###### Concepts of Arbitrage, Replication and ...

The concepts of arbitrage, replication, and risk neutrality are important to comprehend when... **Read More**

The factors that affect the value of an option include the value of the underlying, exercise price, time to maturity, risk-free rate, volatility, and income or cost associated with the underlying.

The value of the underlying has a direct impact on the right to exercise an option. For a call option, it is exercisable if \(S_T>X\). As such, the value of the call option (and long forward) appreciates when the **spot price **of the underlying increases.

In contrast, the put option (and short forward) appreciates when the spot price of the underlying declines. Recall that the put option is in the money if \(S_T<X\).

The exercise price determines whether an option buyer will exercise the option at the expiration. Remember that the payoff of a call option at maturity is \(\text{max}(0, S_T-X)\). Intuitively, a lower exercise price will increase both the likelihood of exercise and settlement value if it is in the money.

For the put option, the exercise price is the upper bound of the option price. Moreover, the payoff of a put option is \(\text{max}(0, X-S_T)\). As such, a high exercise price increases the value of the put option.

The time value of an option represents the likelihood that favorable changes to the underlying price will increase the profitability of the exercise. For both call and put options, a longer time to maturity increases the likelihood of the option finishing in the money, thus increasing the option’s value

A risk-free rate can be seen as the opportunity cost of holding an asset. A risk-free rate is used in the no-arbitrage valuation of derivatives. Note that the value of a call option at any time before maturity \((t<T)\) is given by: $$c_t=\text{max}(0,S_t-X(1+r)^{-(T-t)})$$

It is easy to see that a higher risk-free rate increases the value of the call option. This is because a higher risk-free rate lowers the present value of the exercise price, provided the option is in the money. For a put option, its value at any time before maturity \((t<T)\) is given by: $$p_t=\text{max}(0,X(1+r)^{-(T-t)}-S_t)$$

Intuitively, a higher risk-free rate decreases the exercise value of a put option due to the same explanation in the call option.

Volatility measures the expected dispersion of future movements of an underlying asset. Higher volatility of the underlying asset increases the chances of call and put options finishing in the money without affecting the downside case – the option expires worthless. For instance, as volatility increases, a broader possibility of underlying prices increases the time value of an option and the likelihood of being in the money.

In contrast, lower volatility decreases the time value of both put and call options.

Income (or other non-cash benefits) accrue to the underlying asset owner, not the derivative owner. In other words, the present value of the income or benefits is subtracted from the underlying price. As such, income decreases the value of a call option and increases the value of a put option.

If the asset owner incurs costs (in addition to opportunity cost), compensation is done to cover the added costs. As such, the present value of the costs is added to the underlying price. Therefore, cost increases the value of the call option and decreases the value of the put option.

The table below summarizes the factors that affect the value of an option.

$$\small{\begin{array}{l|l|l} \textbf{Factor} & \textbf{Value of European Call option} & \textbf{Value of European Put option}\\ \hline\text{Value of the Underlying} & \text{Directly proportional} & \text{Inversely proportional} \\ \hline\text{Exercise price} & {\text{Inversely proportional}\\ \text{(as the exercise price increases,}\\ \text{value decreases)}} & {\text{Directly proportional}\\ \text{(as exercise price increases,}\\ \text{value increases)}}\\ \hline\text{Time to Maturity} & \text{Directly proportional} & \text{Directly proportional}\\ \hline\text{Risk-free rate} & \text{Directly proportional} & \text{Inversely proportional}\\ \hline \text{Volatility} & \text{Directly proportional} & \text{Directly proportional}\\ \hline \text{Benefits} & \text{Inversely proportional} & \text{Directly proportional} \\ \hline\text{Costs} & \text{Directly proportional} & \text{Inversely proportional} \end{array}}$$

## Question

Which of the following ismost likely to havethe same effect on the value of a call option? A. High risk-free rate and negative cost of carry. B. Low exercise price and positive cost of carry. C. Longer time to maturity and low volatility.## Solution

The correct answer isA. Both a high risk-free rate and low cost of carry increase the value of a call option. The risk-free rate increases the value of the call option because a higher risk-free rate lowers the present value of the exercise price, provided the option is in the money. Recall that cost of carry is the net of the costs and benefits associated with owning an underlying asset for a period. Therefore, the negative cost of carry implies that the cost associated with the underlying is higher than the benefits. The present value of the costs is added to the underlying price. Therefore, cost increases the value of the call option and decreases the value of the put option.B is incorrect. A low exercise price will increase both the likelihood of exercise and settlement value if it is in the money. A positive cost of carry implies that the present value of the benefits associated with the underlying is higher than the present value of the costs. The present value of the income or benefits is subtracted from the underlying price. As such, income or other non-cash decreases the value of a call option and increases the value of a put option.C is incorrect. The time value of an option represents the likelihood that favorable changes to the underlying price will increase the profitability of the exercise. Therefore, a longer time to maturity for a call option increases the option’s value. Lower volatility decreases the time value of both put and call options.