Johnson binomial tree

What Is Johnson Binomial Tree?

The Johnson binomial tree is a specific variant within the broader field of financial modeling used for option pricing. Like other binomial model approaches, it provides a discrete-time framework to approximate the evolution of an underlying asset price over time. A key characteristic of the Johnson binomial tree is its ability to incorporate more flexible distributional assumptions for asset returns, particularly to account for phenomena like the volatility smile observed in market prices. This differentiates it from simpler binomial models by allowing for non-normal or skewed asset price distributions, providing a potentially more accurate valuation of complex financial instruments.

History and Origin

The concept of binomial trees for option valuation was formalized by Cox, Ross, and Rubinstein in 1979, building upon earlier work by William Sharpe in 1978. While the widely known Cox-Ross-Rubinstein (CRR) model typically assumes a lognormal distribution for asset prices, variations of the binomial tree model emerged to address specific market realities or theoretical refinements. The "Johnson binomial tree" indirectly references the work of N.L. Johnson, who developed systems of frequency curves (Johnson distributions) in the mid-20th century, allowing for transformations of normal distributions to fit various shapes, including those with skewness and kurtosis. These Johnson distributions have been applied in finance to model asset returns, offering a superior alternative to the normal distribution, particularly in areas like portfolio management and, crucially, in option pricing to accommodate observed market anomalies such as the volatility smile. The application of Johnson's distributional frameworks to the binomial tree structure allows for the generation of asset price paths that are consistent with empirical observations beyond the standard lognormal assumption.

Key Takeaways

• The Johnson binomial tree is an option pricing model that uses a discrete-time framework.
• It is designed to accommodate non-normal distributions of underlying asset prices, making it suitable for pricing options where a volatility smile is observed.
• It is a refinement of general binomial tree models, incorporating distributional flexibility derived from Johnson's systems of frequency curves.
• The model enhances the accuracy of European options and American options valuations in markets exhibiting skewness and kurtosis in asset returns.
• Its complexity is generally higher than simpler binomial models due to the advanced statistical fitting required.

Formula and Calculation

The specific formulas for the up ($u$) and down ($d$) factors, as well as the risk-neutral probabilities ($p$), in a Johnson binomial tree are adapted to ensure that the discrete price movements align with a target Johnson distribution (e.g., Johnson SU-distribution) for the underlying asset's returns over the option's life. Unlike simpler models such as the Cox-Ross-Rubinstein (CRR) model where (u) and (d) are often symmetric and (u \cdot d = 1), the Johnson binomial tree aims to match the first few moments (mean, variance, skewness, kurtosis) of the implied distribution of the underlying asset.

The general framework for calculating option value in a binomial tree involves working backward from expiration. At each node, the option value is calculated as the discounted expected value of the option in the next period:

[
C = e^{-r\Delta t} [p C_u + (1-p) C_d]
]

Where:

• (C) = Option value at the current node
• (r) = Risk-free interest rate
• (\Delta t) = Time step duration
• (p) = Risk-neutral probability of an upward move
• (C_u) = Option value if the underlying asset moves up
• (C_d) = Option value if the underlying asset moves down

The precise determination of (u), (d), and (p) in a Johnson binomial tree would involve fitting parameters of the chosen Johnson distribution to market data, such as implied volatility, to reflect the observed stochastic volatility and non-normal characteristics.

Interpreting the Johnson Binomial Tree

Interpreting the Johnson binomial tree involves understanding that it constructs a probabilistic pathway for the underlying asset's price, reflecting a more nuanced view of market dynamics than traditional models. By allowing for asset returns to follow a Johnson distribution, the model acknowledges that large price movements (fat tails) and asymmetrical distributions (skewness) are realistic features of financial markets. This means that the implied probabilities of extreme upward or downward movements, and the overall shape of the price distribution at maturity, are not forced into a symmetrical lognormal pattern. Instead, the Johnson binomial tree is calibrated to align with how market participants perceive risk and potential outcomes, often inferred from the prices of other options across different strike prices—a phenomenon known as the volatility smile or skew. The final option price derived from a Johnson binomial tree reflects these richer distributional assumptions, providing a more robust valuation, particularly for out-of-the-money or in-the-money options where the volatility smile has a significant impact.

Hypothetical Example

Consider valuing a 6-month European call option on a stock currently trading at $100. Assume a risk-free interest rate of 2% per annum, and a strike price of $105. A standard binomial tree might assume a lognormal distribution for the stock price. However, if market data suggests a strong negative skew in the stock's returns, indicating a higher probability of large downward movements than large upward movements (beyond what a lognormal model would suggest), a Johnson binomial tree would be employed.

1. Parameter Estimation: Using historical data and observed market option pricing, parameters for a Johnson SU-distribution are estimated that best fit the observed skewness and kurtosis of the stock's returns.
2. Tree Construction: A multi-step Johnson binomial tree is constructed. Unlike a standard CRR tree, the up and down factors (and potentially the probabilities at each node) are adjusted at each step to ensure that the overall price distribution generated by the tree at maturity approximates the fitted Johnson SU-distribution, rather than a simple lognormal one. For instance, the down-move factor might be calibrated to reflect a higher probability of larger negative jumps, while the up-move factor reflects smaller positive jumps, consistent with the observed negative skew.
3. Payoff Calculation: At the final nodes of the tree (at 6 months), the payoff of the call option is calculated: (\max(S_T - K, 0)), where (S_T) is the stock price at expiration and (K) is the strike price.
4. Backward Induction: The option value is then calculated by working backward through the tree, discounting the expected future payoffs at each node using the risk-free rate and the modified risk-neutral probabilities that are consistent with the Johnson distribution's parameters. This process yields the fair value of the option today, reflecting the non-normal return distribution.

This approach provides a more realistic valuation by accounting for market quirks like the volatility smile, which simpler models might miss.

Practical Applications

The Johnson binomial tree finds its primary utility in the nuanced world of derivative securities valuation, particularly when dealing with options that exhibit characteristics not well-captured by traditional models. One significant application is the accurate pricing of options in markets where the assumption of lognormally distributed asset returns is demonstrably violated. This is often evident through the presence of a "volatility smile" or "skew" in implied volatilities across different strike prices or maturities.

Specifically, the Johnson binomial tree can be used for:

• Pricing Exotic Options: For complex derivatives whose payoffs depend on the path taken by the underlying asset, or those with non-standard exercise features, the ability to model non-normal distributions can provide more precise valuations.
• Risk Management: By incorporating realistic return distributions, the model can offer more accurate assessments of option Greeks such as Delta, Gamma, and Vega, which are crucial for delta hedging and managing portfolio risk.
• Arbitrage Detection: In theory, pricing discrepancies between model values and market prices could indicate potential arbitrage opportunities, although sophisticated models like the Johnson binomial tree aim to reduce such discrepancies by more accurately reflecting market realities.

The approach contributes to a more robust framework for quantitative finance, especially in scenarios where market imperfections or specific distributional patterns are significant.

Limitations and Criticisms

While the Johnson binomial tree offers enhanced flexibility in modeling asset price distributions, it is not without limitations. A primary criticism, common to all binomial tree models, is their discrete nature. While increasing the number of time steps can make the model converge to a continuous-time process, this also significantly increases computational complexity. For the Johnson binomial tree specifically, the challenge lies in accurately calibrating the parameters of the Johnson distribution to market data. This process can be complex and data-intensive, requiring robust statistical methods.

Furthermore, all option pricing models, including the Johnson binomial tree, rely on certain assumptions about market efficiency and the behavior of underlying assets. For instance, models generally assume no transaction costs and the ability to borrow and lend at the risk-free interest rate, assumptions that do not perfectly hold in real markets. T⁷he assumption of constant volatility, though relaxed by the Johnson model's ability to fit a volatility smile, still presents a challenge for many models. The issue of constant volatility is a widely discussed parameter problem, as volatility is rarely constant in practice. T⁶herefore, while the Johnson binomial tree provides a more refined approach to handling non-normal distributions, it does not fully eliminate all the inherent limitations of mathematical models attempting to capture complex financial market dynamics.

Johnson Binomial Tree vs. Cox-Ross-Rubinstein (CRR) Model

The Johnson binomial tree and the Cox-Ross-Rubinstein model are both variants of the binomial model used for option pricing, but they differ significantly in their underlying assumptions about the asset price distribution.

The CRR model, introduced in 1979, is a foundational discrete-time lattice model that assumes that in each time step, the price of an underlying asset can only move up or down by a specific factor. I²t is known for its simplicity and its convergence to the Black-Scholes formula as the number of time steps increases. T¹he Johnson binomial tree, on the other hand, represents a more advanced application, aiming to overcome some of the CRR model's limitations by allowing the asset price path to conform to a wider range of empirical distributions, thus providing a more nuanced and potentially more accurate valuation in complex market scenarios where the lognormal assumption does not hold.

FAQs

What problem does the Johnson binomial tree try to solve?

The Johnson binomial tree attempts to address the limitation of standard option pricing models, such as the basic binomial model or the Black-Scholes model, in adequately pricing options when the underlying asset's returns do not follow a perfectly symmetrical normal or lognormal distribution. It specifically helps in situations where a "volatility smile" or skew is observed in the market, indicating that implied volatility varies with the strike price of options.

Is the Johnson binomial tree more accurate than the standard binomial model?

The Johnson binomial tree can be more accurate than a standard binomial model (like CRR) in situations where the underlying asset's return distribution exhibits significant skewness or kurtosis (fat tails), as it is designed to incorporate these characteristics. For simpler scenarios or options on assets with lognormally distributed returns, a standard binomial model may suffice.

How does the Johnson binomial tree account for the volatility smile?

The volatility smile indicates that options with different strike prices (and thus different implied volatilities) suggest that the market does not believe the underlying asset's returns are lognormally distributed. The Johnson binomial tree addresses this by using Johnson's system of distributions, which can model distributions with varying degrees of skewness and kurtosis. By calibrating the tree's parameters to match these observed distributional properties, the model can generate option prices that are consistent with the market's implied volatilities across different strike prices, thereby accounting for the volatility smile.

Is the Johnson binomial tree widely used?

While the underlying principles of Johnson distributions are applied in various areas of quantitative finance, the "Johnson binomial tree" specifically is a more specialized and computationally intensive variant compared to the more commonly used Cox-Ross-Rubinstein model. Its use is typically in situations requiring a more precise fit to empirical distributions and observed market phenomena like the volatility smile, often by institutional traders or quantitative analysts dealing with complex financial instruments.