Options are an asset to the buyer and a liability to the seller. Hence, the buyer must pay a premium to the seller up front to compensate him for taking the risk. A call option gives the buyer an option to buy the underlying at the strike/exercise price while a put option enables him to sell the underlying.

A European option is an option that can be exercised only at expiration. This simplifies the valuation because we only need to determine the option value at expiry.

A call option has value to a buyer only when the spot price at expiration (ST) is greater than the exercise price (X) otherwise its value is zero. In other words, the value of a call option to the buyer is a maximum of 0 or any excess of future spot price over exercise price (ST – X). This is called intrinsic value or exercise value of the option, i.e. Max (0, ST – X).

A put option has value only if the buyer can sell the underlying at an exercise price (X) which is greater than the spot market price (ST) otherwise it is better to let the option expire and sell the underlying in open market. A put option value at expiration is Max (0, X – ST).

Effect of the value of underlying

The value of a European call option increases while a European put option decreases with an increase in the price of the underlying and vice versa. This is because a call option is equivalent to buying the underlying while the put option is equivalent to selling it.

Effect of the exercise price

The level of an option’s exercise price with respect to the spot price of the underlying at expiration determines the option’s moneyness, an indication of whether the option is worth exercising or not.

If the exercise price is low, there is a greater chance that a call option shall be in-the-money (i.e. future spot price shall be higher than the exercise price) and that the difference shall be greater. This shows that there is an inverse relationship between a call option’s value and its exercise price. This is exactly the opposite in case of a put option, whose value varies directly with the exercise price, but exercise price also sets a maximum cap on a put option’s value because even if the spot price goes to zero, X – ST cannot exceed the exercise price, X.

Effect of time to expiration

A European call option is more valuable when the time to expiration is longer. It is because, over a longer duration, there is a greater chance that the spot price will rise about the exercise price. Further, since the holder of a call option is required to make payment, delaying that payment has time-value-of-money benefits.

A European put option benefits from a longer time period because there is a greater probability of spot price falling below the exercise price, but this effect may be offset by the time value of money effect of the proceeds from the exercise of put. This is because when a buyer exercises a put, he sells the underlying and receives cash whose present value is lower if the time period is longer, the interest rate is higher and where the option is deep-in-the-money (when the difference between X and ST is larger).

Effect of the risk-free rate of interest

Since the buyer of a European option can earn interest on the money which will be used to buy the underlying when he eventually exercises the option, a higher risk-free interest rate over a longer time period means higher option value.

The value of a European put option is inversely related to the risk-free interest rate. It is because a higher interest rate means a lower present value of the exercise price when the option is exercised.

When the options are not exercised, the level of risk-free interest rate becomes irrelevant.

Effect of volatility of the underlying

When volatility of the underlying is high, (a) there is a greater chance that the spot price will go above the exercise price in case of a European call option, and (b) the difference will be larger. However, if the spot price declines further below the exercise price, it does not matter regarding an option-holder decision. This is why volatility and option value are directly related.

The value of a European put option responds similarly to the volatility of the underlying. Due to high dispersion (as measured by the range), there is a greater chance that the spot price goes below the exercise price by a bigger margin. If the spot price is above the exercise price and it goes further up, it does not affect the option value adversely because it is already zero.

Time to expiration and the volatility of underlying results in a difference between an option’s market value and its exercise value called the time value of options. It represents the market’s assessment of the potential for an increase in exercise value of an option due to time to expiration is longer and volatility versus any decrease in exercise value. However, as the option approaches its expiry, its time value shrinks and the option’s market value gravitates towards the exercise value through a process called time value decay.

Effect of payments on the underlying and cost of carry

Since the value of underlying falls as soon as it pays dividends or interest, the value of a European call option responds negatively to any such payments. However, they benefit from any cost of carry because the option holders do not have to bear those costs.

The value of a European put option benefits from a decrease in the underlying’s price due to payment of benefits. However, its value suffers from any increase in the cost of holding the underlying.

Lowest prices of calls and puts

The lowest value of a call option is the greater of zero or the excess of underlying’s price over the present value of the exercise price. Similarly, the lowest value of a put option is the greater of zero or the excess of the present value of the exercise price over the underlying’s price.

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