Reading 73

Moneyness refers to whether an option is in the money or out of the money. If immediate exercise of the option would generate a positive payoff, it is in the money. If immediate exercise would result in a loss (negative payoff), it is out of the money. When the current asset price equals the exercise price, exercise will generate neither a gain nor loss, and the option is at the money.

The following describes the conditions for a call option to be in, out of, or at the money. $S$ is the price of the underlying asset and $X$ is the exercise price of the option.

• In-the-money call options. If $S - X > 0$, a call option is in the money. $S - X$ is the amount of the payoff a call holder would receive from immediate exercise — buying a share for $X$ and selling it in the market for a greater price $S$.
• Out-of-the-money call options. If $S - X < 0$, a call option is out of the money.
• At-the-money call options. If $S = X$, a call option is said to be at the money.

The following describes the conditions for a put option to be in, out of, or at the money.

• In-the-money put options. If $X - S > 0$, a put option is in the money. $X - S$ is the amount of the payoff from immediate exercise — buying a share for $S$ and exercising the put to receive $X$ for the share.
• Out-of-the-money put options. When the stock's price is greater than the exercise price, a put option is said to be out of the money. If $X - S < 0$, a put option is out of the money.
• At-the-money put options. If $S = X$, a put option is said to be at the money.

We define the exercise value (or intrinsic value) of an option as the maximum of zero and the amount that the option is in the money. That is, the exercise value is the amount an option is in the money, if it is in the money, or zero if the option is at or out of the money. The exercise value is the value of the option if exercised immediately.

Prior to expiration, an option has time value in addition to its exercise value. The time value of an option is the amount by which the option premium (price) exceeds the exercise value and is sometimes called the speculative value of the option. This relationship can be written as:

option premium = exercise value + time value

At any point during the life of an option, its value will typically be greater than its exercise value. This is because there is some probability that the underlying asset price will change in an amount that gives the option a positive payoff at expiration greater than the current exercise value. Recall that an option's exercise value (to a buyer) is the amount of the payoff at expiration and has a lower bound of zero.

When an option reaches expiration, there is no time remaining and the time value is zero. This means the value at expiration is either zero, if the option is at or out of the money, or its exercise value, if it is in the money.

To model forward commitments, we used no-arbitrage pricing based on an initial value of zero to both parties. With options, however, the initial values of options are positive; the buyer pays a premium (the option price) to the writer (seller). Another difference is that where forward commitments have essentially unlimited gains or losses for both parties (except to the extent that prices are constrained by zero), options are one-sided: potential losses for the buyer, and potential gains for the writer, are limited to the premium paid. For these reasons, the no-arbitrage approach we use for pricing contingent claims is different from the model we use for forward commitments.

The following is some terminology that we will use to determine the minimum and maximum values for European options:

• $S_t$ = price of the underlying stock at time $t$
• $X$ = exercise price of the option
• $T - t$ = time to expiration
• $c_t$ = price of a European call at any time $t$ prior to expiration at time $T$
• $p_t$ = price of a European put at any time $t$ prior to expiration at time $T$
• $R_f$ = risk-free rate

Upper Bound for Call Options. The maximum value of a European call option at any time $t$ is the time-$t$ share price of the underlying stock. No one would pay more for the right to buy an asset than the asset's market value — it would be cheaper simply to buy the underlying asset. At time $t = 0$, the upper boundary is $c_0 \le S_0$, and at any time $t$ the upper boundary is $c_t \le S_t$.

Upper Bound for Put Options. The value of a put cannot exceed its exercise price (the payoff if the underlying drops to zero). However, because European puts cannot be exercised prior to expiration, their maximum value is the present value of the exercise price discounted at the risk-free rate. Even if the stock price goes to zero and is expected to stay at zero, the put buyer will not receive the intrinsic value $X$ until expiration.

At time $t = 0$, for European puts:

$p_0 \le X(1 + R_f)^{-T}$

At any time $t$ during the put's life:

$p_t \le X(1 + R_f)^{-(T-t)}$

Lower Bounds for Options. Theoretically, no option will sell for less than its intrinsic value, and no option can take on a negative value. For European options, however, the lower bound is not so obvious because these options are not exercisable immediately. To determine the lower bounds, we examine the value of a portfolio in which the option is combined with a long or short position in the stock and a pure-discount bond.

For a European call option, construct the following portfolio at $t = 0$:

• A long at-the-money European call option with exercise price $X$, expiring at time $T$.
• A long discount bond priced to yield the risk-free rate that pays $X$ at option expiration.
• A short position in one share of the underlying stock priced at $S_0 = X$.

The current value of this portfolio is $c_0 - S_0 + X(1 + R_f)^{-T}$.

At expiration time $T$, the portfolio pays $c_T - S_T + X$. We collect $c_T = \max[0, S_T - X]$ on the call, pay $S_T$ to cover the short stock, and collect $X$ from the bond.

• If $S_T \ge X$, the call pays $S_T - X$, the bond pays $+X$, and we pay $-S_T$ to cover the short. Payoff: $S_T - X + X - S_T = 0$.
• If $S_T < X$, the call value is zero, we collect $X$ on the bond, and pay $-S_T$ to cover the short. Payoff: $0 + X - S_T = X - S_T \ge 0$.

No matter the outcome, the portfolio value is non-negative. To prevent arbitrage, a portfolio with no possibility of a negative payoff cannot have a negative value:

$c_0 - S_0 + X(1 + R_f)^{-T} \ge 0$

which gives $c_0 \ge S_0 - X(1 + R_f)^{-T}$. Combined with the lower limit of zero:

$c_0 \ge \max\!\left[0,\; S_0 - X(1 + R_f)^{-T}\right]$

Note that $X(1 + R_f)^{-T}$ is the present value of a pure-discount bond with face value $X$.

For a European put option, form the following portfolio at $t = 0$:

• A long at-the-money European put option with exercise price $X$, expiring at $T$.
• A short position on a risk-free bond priced at $X(1 + R_f)^{-T}$ (equivalent to borrowing $X(1 + R_f)^{-T}$).
• A long position in a share of the underlying stock priced at $S_0$.

At expiration $T$, the portfolio pays $p_T + S_T - X$. We collect $p_T = \max[0, X - S_T]$ on the put, receive $S_T$ from the stock, and pay $X$ on the bond (loan).

• If $S_T \ge X$, payoff equals $p_T + S_T - X = S_T - X \ge 0$.
• If $S_T < X$, payoff equals $0$.

The portfolio has no negative payoff, so its value at $t = 0$ must be non-negative:

$p_0 + S_0 - X(1 + R_f)^{-T} \ge 0$

Rearranging gives the minimum value:

$p_0 \ge \max\!\left[0,\; X(1 + R_f)^{-T} - S_0\right]$

There are six factors that determine option prices.

1. Price of the underlying asset. For call options, the higher the price of the underlying, the greater its exercise value and the higher the value of the option. Conversely, the lower the price of the underlying, the less its exercise value and the lower the value of the call. In general, call values increase when the underlying value increases. For put options the relationship is reversed — an increase in the underlying reduces the value of a put.

2. The exercise price. A higher exercise price decreases the values of call options; a lower exercise price increases the values of calls. A higher exercise price increases the values of put options; a lower exercise price decreases the values of puts.

3. The risk-free rate of interest. An increase in the risk-free rate will increase call option values, and a decrease in the risk-free rate will decrease call option values. An increase in the risk-free rate will decrease put option values, and a decrease in the risk-free rate will increase put option values.

4. Volatility of the underlying. Volatility is what makes options valuable. If there were no volatility in the price of the underlying (its price remained constant), options would always equal their exercise values and time/speculative value would be zero. An increase in the volatility of the underlying increases the values of both put and call options; a decrease in volatility decreases both put and call values.

5. Time to expiration. Because volatility is expressed per unit of time, longer time to expiration effectively increases expected volatility and increases the value of a call option. Less time to expiration decreases the time value of a call so that at expiration its value is simply its exercise value.

For most put options, longer time to expiration will increase option values for the same reason. However, for some European put options, extending the time to expiration can decrease the value of the put. In general, the deeper a put is in the money, the higher the risk-free rate, and the longer the current time to expiration, the more likely that extending the option's time to expiration will decrease its value.

To understand this, consider a put at $\$20$ on a stock whose value has decreased to $\$1$. The exercise value of the put is $\$19$, so the upside is very limited and the downside (if the underlying subsequently rises) is significant; because no payment will be received until expiration, the option value reflects the present value of any expected payment, and extending the time to expiration would decrease that present value. While we generally expect a longer time to expiration to increase the value of a European put, in the case of a deep in-the-money put, a longer time to expiration could decrease its value.

6. Costs and benefits of holding the asset. If there are benefits of holding the underlying (dividends, interest payments, or convenience yield), call values are decreased and put values are increased. The reason is most easily understood by considering cash benefits: a dividend or coupon reduces the value of the underlying, and a lower underlying value decreases call values and increases put values.

Positive storage costs make it more costly to hold an asset. We can think of this as making a call option more valuable because call holders can have long exposure to the asset without paying the costs of actually owning it. Puts are less valuable when storage costs are higher.