Pricing and Valuation of Options

An option is a derivative contract that gives the holder the right, but not the obligation, to transact in an underlying asset at a predetermined price on or before a future date.

• A call option gives the holder the right to buy the underlying asset
• A put option gives the holder the right to sell the underlying asset

Option Exercise Styles

There are two primary styles for exercising an option:

• European Options: Can be exercised only on the expiration date
• American Options: Can be exercised at any time up to and including the expiration date

The total price of an option can be broken down into three key components: its exercise value, moneyness, and time value.

Exercise Value (Intrinsic Value)

The exercise value is the payoff an investor would receive from exercising the option immediately. It can never be less than zero.

Call Option Exercise Value

• At expiration (t=T), the value is the underlying price minus the exercise price, or zero.
  C_T = max(0, S_T – X)
• Before expiration (t < T), the value is the underlying price minus the present value of the exercise price, or zero.
  c_t = max S_t – X (1 + r)^(T–t)

Put Option Exercise Value

• At expiration (t=T), the value is the exercise price minus the underlying price, or zero.
  P_T = max(0, X – S_T)
• Before expiration (t < T), the value is the present value of the exercise price minus the underlying price, or zero.
  p_t = max X (1 + r)^(T–t) – S_t

Where: S is the spot price of the underlying, X is the exercise price, r is the risk-free rate, and (T–t) is the time remaining to expiration.

Moneyness

Moneyness describes the relationship between the underlying asset's price and the option's exercise price.

• In-the-Money (ITM): The option has a positive exercise value. For a call, S > X; for a put, X > S
• Out-of-the-Money (OTM): The option has zero exercise value. For a call, S < X; for a put, X < S
• At-the-Money (ATM): The underlying price is equal to the exercise price (S = X)

Time Value (Extrinsic Value)

Time value is the portion of an option's price that is above its exercise value. It represents the potential for the option to become more profitable before expiration.

Key characteristics of time value:

• It decreases as the time to expiration approaches, a phenomenon known as time decay
• It is zero at the moment of expiration
• Higher volatility in the underlying asset's price leads to a higher time value for the option, and lower volatility leads to a lower time value

Several key factors influence the price of call and put options. The table below summarizes the effect of an increase in each factor.

Factor	Effect on Call Value	Effect on Put Value	Reasoning
Value of the Underlying (S)	Increase	Decrease	A higher spot price is favorable for the right to buy (call) and unfavorable for the right to sell (put).
Exercise Price (X)	Decrease	Increase	A lower exercise price makes a call more valuable, while a higher exercise price makes a put more valuable.
Time to Expiration (T)	Increase	Increase	More time provides more opportunity for favorable price movements for both option types.
Risk-Free Interest Rate (r)	Increase	Decrease	A higher rate lowers the present value of the exercise price, which benefits calls (pay less in today's dollars) and hurts puts (receive less in today's dollars).
Volatility of the Underlying	Increase	Increase	Higher volatility increases the chance of a large price swing, which increases the time value for both calls and puts. The effect is the same for both option types.
Income/Benefits on Underlying	Decrease	Increase	Income (like dividends) lowers the underlying's price, which hurts calls and helps puts.
Costs on Underlying	Increase	Decrease	Costs of carry (like storage) increase the underlying's forward price, which helps calls and hurts puts.