There are primarily six factors that determine the value of an option.

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.

Exercise Price

The lower the exercise price, the higher 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 the exercise price; the higher 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.

Time to Expiration

The more the time to expiration, the greater 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.

Risk-Free Interest Rate

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. 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.

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.

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.