American Options vs. European Options (Mathematical Modeling)

Mathematical modeling of options is used heavily in finance.

Both American and European options are types of financial derivatives, but they differ in their exercise rights, which leads to differences in their mathematical modeling.

Let’s look into these differences:

Key Takeaways – American Options vs. European Options (Mathematical Modeling)

• American options can be exercised any time up to expiration.
• But European options can be exercised only at expiration.
• This makes American options’ modeling more complex due to the early exercise feature.
• The Black-Scholes model is predominant for European options, offering a direct formula for pricing.
• American options often employ the binomial tree model or solve a free boundary problem to account for early exercise.
• Both option types use “Greeks” to measure sensitivities, but values can differ between them.
  • For instance, the early exercise potential in American options can influence their Delta and Theta compared to European counterparts.

European Options

Let’s look at the basics of mathematical modeling for European options:

Exercise Rights

European options can only be exercised at their expiration date.

This means if you buy a European call option, you can only exercise the right to buy the underlying asset at the specified strike price on the expiration date.

Mathematical Modeling

The most famous model for European options is the Black-Scholes model.

The Black-Scholes equation is a partial differential equation (PDE) that describes the price evolution of the option.

The solution to the Black-Scholes PDE gives the option’s price as a function of the underlying asset price, time, volatility, interest rate, and strike price.

Given no early exercise, Black-Scholes and similar PDEs are used.

Alternatives to Black-Scholes

Some alternatives to Black-Scholes (or serve as alternative to modeling a certain component – e.g., volatility) popularly include:

• Binomial Tree Model
• Trinomial Tree Model
• Monte Carlo Simulation
• Finite Difference Methods
• Bachelier Model
• Heston Model
• Feynman-Kac Model
• Cox-Ross-Rubinstein Model
• Garman-Kohlhagen Model
• Hull-White Model
• Jump Diffusion Model
• CEV (Constant Elasticity of Variance) Model
• GARCH (Generalized Autoregressive Conditional Heteroskedasticity) Model
• SABR Model
• Bates Model

Many traders and investment firms will also use their own proprietary models.

American Options

Now let’s look at the basics of mathematical modeling for American options:

Exercise Rights

American options can be exercised at any time up to their expiration date.

This early exercise feature makes their valuation more complex.

Mathematical Modeling

Valuing American options is more challenging due to the early exercise feature.

Binomial Tree Method

One popular method is the binomial tree model.

This is where the price of the underlying asset is modeled to move up or down at each step, and the option’s value is determined at each node by considering the possibility of early exercise.

The binomial tree model is an example of Markov chains and decision trees in finance.

Free Boundary Problem

Another approach involves solving a free boundary problem, where the boundary represents the early exercise boundary.

The option’s holder will exercise the option when it’s optimal. So determining this boundary is important.

Finite Difference Methods & Monte Carlo Simulations

Finite difference methods and Monte Carlo simulations are also used – especially for more complex American options or when considering dividends.

Key Differences in Modeling American & European Options

• The early exercise feature of American options introduces additional complexity into their mathematical modeling.
• European options have closed-form solutions like the Black-Scholes formula, making them simpler to value.
• American options often require iterative numerical methods, like the binomial tree, to account for the possibility of early exercise at each point in time.

Early Exercise in American Options

The early exercise feature of American options introduces complexity to their mathematical modeling.

This is because, at any given time before expiration, a decision must be made on whether it’s optimal to exercise the option or keep it.

This necessitates the use of models that can account for this feature, such as the binomial tree model or methods that address the free boundary problem.

Binomial Tree in American Options

The binomial tree model is a method that breaks down the option’s life into a series of discrete time intervals.

At each interval, the underlying asset price can move up or down, creating a tree of possible prices.

The model is particularly relevant for American options because it allows for the consideration of early exercise at each node in the tree.

Determining the option’s value by back-calculating from expiration to the present.

Partial Differential Equations (PDEs) and Option Pricing

PDEs are foundational to option pricing because they describe the evolution of the option’s price concerning changes in the underlying asset and time.

The Black-Scholes equation, for instance, is a PDE that gives the price of a European option as a function of various parameters.

Free Boundary Problem in American Options

The free boundary problem is important for American options because it determines the optimal early exercise boundary.

This boundary represents the asset prices where it becomes optimal to exercise the option before expiration.

Solving for this boundary is essential to accurately value American options.

Monte Carlo Simulations in European & American Option Pricing

Monte Carlo simulations involve generating a large number of random sample paths for the underlying asset’s price and then computing the option’s payoff for each path.

By averaging these payoffs and discounting back to the present, the option’s price can be estimated.

This method is flexible and can be applied to both American and European options, especially when analytical solutions are challenging to derive.

Volatility Modeling in European & American Option Pricing

Volatility represents the degree of variation in the underlying asset’s price.

In the mathematical models for both option types, volatility is a key parameter.

Higher volatility typically increases the option’s value – all else equal – because it implies a greater chance of the option ending in-the-money.

A unique aspect of European options is that they can’t be more valuable than American options due to American options’ having the early exercise feature.

Dividends in European & American Option Pricing

Dividends decrease the value of call options and increase the value of put options.

For American options, expected dividends can make early exercise more attractive for calls before a dividend payout.

For European options, dividends are typically factored into the Black-Scholes model by adjusting the underlying asset’s price.

Interest Rates in European & American Option Pricing

Interest rates are an important parameter in option pricing models.

They represent the time value of money, affecting the discounting of future payoffs to the present.

An increase in interest rates will generally increase the value of call options and decrease the value of put options.

Hedging in European & American Option Pricing

Hedging American options is more complex because of the possibility of early exercise.

This unpredictability means that the hedging strategy must be adjusted dynamically and more frequently, ensuring that the portfolio remains delta-neutral or protected against price movements.

For instance, let’s say a trader is following a covered call or covered put strategy.

If the option is in the money, the transaction can close out before the expiration date.

The closing out of the position can affect the diversification/balance of a portfolio, the cash balance in the account, and so on, which can in turn lead to an unoptimized portfolio, interest costs, etc.

Greeks in European & American Option Pricing

The “Greeks” in option pricing represent sensitivities of the option’s price to various factors:

Delta

Measures the option’s price sensitivity to changes in the underlying asset’s price.

It’s similar for both American and European options but can vary due to early exercise in American options.

Gamma

Captures the rate of change in Delta.

Consistent for both option types, reflecting acceleration of price change.

Vega

Indicates sensitivity to volatility.

Generally similar for both, but nuances exist due to differing expiration dynamics.

Theta

Represents price sensitivity to time decay.

American options, due to early exercise rights, might have a different Theta profile.

Rho

Measures sensitivity to interest rate changes.

While fundamental behavior is consistent, the magnitude can vary between American and European options due to early exercise possibilities.

Terms & Definitions

Some terms and definitions as it pertains to American options, European options, and their mathematical modeling:

• Option pricing: The process of determining the fair market value of an option.
• Financial derivatives: Financial instruments whose value is derived from an underlying asset or group of assets.
• Black-Scholes model: A mathematical model for pricing European-style options.
• Early exercise: The action of exercising an option before its expiration date.
• Binomial tree model: A method to value options by modeling multiple possible future price outcomes.
• Partial differential equation (PDE): An equation involving multiple variables and their partial derivatives.
• Free boundary problem: A PDE problem where the boundary is unknown and must be determined as part of the solution.
• Monte Carlo simulation: A computational method using random sampling to estimate numerical results.
• Finite difference methods: Numerical methods for solving differential equations using discrete approximations.
• Volatility: A measure of the price variation of a financial instrument over time.
• Expiration date: The date on which an option contract becomes void.
• Strike price: The predetermined price at which an option can be exercised.
• Risk-neutral valuation: Pricing financial derivatives assuming no preference between risk and reward. (See Q World vs. P World)
• Dividends: Payments made by corporations to their shareholders from profits.
• Option premium: The price paid to acquire an option.
• Intrinsic value: The difference between the underlying asset’s price and the strike price of an option.
• Time value: The portion of an option’s premium attributed to the time remaining until expiration.
• Hedging: Making investments to reduce the risk of adverse price movements.
• Greeks (Delta, Gamma, Vega, Theta, Rho): Measures capturing the sensitivity of an option’s price to various factors.
• Implied volatility: The market’s forecast of a likely movement in a security’s price. Inferred from option prices.
• Arbitrage opportunities: Situations where one can buy and sell assets simultaneously to profit from price differences.
• Interest rate: The amount charged by lenders to borrowers, expressed as a percentage of the principal.
• Underlying asset: The financial instrument (e.g., stock) on which a derivative’s price is based.
• Call option: A financial contract giving the holder the right, but not the obligation, to buy an asset at a set price.
• Put option: Gives the holder the right, but not the obligation, to sell an asset at a set price.

FAQs – American Options vs. European Options (Mathematical Modeling)

What are the primary differences between American and European options?

The main difference between American and European options lies in their exercise rights.

American options can be exercised at any time up to their expiration date, whereas European options can only be exercised at their expiration date.

Why is the Black-Scholes model specifically used for European options?

The Black-Scholes model is specifically used for European options because it provides a closed-form solution for options that can only be exercised at expiration.

Given the lack of an early exercise feature in European options, the model simplifies their valuation with a direct formula.

Can European options ever be more valuable than American options?

No, European options cannot be more valuable than their American counterparts.

This is because American options offer more flexibility with the added early exercise feature.

At best, when there’s no advantage to early exercise, the European option’s value will equal the American option’s value.

What are the computational challenges associated with modeling American options?

Modeling American options is computationally intensive because of the early exercise feature.

Determining the optimal exercise strategy requires considering multiple paths and scenarios, especially in methods like the binomial tree.

Additionally, solving the free boundary problem or running extensive Monte Carlo simulations can be computationally expensive.

Are there real-world scenarios where the choice between American and European options significantly impacts financial outcomes?

Yes, the choice between American and European options can have financial implications.

For instance, in markets with expected dividends, the early exercise of American call options might be optimal before a dividend payout.

This leads to different financial outcomes compared to holding a European option.

Additionally, the flexibility of American options can provide strategic advantages in certain market conditions, affecting hedging, trading strategies, and risk management.

What are the “Greeks” in option pricing?

The “Greeks” are measures of sensitivity in option pricing. They include:

• Delta: Sensitivity of option price to changes in the underlying asset’s price.
• Gamma: Sensitivity of Delta to changes in the underlying price.
• Vega: Sensitivity to changes in volatility.
• Theta: Sensitivity to the passage of time.
• Rho: Sensitivity to changes in the interest rate.

While the definitions of the Greeks remain consistent, their values can differ for American and European options due to the early exercise feature in American options.

Conclusion

While both types of options share some fundamental mathematical principles, the distinct exercise rights of European and American options lead to different modeling approaches and complexities.