Option Valuation

Option Valuation Study Notes

16.1 Introduction

  • Intrinsic Value

    • Definition: The intrinsic value of an option is determined as the stock price minus the exercise price. It represents the profit that could be attained by immediately exercising an in-the-money call option.

  • Time Value

    • Definition: The time value of an option is defined as the difference between the option’s market price and its intrinsic value.

  • Determinants of Option Value

    • The value of an option is influenced by the following factors:

    • Stock Price (S): Higher stock prices increase the value of an option.

    • Exercise Price (X): Higher exercise prices decrease the value of an option.

    • Volatility of Price (σ): Increased volatility raises the value of an option.

    • Time to Expiration (T): Longer time until expiration increases the value of an option.

    • Interest Rate (r): Higher interest rates contribute to an increase in the option's value.

    • Dividend Rate: Higher dividend payouts tend to decrease the value of an option.

16.2 Call Option Value Before Expiration

  • Figure 16.1 Breakdown

    • The value of a call option before expiration can be categorized into:

    • Intrinsic Value: The value at expiration equals to the intrinsic value of the option.

    • Time Value: The portion of the option's price that exceeds its intrinsic value, representing the potential for future profit.

  • Positioning of Options

    • In-the-money: When the stock price exceeds the exercise price.

    • Out-of-the-money: When the stock price is below the exercise price.

16.3 Black-Scholes Option Valuation

  • Black-Scholes Pricing Formula

    • Formula: C0 = S0 e^{(rf - d)T} N(d1) - Xe^{-rf T} N(d2)

    • Where:

    • d1 = rac{ ext{ln}(S0/X) + (r_f - d + rac{σ^2}{2}) T}{σ ext{sqrt}(T)}

    • d2 = d1 - σ ext{sqrt}(T)

    • Constants Explained:

    • C_1: Current call option value

    • S_0: Current stock price

    • N(d): Probability function used to find the area under the normal distribution curve up to the value of $d$

    • X: Exercise price

    • e: Base of the natural logarithm, approximately equal to 2.71828

    • d: Annual dividend yield of the underlying stock

    • r: Risk-free interest rate

    • T: Time until expiration

    • ln: Natural logarithm

    • σ: Standard deviation of annualized continuously compounded rate of return

  • Implied Volatility

    • Definition: The implied volatility is the standard deviation of stock returns that aligns with the market value of the option. It reflects the market's expectations of future volatility.

16.4 Implied Volatility Trend Analysis

  • Figure 16.5

    • Displays the historical trend of implied volatility for the S&P 500 over various time periods, highlighting key events:

    • Gulf War

    • Long Term Capital Management (LTCM) collapse

    • September 11 terrorist attacks

    • Iraq War

    • Subprime and credit crises

    • U.S. debt downgrade

16.5 Put-Call Parity Relationship

  • Put-Call Parity Formula

    • Represents the relationship between put and call prices:

    • C + PV(X) = P + S_0

    • Where:

      • C: Call Price

      • PV(X): Present value of exercise price

      • P: Put Price

      • S_0: Current stock price

  • Implications of Put-Call Parity

    • Indicates that there is a systematic relationship between the prices of puts and calls, allowing arbitrage opportunities in efficient markets.