Glossary

14.2 Options Valuation and Strategies

3 min read•july 18, 2024

Option pricing models are essential tools for valuing financial derivatives. The binomial and Black-Scholes models provide frameworks for calculating option prices based on key inputs like asset price, , and .

Several factors influence option prices, including the underlying asset price, strike price, , volatility, and interest rates. Understanding these factors helps investors analyze and construct various option strategies to manage risk and potentially enhance returns.

Option Pricing Models

Binomial and Black-Scholes option pricing

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  • Binomial option pricing model assumes the underlying asset moves up or down by a fixed percentage in discrete time steps, creating a binomial tree of possible price paths
    • Option value is calculated by working backward through the tree, discounting the expected payoff at each node
  • Black-Scholes-Merton (BSM) model assumes the underlying asset follows a geometric Brownian motion in continuous time
    • Uses a partial differential equation to determine the option price based on key inputs: current underlying asset price (S0S_0), strike price (KK), time to expiration (TT), (rr), and volatility (σ\sigma)
    • price formula: C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) - Ke^{-rT} N(d_2)
    • price formula: P=Ke−rTN(−d2)−S0N(−d1)P = Ke^{-rT} N(-d_2) - S_0 N(-d_1)
      • d1=ln⁡(S0K)+(r+σ22)TσTd_1 = \frac{\ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}} and d2=d1−σTd_2 = d_1 - \sigma \sqrt{T}
      • N(x)N(x) represents the cumulative standard normal distribution function

Factors Influencing Option Prices

Key factors in option pricing

  • Underlying asset price (S0S_0) directly impacts option prices
    • Higher S0S_0 increases call option price and decreases put option price
  • Strike price (KK) sets the price at which the option can be exercised
    • Higher KK decreases call option price and increases put option price
  • Time to expiration (TT) affects option prices due to uncertainty and time decay
    • Longer TT generally increases both call and put option prices
    • Time decay () accelerates as expiration approaches, eroding option value
  • Volatility (σ\sigma) measures the uncertainty of the underlying asset's returns
    • Higher σ\sigma increases both call and put option prices (options on stocks like Tesla vs. utility companies)
  • Risk-free interest rate (rr) influences the present value of the strike price
    • Higher rr increases call option prices and decreases put option prices

Option Strategies

Construction of option strategies

  • : long the underlying asset and short a call option on the same asset
    • Limits upside potential but provides downside protection and generates income from the call premium (suitable for investors with a neutral to slightly bullish outlook)
  • : long the underlying asset and long a put option on the same asset
    • Provides downside protection at the cost of the put premium (acts as an insurance policy)
  • : long a call and a put option with the same strike price and expiration date
    • Profits from significant price movements in either direction (bullish or bearish)
    • Maximum loss limited to the total premium paid
  • Spreads involve simultaneously buying and selling options with different strike prices or expiration dates
    1. : long a lower strike call and short a higher strike call (or long a lower strike put and short a higher strike put)
    2. : short a lower strike call and long a higher strike call (or short a lower strike put and long a higher strike put)
    3. Profits from moderate price movements in the desired direction with limited risk and reward (credit spreads vs. debit spreads)

Risk-return analysis of options

  • Evaluate the maximum profit, maximum loss, and breakeven points for each strategy
    • Breakeven points occur when the underlying asset price equals the strike price plus (minus) the premium paid (received) for calls (puts)
  • Consider the impact of different market scenarios on the strategy's performance
    • Bullish, bearish, or neutral market conditions (straddle performs best in highly volatile markets)
    • High or low volatility environments (high volatility favors long option positions)
  • Analyze the strategy's sensitivity to changes in the underlying asset price, time to expiration, and volatility using Greeks
    • : change in option price per unit change in the underlying asset price (delta )
    • : change in delta per unit change in the underlying asset price (high gamma indicates high sensitivity to price changes)
    • Theta: change in option price per unit of time decay (short options benefit from time decay)
    • : change in option price per unit change in volatility (long options have positive vega)

Key Terms to Review (26)

Bear spread: A bear spread is an options trading strategy designed to profit from a decline in the price of an underlying asset. It involves simultaneously buying and selling options with different strike prices but the same expiration date, typically using put options to limit potential losses while capitalizing on downward movement. This strategy allows investors to minimize their risk while still benefiting from bearish market conditions.
Binomial model: The binomial model is a mathematical method used for pricing options, allowing the evaluation of an option's value by considering two possible future states for the underlying asset at each time step. This model breaks down the time until expiration into discrete intervals, creating a binomial tree that represents the potential paths an asset's price could take, which is particularly useful for understanding options valuation and strategies.
Black-Scholes Model: The Black-Scholes Model is a mathematical model used to calculate the theoretical price of options, considering various factors such as the underlying asset price, exercise price, time to expiration, risk-free interest rate, and volatility. It plays a vital role in options valuation by providing a framework to understand how these factors influence option pricing and helps traders devise effective strategies.
Bull Spread: A bull spread is an options trading strategy that involves buying and selling options at different strike prices but with the same expiration date to capitalize on a moderate rise in the underlying asset's price. This strategy can be executed using either call options or put options and is designed to limit potential losses while also capping potential gains. Traders typically use bull spreads when they expect a bullish market movement but want to reduce the cost of entering the trade.
Call option: A call option is a financial contract that gives the holder the right, but not the obligation, to buy a specific amount of an underlying asset at a predetermined price, known as the strike price, within a specified time period. This financial instrument is commonly used in options trading strategies to leverage potential gains from an increase in the asset's price while limiting risk to the premium paid for the option.
Covered call: A covered call is an options strategy where an investor holds a long position in an asset and simultaneously sells call options on that same asset. This strategy is often used to generate additional income from the premiums received from selling the call options while also providing some downside protection. Investors who employ this strategy typically have a neutral to slightly bullish outlook on the underlying asset.
Delta: Delta is a measure of the sensitivity of an option's price to changes in the price of the underlying asset. It represents the expected change in the option's price for a $1 change in the price of the underlying security, making it a critical factor in options valuation and trading strategies. Understanding delta allows traders to gauge how an option will react to price movements in the underlying asset, helping them make informed decisions on their trades.
Dot-com bubble: The dot-com bubble refers to a period of excessive speculation in the late 1990s and early 2000s, during which the stock prices of internet-based companies soared to unsustainable levels. This phenomenon was driven by the rapid growth of the internet, an influx of venture capital, and widespread public enthusiasm for technology stocks, leading to a market crash in 2000 when valuations plummeted.
Gamma: Gamma is a second-order derivative of an option's price with respect to the underlying asset's price, representing the rate of change in delta for a one-unit change in the underlying asset. It provides insights into how the delta of an option will change as the price of the underlying moves, making it a crucial factor in options valuation and risk management strategies.
Hedging: Hedging is a risk management strategy used to offset potential losses or gains that may be incurred by an investment. By taking an opposite position in a related asset, investors can protect themselves from adverse price movements. This strategy often involves using derivatives like options, futures, and forwards to create a safety net against unpredictable market conditions.
Intrinsic Value: Intrinsic value refers to the perceived or calculated value of an asset, independent of its market price. It often reflects the true worth based on fundamental analysis, including factors like cash flow, earnings, and growth potential. Understanding intrinsic value helps investors determine whether an asset is overvalued or undervalued in the market.
Leverage: Leverage refers to the use of borrowed funds to amplify potential returns on investment. It plays a crucial role in finance as it allows companies to increase their investment capacity and potentially enhance profitability, but it also raises the risk of losses when investments do not perform as expected.
Market sentiment: Market sentiment refers to the overall attitude of investors toward a particular security or financial market. It reflects the collective feelings and emotions of investors, which can drive market trends and influence prices. This sentiment can be bullish (optimistic) or bearish (pessimistic) and is often influenced by various factors such as news, economic indicators, and social media.
October 1987 Crash: The October 1987 Crash, also known as Black Monday, refers to the sudden and severe stock market crash that occurred on October 19, 1987, when the Dow Jones Industrial Average plummeted by 22.6% in a single day. This unprecedented drop raised concerns about market volatility and the effectiveness of existing risk management strategies, leading to significant discussions about options valuation and strategies in the aftermath of the crash.
Options exchange: An options exchange is a marketplace where options contracts are traded, allowing buyers and sellers to transact based on the underlying asset's price movements. These exchanges provide a platform for standardization, liquidity, and transparency, making it easier for investors to execute their trading strategies and manage risk. The existence of options exchanges plays a vital role in the broader financial markets by enabling price discovery and offering various trading strategies.
Protective Put: A protective put is an options trading strategy where an investor buys a put option for a stock they already own, providing insurance against a potential decline in the stock's price. This strategy allows the investor to maintain ownership of the stock while limiting potential losses, as the put option gives them the right to sell the stock at a predetermined price, thus acting as a safety net in case of adverse price movements.
Put option: A put option is a financial contract that gives the holder the right, but not the obligation, to sell a specific quantity of an underlying asset at a predetermined price, known as the strike price, within a specified time period. This instrument is particularly useful for investors looking to hedge against potential declines in the price of the underlying asset or to speculate on price movements. By understanding put options, investors can develop strategies that manage risk and capitalize on market fluctuations.
Risk-free interest rate: The risk-free interest rate is the theoretical return on an investment with zero risk of financial loss, often represented by the yield on government securities like Treasury bonds. It serves as a baseline for evaluating other investments, providing a benchmark against which the risk and potential return of various assets can be assessed. This concept is crucial in finance for pricing derivatives, especially options, where it helps determine the time value of money and the fair value of riskier securities.
SEC Regulations: SEC regulations refer to the rules and guidelines established by the U.S. Securities and Exchange Commission (SEC) to govern the securities industry, ensuring transparency, fairness, and protection for investors. These regulations play a crucial role in shaping corporate behaviors, influencing dividend policies, managing cash and marketable securities, and guiding options valuation strategies, ultimately maintaining market integrity and investor confidence.
Straddle: A straddle is an options trading strategy that involves buying both a call option and a put option for the same underlying asset, with the same strike price and expiration date. This strategy is typically employed when an investor expects a significant price movement in the underlying asset but is uncertain about the direction of that movement. By utilizing a straddle, traders can potentially profit from volatility in the market, regardless of whether prices rise or fall.
Strike Price: The strike price, also known as the exercise price, is the predetermined price at which the holder of an option can buy or sell the underlying asset when exercising the option. This price is crucial in determining the potential profitability of options and serves as a key factor in options pricing, influencing both the risk and reward profile of these financial derivatives.
Theta: Theta is a measure of the rate of decline in the value of an option as it approaches its expiration date. This concept is crucial for understanding options valuation, as it quantifies time decay, which impacts the premium of options contracts. Investors use theta to make informed decisions about when to buy or sell options based on their time-sensitive nature.
Time to expiration: Time to expiration refers to the period remaining until an options contract becomes void. This concept is crucial in options valuation and strategies, as the length of time affects the option's premium and potential profitability. The closer an option gets to its expiration date, the more it is influenced by time decay, which erodes the extrinsic value of the option.
Time value: Time value refers to the concept that the value of money changes over time due to potential earning capacity. This idea is crucial in finance as it emphasizes that a specific amount of money today is worth more than the same amount in the future because of its ability to earn interest or generate returns over time. This principle is particularly important in the evaluation of options, as the time remaining until expiration affects an option's price and strategic decisions.
Vega: Vega is a measure of an option's sensitivity to changes in the volatility of the underlying asset. It indicates how much the price of an option is expected to change when the volatility of the underlying asset increases or decreases by 1%. Vega is an important concept in options valuation as it helps traders understand how fluctuations in market conditions may impact the value of their options positions.
Volatility: Volatility refers to the degree of variation in the price of a financial asset over time, commonly measured by the standard deviation of returns. It indicates the extent to which an asset's price can fluctuate, making it a crucial concept in understanding risk and uncertainty in financial markets. High volatility suggests that an asset's price can change dramatically in a short period, while low volatility implies steadiness. This concept is essential for portfolio management and options trading strategies.
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