Lattice-Based Model

What Is A Lattice-Based Model?

A Lattice-based Model refers to a specific type of computational model in the form of a binomial tree used to value financial derivatives and other complex financial instruments. The main goal is to determine the fair value of financial derivatives, such as options, swaps, and exotic options.

Lattice-based models are particularly useful for valuing derivatives with complex features, early exercise options, and situations where the underlying asset's price follows stochastic processes. They can handle multiple sources of uncertainty and are well-suited for options with path-dependent payoffs.

• Lattice-based model is a computational technique that values financial derivatives and other complex financial instruments.
• It involves breaking down time into discrete intervals (time steps) and constructing a lattice or tree structure to model the possible states of the underlying asset or financial instrument at each time step.
• The most well-known lattice-based model in finance is the binomial option pricing model, which is used to value options.
• The fundamental difference between the lattice-based model and the Black Scholes model lies in how they handle time.

How Does The Lattice-Based Model In Finance Work?

A Lattice-based model in finance works by discretizing time and constructing a lattice of possible asset prices over time to value financial derivatives and complex financial instruments. The model simulates the behavior of the underlying asset. It then calculates the fair value of derivatives based on their potential future outcomes. One of the most well-known lattice models is the binomial option pricing model. It uses a two-step lattice, but more sophisticated models can use multiple steps for increased accuracy.

Steps

Here's a step-by-step explanation of how a Lattice-based model works:

1. Discretization of Time: The continuous time interval is divided into smaller discrete time steps. Each time step represents a short period. One can adjust the number of steps to achieve the desired level of precision. For example, if the derivative's time to expiration is one year, we can divide it into 12 monthly steps.
2. Construction of the Lattice: At each time step, one can generate a set of possible asset prices depending on the assumed volatility of the underlying asset and other relevant market factors. They can typically do this using a stochastic process, such as the geometric Brownian motion. Here, the asset price can move up or down at each time step, with the volatility determining the probabilities. Here, one can construct the lattice by connecting these possible prices over time.
3. Valuation of the Derivative: Starting from the final time step (expiration date of the derivative), one can compute the value of the derivative at each node of the lattice, working backward through time. This process is often called backward induction. The value at each node depends on the derivative's payoff at that point and the expected future values at the next time step, considering the probabilities of different price movements.
4. Option Exercise Decisions: For American options, the lattice model allows for early exercise decisions. At each node, we can compare the option's value to the immediate exercise value to select the higher value. This accounts for the possibility of early exercise, which can affect the option's overall value.
5. Risk-Neutral Valuation: The lattice model relies on the concept of risk-neutral valuation. Here, we can match the risk-free interest rate to calculate future asset prices. This ensures that the model adheres to the no-arbitrage principle. Thus, we can derive the risk-neutral probabilities from the risk-free rate and the up and down movement factors in the lattice.

Examples

Let us look at a few examples to understand the concept better:

Example #1

Let us consider a hypothetical example of a lattice-based model applied to value a European call option on a stock.

Suppose the current stock price is $100, and the option's strike price is $110. The risk-free interest rate is 5% per annum, and the time to expiration is one year, which is discretized into 12 monthly steps. It is assumed that the stock price can either go up by 10% or down by 10% at each step, with equal probabilities of 0.5 for each movement.

Using the lattice-based model, one constructs a binomial tree with the stock price nodes at each step. In the final step, they calculate the option's payoff. This is the difference between the stock price and the strike price if it is positive (max) or zero if it's negative. Then, they work backward through the tree, calculating the option's value at each node using risk-neutral valuation.

At each node, users calculate the option's expected value as a discounted average of the values of its two child nodes. They weigh it by the risk-neutral probabilities of the stock price going up or down and repeat this process until the initial node is reached. Here, one obtains the fair value of the European call option.

Suppose the lattice-based model determines that the option's fair value is $6.50. This means that, under the assumptions of the model, the theoretical fair price of the European call option is $6.50, given the current stock price, the strike price, the interest rate, and the assumed stock price movements.

Example #2

A real-life example of a lattice-based model in finance is the application of the binomial option pricing model to value options. The binomial option pricing model is one of the most well-known lattice-based models. In addition, it has been widely used in the financial industry to calculate the fair value of options.

Suppose a company's stock is currently trading at $50, and a European call option with a strike price of $55 and an expiration date in three months is available. In addition, the risk-free interest rate is 4% per annum. To value this option using the binomial option pricing model, one discretizes time into monthly steps (3 months) and assumes that the stock price can either go up by 10% or down by 10% in each step.

One historical example is the Black-Scholes-Merton model, which economists Fischer Black, Myron Scholes, and Robert Merton introduced. They developed it in the 1970s, and although the Black-Scholes-Merton model finds common association with continuous models, one can discretize it into a binomial model, and this version plays a crucial role in its development.

The model assumes that the underlying stock obeys geometric Brownian motion, which is a stochastic process where the stock price can change continuously over time. By discretizing time into small intervals and assuming that the stock price can either go up or down by a certain percentage at each interval, a binomial tree helps to approximate a continuous model.

Lattice-Based Model vs Black Scholes

The differences between the lattice-based Model vs Black Scholes Model are as follows:

Lattice based model	Black Scholes model
Lattice-based models offer more flexibility in handling certain complexities, such as dividends, American-style options with early exercise, and path-dependent options.	The Black-Scholes model is based on continuous time, where the underlying asset's price follows geometric Brownian motion, allowing for a continuous range of possible stock prices.
Lattice-based models offer more flexibility in handling certain complexities, such as dividends, American-style options with early exercise, and path-dependent options.	The Black-Scholes model, being based on continuous time, assumes that the option is held until expiration, which is appropriate for European-style options without early exercise features.

Frequently Asked Questions (FAQs)

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