75. Pricing and Valuation of Options

• Option values cannot be negative; they can only be zero or positive.
• Nonnegative values are the lower and upper bounds on options.
• Calculating the upper bound on a European call option includes the present value calculation of the exercise price except when simply using the underlying asset price.

Option Bounds

• The minimum value of a European call option is: ct ≥ Max[0, St – X(1 + Rf)-(T-t)]

• The maximum value of a European call option is: St

• The minimum value of a European put option is: pt ≥ Max[0, X(1 + Rf)-(T-t) – St]

• The maximum value of a European put option is: X(1 + Rf)-(T-t)

• Options have upper and lower no-arbitrage price bounds.

• Forward commitments represent obligations, and therefore, there are no price bounds, except a lower bound when the underlying asset cannot have a negative value.

• Option buyers pay a premium upfront, while forward buyers do not.

• An option's price (or premium) comprises its intrinsic and time values.

• An out-of-the-money option's entire premium is its time value due to having zero intrinsic value.

• Time value is zero at an option's expiration date.

• The amount by which an option's premium exceeds its intrinsic value equals the time value.

• When the stock price is above the strike price, a put option is out-of-the-money.

• When the stock price is below the strike price, a call option is out-of-the-money.

• An increase in the risk-free interest rate increases call option values and decreases put option values.

• A higher exercise price increases the value of a put option at expiration.

• The risk-free interest rate and volatility only affect the time value of options, which is zero at expiration.

• At expiration, an investor will exercise a European put option on a stock if the stock price is less than the exercise price.

• A put option gives its owner the right to sell the underlying good at a specified exercise price for a specified time period.

• When the stock's price is less than the exercise price, a put option has value and is said to be in-the-money.

• Dividends or interest paid by the underlying asset decrease the value of call options.

• For a European style put option, time value equals its market price minus its exercise value.

• Exercise value is synonymous with intrinsic value.

• A put's exercise value is the maximum of zero or its exercise price minus the stock price.

• A call option is in the money when the exercise price is less than the market price of the asset.

• If the market price of a stock is 27, a call option with an exercise price of 25 and a put option with an exercise price of 30 are both in the money.

• The put option is in the money because the option holder has the right to sell the stock for more than its market price.

• The call option is in the money because the option holder has the right to buy the stock for less than its market price.

• A European put option's lower bound cannot exceed the difference between the present value of the exercise price and the underlying asset price.

• The lower bound on a European put option is always zero or positive but can never be negative.

• pt ≥ Max[0, X(1 + Rf)¯(T-t) – St]. is the lower bound of a European put option

• A one-year European call option has an exercise price of X = $500 when the underlying asset trades at So = $485, and the risk-free rate is r = 1.25%.

• With the underlying asset price at St = $510, the no-arbitrage upper bound of this option in six months is $510.

• The upper bound of this option is the underlying asset price because no call buyer would pay more for the option than the asset's market price.

• A decrease in the risk-free rate would decrease call option values and increase put option values.

• A decrease in the risk-free rate will increase the value of both positions since the investor is short calls and long puts.

• A short put position will be more valuable as a result of an increase in the price of the underlying asset.

• An increase in the price of the underlying asset will decrease the value of a put option, which makes a long position in the put less valuable and a short position more valuable.

• An increase in either the volatility of the underlying asset price or time to expiration would increase the put value and decrease the value of a short position.

• At expiration, exercise value is equal to time value for an out-of-the-money call or an out-of-the-money put.

• The time value of an option is zero at expiration.

• For an out-of-the-money option, the exercise value is zero at expiration.

• An increase in volatility will increase the value of a call option.

• Increased volatility of the underlying asset increases both put values and call values.

• A higher exercise price or an increase in cash flows on the underlying asset decrease the value of a call option.

• Profits from a long call are positively correlated with the stock price, and profits from a long put are negatively correlated with the stock price.

• For a call, the buyer's (or the long position's) potential gain is unlimited and is in-the-money when the stock price (S) exceeds the strike price (X).

• For a put, the buyer's (or the long position's) potential gain is equal to the strike price less the premium. A put option is in-the-money when X > S.

• The time value of a European call option with 30 days to expiration will most likely be greater than the current option premium if the option is currently out-of-the-money.

• The option premium is made up of time value and intrinsic value.

• Intrinsic value is positive if an option is in-the-money and zero otherwise.

• Time value is always positive for call options.

• A call option's intrinsic value increases as the stock price increases above the strike price, while a put option's intrinsic value increases as the stock price decreases below the strike price.

• For a call option, as the underlying stock price increases above the strike price, the option moves farther into the money, and the intrinsic value is increasing.

• For a put option, as the underlying stock price decreases below the strike price, the option moves farther into the money, and the intrinsic value is increasing.

• A European put option that has a longer time to expiration may be worth less than the put that is nearer to expiration.

• Options with greater time to expiration are worth more than otherwise identical options that are nearer to expiration normally.

• However, this relationship may not hold for European puts.

• An option's intrinsic value is equal to the amount the option is in the money, and the time value is the market value minus the intrinsic value.

• Intrinsic value is the amount the option is in the money and is the value that would be realized if the option were at expiration.

• Prior to expiration, the option's market value will normally exceed its intrinsic value, and the difference between market value and intrinsic value is called time value.

• The upper bound of a European put option is the exercise price.

• Because European puts cannot be exercised prior to expiration, their maximum value (or upper bound) is the present value of the exercise price, discounted at the risk-free rate, or X / (1 + Rf)(T-t).