###### Swap Contracts

Since a swap involves a series of payments over a fixed period of... **Read More**

Aside from the moneyness, time to expiration, and exercise price, other factors determine the value of an option. The risk-free rate, volatility of the underlying and cash flows from the underlying and cost-of-carry impact option values.

With American-style options, as the time to expiration increases, the value of the option increases. This makes perfect sense: with more time before the expiration date; there are higher chances of the option moving in-the-money.

- As the time to expiration
*increases*, the value of a call option*increases*. - As the time to expiration
*increases*, the value of a put option also*increases*.

Here, the simplest way to think about this is as a rate of return on a stock. Let’s say you have the choice between buying a bond worth $1000 or one share of stock priced at $1000. If you know the risk-free interest rate is a known 5%, you would expect the stock price to increase by more than 5% on average. Otherwise, why would you buy a share of stock instead of investing in a risk-free bond? Therefore,

- As time the risk-free rate
*increases*, the value of a call option*increases*. - However, as the risk-free rate
*increases*, the value of a put option*decreases*.

Volatility is considered the most significant factor in the valuation of options. As volatility increases, the value of all options increases. Since the maximum loss for the buyer of a call or put option is limited to the premium paid, we can conclude that there are higher chances of the option expiring in-the-money as volatility increases.

- As volatility
*increases*, the value of a call option*increases*. - As volatility
*increases*, the value of a put option*increases*.

The combined effects of time to expiration and volatility of the underlying give rise to the concept of the time value of an option. This reflects the value of the uncertainty that arises from the volatility of the underlying. Thus, the value of an option declines as expiration approaches and can be measured as a time value decay factor.

Payments from an underlying may include dividends. As we’ve seen previously, immediately after paying a dividend, the stock price falls by the dividend amount. However, the benefits of these cash flows to the holders of the underlying security do not pass to the holder of a call option. Therefore,

- As dividends
*increase*, the value of a call option*decreases*. - However, as dividends
*increase*, the value of a put option*increases*.

Some options can also be written on commodities such as oil, gold, corn, etc. Holders of a call option can participate in the upside movement of the underlying without incurring the asset’s carrying costs (storage costs, insurance costs, etc.). As such:

- As carrying costs
*increase*, the value of a call option*increases*. - However, as carrying costs
*decrease*, the value of a put option*decreases*.

A call option gives the holder the right to buy the stock at a specified price. The value of the call is **always less** than the value of the underlying stock. Thus,

$$ c\le { S }_{ 0 }$$

If the value of a call were to be higher than the value of the underlying stock, arbitrageurs would sell the call and buy the stock, earning an instant risk-free profit in the process.

A put option gives the holder the right to sell the underlying stock at a specified price. The value of a put is always less than the strike price. Thus,

$$ p\le K\quad $$

If the value of a put were to be higher than the strike price, everyone would move swiftly to sell the option and then invest the proceeds at a risk-free rate throughout the option’s life.

Call options can never be worth less than zero as the call option holder cannot be forced to exercise the option. The lowest value of a call option has a maximum price of zero, and the underlying price less than the present value of the exercise price. This is written as follows:

$$c_0 \geq max(0, S_0 – \frac{X}{(1+r)^T}) $$

A put option has an analogous result. A put option can never be worth less than zero as the option owner cannot be forced to exercise the option. The lowest value of a put option is the maximum of zero, and the present value of the exercise price less the value of the underlying. This is expressed as follows:

$$p_0 \geq max(0, \frac{X}{(1+r)^T} – S_0) $$

QuestionBoth a European call and European put options expire in 90 days, with the same exercise price of $70 and the same underlying asset. The current price of the underlying asset is $60. The risk-free rate of return is 5%. Find the lower bounds of both options.

A. European call = max(0, -9.13) = 0; European Put = max(0,9.13) = 9.13

B. European Call = max(0, -9.43) = 0; European Put = max(0,9.43) = 9.43

C. European Call = max(0, -9.13) = 0; European Put = max(0,9.13) = -9.13

SolutionThe correct answer is A.

European call (c

_{0}): max[0, 60 – 70/(1 + 5%)^{0.2465}] = max[0, -9.13] = 0European put (p

_{0}): max[0, 70/(1 + 5%)^{0.2465}– 60] = max[0, 9.13] = 9.13