly proportional with the exception of long-dated options, deep-in-the-money options, higher-risk free rate Risk-free rate Directly proportional Inversely proportional Volatility Directly proportional Directly proportional Benefits Inversely proportional Directly proportional Costs Directly proportional Inversely proportional

(58.18) Which of the following factors is least likely to impact the call and put options differently?

(a) Time to expiration
(b) Risk-free rate
(c) Costs associated with holding the asset

(58.19) What will be the most likely impact on the value of a European call option with a decrease in the risk-free rate?

(a) Increase
(b) Decrease
(c) No impact

(58.20) Which of the following options is likely to be most valuable on the same underlying trading at \$100?

(a) European put option with exercise price of \$80 and two months to expiry
(b) European put option with exercise price of \$80 and three months to expiry
(c) European put option with exercise price of \$75 and two months to expiry

(58.21) Which of the following factors is least likely to be favorable for the value of a European call option?

(a) Increase in exercise price
(b) Increase in underlying price
(c) Increase in risk-free rate

Solutions:

(58.18) Correct Answer is A: The risk-free rate impacts the call and put options in opposite way. The costs of holding the asset also impact the options in opposite way. Both call and put options increase in value with increase in the time to expiration with the exception of European deep-in-the-money put option with a higher risk-free rate, low volatility and longer time to expiration.

(58.19) Correct Answer is B: The European call option will decrease in value with a decrease in the risk-free rate as the call option value is directly proportional to the risk-free rate.

(58.20) Correct Answer is B: The European put options increase in value with increase in time to expiration with exception of deep-in-the-money put options or when the time period is too long. The lower is the exercise price of the European put option, the lower is its value. Therefore, the European put option with an exercise price of \$80 and three months to expiration will have the maximum value.

(58.21) Correct Answer is A: An increase in risk-free rate and the underlying price will lead to an increase in the value of a European call option. An increase in the exercise price will decrease the value of a European call option.

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