Lattice Models in Options Pricing

Lattice models, also referred to as tree models, are a fundamental tool in quantitative finance, specifically in the pricing of options and other derivative securities. These models are designed to allow for a discrete approximation of an underlying asset’s price evolution over time. They can be particularly useful in scenarios where closed-form solutions are not readily available, providing flexibility and computational efficiency.

Binomial Model

The binomial options pricing model, introduced by John Cox, Stephen Ross, and Mark Rubinstein in 1979, is one of the most commonly used lattice models. It is a discrete-time model for the valuation of options, presenting a simplified view of the possible paths that the price of the underlying asset can take.

Model Construction

In a binomial model, the price of the underlying asset can either move up or down by specific factors, u (up factor) and d (down factor), over each discrete time step, Δt. The probabilities of these movements are p (probability of an upward move) and (1-p) (probability of a downward move).

The factors u and d can be determined based on the volatility σ and the length of the time step Δt:

[ u = e^{σ \sqrt{Δt}} ] [ d = e^{-σ \sqrt{Δt}} ]

The probability p of an upward movement in a risk-neutral world is given by:

[ p = \frac{e^{rΔt} - d}{u - d} ]

Where r is the risk-free rate.

Valuation Process

1. Lattice Construction: Start by creating a multi-period tree where each node represents a possible price of the underlying asset at a specific time.
2. Option Value Calculation: Work backwards from the terminal nodes (maturity) of the tree to calculate the value of the option at each node.
3. Payoff at Maturity: The value at each terminal node is the payoff from the option (e.g., for a call option, ( \max(S_T - K, 0) ), where ( S_T ) is the underlying asset price at maturity and ( K ) is the strike price).
4. Discounted Expected Value: The value at each preceding node is the discounted expected option value, considering the probabilities of moving to adjacent nodes in the next period.

[ V_i = e^{-rΔt} [p V_{i+1} + (1-p) V_{i-1}] ]

Advantages and Disadvantages

Advantages:

• Flexibility: Can handle a variety of option types and incorporate features like dividends and American-style exercise.
• Intuitive: Easy to conceptualize and implement, making it accessible for educational purposes.

Disadvantages:

• Computationally Intensive: For a large number of time steps and high-dimensional problems, the binomial model can become computationally expensive.
• Approximations: The model being discrete in nature introduces approximation errors, although these errors can be minimized by increasing the number of time steps.

Trinomial Model

The trinomial options pricing model is an extension of the binomial model that considers three possible price movements: up, down, and unchanged. This model offers more accuracy and stability compared to the binomial model, particularly when dealing with large time steps.

Model Construction

In a trinomial model, the price of the underlying asset can move up by a factor u, down by a factor d, or remain unchanged by a factor m (usually set to 1).

[ u = e^{σ \sqrt{2Δt}} ] [ d = e^{-σ \sqrt{2Δt}} ] [ m = 1 ]

The risk-neutral probabilities p_u, p_d, and p_m for up, down, and unchanged movements, respectively, are defined as:

[ p_u = \frac{1}{2} \left( \frac{λ^2 + λ}{2} + λ + 1 \right) ] [ p_d = \frac{1}{2} \left( \frac{λ^2 + λ}{2} - λ \right) ] [ p_m = 1 - p_u - p_d ]

where ( λ = \frac{rΔt}{σ\sqrt{2Δt}} ).

Trinomial Tree Valuation

1. Build the Tree: Create a trinomial tree over the option’s life.
2. Compute Payoffs: Determine the payoff at each final node.
3. Backward Induction: Move backwards from maturity to the present, computing the discounted expected option value at each step.

[ V_i = e^{-rΔt} [p_u V_{i+1,u} + p_m V_{i,m} + p_d V_{i+1,d}] ]

Pros and Cons

Pros:

• Reduced Error: By including an additional price state, the model provides a closer approximation to continuous processes.
• Stability: More stable and accurate compared to the binomial model, especially in approximating high volatility.

Cons:

• Complexity: More complex to implement and visualize compared to the binomial model.
• Computational Expense: Slightly more computationally demanding due to the additional state.

Extensions to Lattice Models

Implied Volatility and Calibration

Both the binomial and trinomial models require proper calibration of volatility to accurately price options. Calibration involves aligning the model-generated prices with market-observed prices by adjusting the volatility parameter.

American Options

Lattice models are particularly well-suited for valuing American options, which can be exercised at any time before maturity. The algorithm checks at each node whether early exercise yields a higher value than holding the option.

Dividend-Paying Stocks

Adjustments can be made to lattice models to incorporate dividends. For discrete dividends, the underlying stock price is reduced by the dividend amount at the ex-dividend date. For continuous dividends, the stock price evolution factor accounts for the dividend yield δ.

[ u = e^{(σ - δ) \sqrt{Δt}} ] [ d = e^{-(σ + δ) \sqrt{Δt}} ]

Practical Applications and Software

Lattice models are widely used in finance for various applications beyond classic options pricing, including:

• Convertible Bonds: Pricing securities that can be converted into a certain number of underlying shares.
• Real Options: Evaluating investment opportunities with managerial flexibility.
• Risk Management: Assessing the risk exposure of portfolios containing derivatives.

Several software platforms and tools provide implementations of lattice models, including but not limited to MATLAB, Python (with libraries such as QuantLib and pandas), and specialized financial software like Bloomberg Terminal and Thomson Reuters Eikon.

Conclusion

Lattice models, through their adaptability and intuitive structure, have become integral in the field of financial derivatives’ pricing. While they come with certain computational limitations, their flexibility in handling various types of derivatives and market conditions continues to make them an invaluable tool in quantitative finance.

For more information on financial modeling software and tools, visit Bloomberg Terminal or Thomson Reuters Eikon.