9.1 Options: Characteristics and Valuation
7 min read•july 30, 2024
Options are financial contracts that give buyers the right to buy or sell assets at set prices. They're a key part of derivative markets, offering flexibility and risk management tools for investors.
Understanding options involves grasping their structure, types, and valuation methods. From basic calls and puts to complex pricing models like Black-Scholes, options play a crucial role in modern finance and investment strategies.
Options Contracts: Structure and Terminology
Basic Structure and Components
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- An option is a financial derivative that represents a contract sold by one party to another party, giving the buyer the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at an agreed-upon price (strike price) within a certain time frame ()
- The buyer of an option pays a premium to the seller for the right to exercise the option
- The premium is the price of the option contract and is determined by various factors such as the underlying asset's price, volatility, time to expiration, and interest rates
- The underlying asset of an option can be a stock (Apple or Tesla), bond, commodity (gold or oil), currency (USD or EUR), or index (S&P 500 or NASDAQ)
Option Styles and Trading Markets
- Options are classified as either American-style or European-style
- American-style options can be exercised at any time before the expiration date
- European-style options can only be exercised on the expiration date
- Options are traded on exchanges and over-the-counter (OTC) markets
- Exchange-traded options are standardized contracts with fixed strike prices and expiration dates (Chicago Board Options Exchange or CBOE)
- OTC options can be customized to suit the needs of the parties involved, offering more flexibility in terms of strike prices, expiration dates, and underlying assets
Call vs Put Options: Payoff Structures
Call Options
- A call option gives the buyer the right to purchase the underlying asset at a predetermined strike price
- The payoff structure of a call option at expiration is max(S - K, 0), where S is the price of the underlying asset and K is the strike price
- If S > K, the call option is in-the-money, and the buyer will exercise the option for a profit
- If S ≤ K, the call option is out-of-the-money or at-the-money, and the buyer will let the option expire worthless
- The maximum loss for the buyer of a call option is limited to the premium paid, while the potential profit is theoretically unlimited
- The seller (writer) of a call option has a potential loss that is theoretically unlimited, while the maximum profit is limited to the premium received
Put Options
- A put option gives the buyer the right to sell the underlying asset at the strike price
- The payoff structure of a put option at expiration is max(K - S, 0)
- If K > S, the put option is in-the-money, and the buyer will exercise the option for a profit
- If K ≤ S, the put option is out-of-the-money or at-the-money, and the buyer will let the option expire worthless
- The maximum loss for the buyer of a put option is limited to the premium paid, while the potential profit is limited to the strike price minus the premium
- The seller of a put option has a maximum loss equal to the strike price minus the premium received, and the maximum profit is limited to the premium received
Option Value: Intrinsic and Time
Intrinsic Value
- The of an option is the amount by which the option is in-the-money
- For a call option, the intrinsic value is max(S - K, 0), where S is the current price of the underlying asset and K is the strike price
- For a put option, the intrinsic value is max(K - S, 0)
- At expiration, an option's value consists entirely of its intrinsic value, as the time value has decayed to zero
- Example: If a call option has a strike price of 55, the intrinsic value of the call option is $5
Time Value
- The time value of an option is the difference between the option's price (premium) and its intrinsic value
- It represents the additional value that investors are willing to pay for the potential of the option to become more profitable before expiration
- The time value of an option is affected by factors such as the time remaining until expiration, the volatility of the underlying asset, and the difference between the underlying asset's price and the strike price
- As the expiration date approaches, the time value of an option decreases, a phenomenon known as time decay
- This occurs because there is less time remaining for the option to become profitable or increase in value
- Example: If a call option with a strike price of 7 when the underlying stock is at 2 (option price of 5)
Black-Scholes Model: Option Valuation
Model Overview and Assumptions
- The Black-Scholes model is a mathematical model used to estimate the theoretical value of European-style options
- It takes into account various factors such as the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset
- The Black-Scholes model assumes that:
- The underlying asset's price follows a geometric Brownian motion
- The risk-free interest rate and volatility are constant
- There are no dividends
- The option can only be exercised at expiration (European-style)
Black-Scholes Formula
- The Black-Scholes formula for a European call option is: C = S * N(d1) - K * e^(-r * T) * N(d2)
- C is the call option price
- S is the current stock price
- K is the strike price
- r is the risk-free interest rate
- T is the time to expiration (in years)
- N(d1) and N(d2) are the standard normal cumulative distribution functions
- The Black-Scholes formula for a European put option is: P = K * e^(-r * T) * N(-d2) - S * N(-d1)
- P is the put option price, and the other variables are the same as in the call option formula
- The variables d1 and d2 are calculated as follows:
- d1 = (ln(S / K) + (r + (σ^2 / 2)) * T) / (σ * √T)
- d2 = d1 - σ * √T
- ln is the natural logarithm, and σ is the volatility of the underlying asset (expressed as a decimal)
Limitations and Applications
- The Black-Scholes model has several limitations, such as the assumption of constant volatility and the absence of dividends
- In reality, volatility can change over time, and many stocks pay dividends, which can affect option prices
- Despite its limitations, the Black-Scholes model is widely used by traders, investors, and financial institutions to price options, manage risk, and develop hedging strategies
- The model's insights have also been applied to other areas of finance, such as the valuation of corporate liabilities and the pricing of exotic options
Option Pricing: Factors and Influences
Underlying Asset Price and Strike Price
- The price of the underlying asset directly affects the value of an option
- As the underlying asset's price increases, the value of a call option increases, and the value of a put option decreases
- Conversely, as the underlying asset's price decreases, the value of a call option decreases, and the value of a put option increases
- The strike price of an option is the price at which the underlying asset can be bought or sold upon exercise
- For a call option, as the strike price increases, the option value decreases
- For a put option, as the strike price increases, the option value increases
Time to Expiration and Volatility
- The time to expiration affects the value of an option through time decay
- As the time to expiration decreases, the time value of an option decreases, leading to a decrease in the option's price
- This effect is more pronounced for at-the-money options and less significant for deep in-the-money or out-of-the-money options
- Volatility is a measure of the underlying asset's price fluctuations
- Higher volatility leads to higher option prices because there is a greater probability that the option will expire in-the-money
- Conversely, lower volatility leads to lower option prices
Interest Rates and Dividends
- Interest rates affect option prices through their impact on the present value of the strike price
- As interest rates increase, the present value of the strike price decreases, leading to an increase in call option values and a decrease in put option values
- Conversely, as interest rates decrease, the present value of the strike price increases, leading to a decrease in call option values and an increase in put option values
- Dividends paid by the underlying asset can affect option prices
- For call options, higher dividends lead to lower option prices, as the expected future price of the underlying asset is reduced
- For put options, higher dividends lead to higher option prices, as the expected future price of the underlying asset is reduced
Key Terms to Review (11)
Assignment: In the context of options trading, an assignment occurs when the holder of an option exercises their right to buy or sell the underlying asset, obligating the writer of the option to fulfill that transaction. This concept is crucial for understanding the mechanics of options trading, as it signifies the point at which an option contract is executed, resulting in a transfer of ownership or rights related to the underlying asset.
Binomial model: The binomial model is a mathematical method used to value options by modeling the possible price movements of an underlying asset over time. It breaks down the time until expiration into discrete intervals, allowing for a simple way to calculate the potential outcomes of holding an option at each interval. This model is particularly useful because it captures the uncertainty of asset prices, making it a vital tool in options valuation.
Covered call: A covered call is an options trading strategy where an investor holds a long position in an asset and sells call options on that same asset to generate income. This strategy is commonly used to enhance returns on stocks the investor already owns while providing some downside protection. By selling the call options, the investor collects a premium, which can offset losses if the asset's price declines.
Exercise Price: The exercise price, also known as the strike price, is the predetermined price at which an option holder can buy or sell the underlying asset when they exercise their option. This price is crucial in determining the potential profit or loss from an option, as it establishes the level at which the underlying asset must move to make exercising the option advantageous.
Expiration Date: The expiration date is the predetermined date on which an options contract becomes void and can no longer be exercised. It is crucial for options trading as it dictates the time frame in which an option holder can exercise their right to buy or sell the underlying asset at the agreed-upon price. The expiration date plays a significant role in the valuation of options, as the time remaining until expiration influences the option's premium and overall market behavior.
Intrinsic value: Intrinsic value is the perceived or calculated true value of an asset, based on its fundamental characteristics, rather than its market price. This concept is essential for evaluating investments, as it helps investors determine whether an asset is overvalued or undervalued based on its future cash flows, risk factors, and growth potential. Understanding intrinsic value enables better decision-making in investment strategies and valuation techniques.
Open Interest: Open interest refers to the total number of outstanding contracts in a given derivatives market that have not yet been settled or closed. It provides insights into market activity and liquidity, as it reflects the level of engagement and interest in particular options or futures contracts. A higher open interest generally indicates that more capital is invested in the market, suggesting a healthy level of trading activity and potential price movements.
Option premium: The option premium is the price that an investor pays to purchase an option contract, representing the cost of acquiring the right but not the obligation to buy or sell an underlying asset at a predetermined price within a specified timeframe. This premium is influenced by various factors such as the underlying asset's price, the strike price of the option, time until expiration, and market volatility, making it a crucial aspect in understanding how options are valued and traded.
Options disclosure document: An options disclosure document is a crucial resource that provides detailed information about the risks and characteristics of trading options. This document helps investors understand the complexities associated with options trading, including potential risks, strategies, and how options can be used to hedge or speculate in financial markets.
SEC Regulations: SEC regulations are the rules and guidelines established by the U.S. Securities and Exchange Commission to govern the securities industry and protect investors. These regulations cover various aspects of trading, disclosure, and reporting to ensure transparency and fairness in the financial markets, including those related to options trading.
Volatility risk: Volatility risk refers to the potential for fluctuations in the price of an asset, which can lead to unexpected gains or losses. This risk is particularly significant in the context of options, as the value of an option is heavily influenced by the underlying asset's price movements. A higher level of volatility generally increases the value of options due to the greater potential for favorable price changes, making understanding this risk crucial for investors using options as part of their strategy.