Options Pricing Under Arithmetic Brownian Motions: A Comprehensive Study

Centrala begrepp

This paper provides a comprehensive study of options pricing under arithmetic Brownian motion (ABM), including deriving pricing formulas for various underlying asset types, exploring the equivalence of risk-neutral valuation and the PDE approach, and discussing the properties and potential arbitrage implications of ABM in options pricing.

Sammanfattning

Liu, Q., & Guo, S. (n.d.). Risk-neutral valuation of options under arithmetic Brownian motions.

This paper aims to provide a comprehensive study of options pricing under arithmetic Brownian motion (ABM), addressing the need for a thorough understanding of this model in light of recent market events like the CME Group's adoption of Bachelier pricing for oil futures options.

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Risk-neutral valuation of options under arithmetic Brownian motions

by Qiang Liu, S... på arxiv.org 11-12-2024

https://arxiv.org/pdf/2405.11329.pdf

Djupare frågor

How will the increasing adoption of ABM in pricing oil futures options impact the broader energy derivatives market and risk management practices?

The increasing adoption of ABM in pricing oil futures options has several potential impacts on the broader energy derivatives market and risk management practices:
Impacts on the Energy Derivatives Market:

More Accurate Pricing for Certain Products: ABM, by allowing for negative prices, can provide more realistic valuations for energy derivatives, especially those with negative underlying prices or strikes. This is particularly relevant for commodities like oil, which experienced negative prices during the COVID-19 pandemic.
Increased Model Complexity: The shift to ABM introduces complexities in pricing and hedging compared to the traditional GBM framework. Traders and risk managers need to adapt to these changes, potentially requiring new pricing models, hedging strategies, and risk management tools.
Potential for Arbitrage Opportunities: As the paper highlights, ABM can lead to seeming arbitrage opportunities, especially with long maturities and high volatilities. While these might not represent true arbitrage under unlimited liability, they still present trading and risk management challenges.
Impact on Volatility Surface: The use of ABM can influence the implied volatility surface, particularly for options with negative or low underlying prices. This can impact hedging costs and strategies for market participants.
Impacts on Risk Management Practices:

Model Risk Management: The adoption of ABM necessitates robust model risk management frameworks. This includes model validation, stress testing, and backtesting to ensure the models accurately capture the dynamics of energy prices under various market conditions.
Hedging Strategies: Hedging strategies need to be adjusted to account for the different price dynamics implied by ABM. This might involve using different option instruments or adjusting hedge ratios.
Counterparty Risk: The potential for negative prices under ABM can impact counterparty risk assessments. Clearing houses and market participants need to ensure adequate margining and collateralization practices to mitigate potential losses from negative prices.
Overall, the increasing adoption of ABM in energy derivatives necessitates a shift in market practices, demanding more sophisticated pricing models, risk management techniques, and a deeper understanding of the implications of ABM for trading and hedging.

Could the potential arbitrage opportunities under ABM, particularly for long maturities and high volatilities, be exploited in practice, and if so, what measures can be taken to mitigate these risks?

While the paper highlights potential arbitrage opportunities under ABM, particularly for long maturities and high volatilities, exploiting them in practice is not straightforward and comes with significant challenges:
Challenges in Exploiting Arbitrage:

Unlimited Liability Assumption: The seeming arbitrage opportunities arise from the unlimited liability assumption of the underlying asset under ABM. In reality, most assets, even commodities like oil, have practical limits on how negative their prices can go.
Liquidity Constraints: Arbitrage opportunities often require trading illiquid options with long maturities or extreme strikes. The lack of liquidity in these options can make it difficult to execute trades at the desired prices, diminishing potential profits.
Transaction Costs: Transaction costs, including brokerage fees and bid-ask spreads, can erode the profitability of arbitrage strategies, especially in less liquid markets.
Model Risk: The arbitrage opportunities rely on the accuracy of the ABM model. Any misspecification or miscalibration of the model can lead to incorrect pricing and potential losses.
Measures to Mitigate Risks:

Realistic Constraints on Underlying Prices: Incorporating realistic constraints on how negative underlying prices can go can help prevent unrealistic option prices and mitigate potential arbitrage opportunities.
Enhanced Model Validation and Backtesting: Rigorous model validation and backtesting procedures are crucial to ensure the ABM model accurately captures market dynamics and minimizes pricing errors.
Stress Testing: Stress testing models under various extreme scenarios, including those with high volatilities and long maturities, can help identify potential weaknesses and areas for model improvement.
Liquidity Risk Management: Market participants should carefully assess liquidity risks associated with trading options with long maturities or extreme strikes. This includes setting appropriate position limits and diversifying trading strategies.
In conclusion, while ABM presents potential arbitrage opportunities, exploiting them is challenging due to practical constraints and risks. Implementing robust risk management measures, including realistic price constraints, enhanced model validation, and liquidity risk management, is crucial to mitigate these risks.

Given the historical dominance of GBM in options pricing, what factors might accelerate or hinder the wider adoption of ABM in other asset classes beyond commodities?

Despite the historical dominance of Geometric Brownian Motion (GBM) in options pricing, several factors could accelerate or hinder the wider adoption of Arithmetic Brownian Motion (ABM) in other asset classes beyond commodities:
Factors Accelerating ABM Adoption:

Empirical Evidence: Increased empirical evidence demonstrating the superior performance of ABM in pricing options for certain asset classes, particularly those exhibiting mean-reversion or negative price possibilities, could drive adoption.
Regulatory Changes: Regulatory changes mandating or encouraging the use of models that better capture the risks associated with negative prices, similar to the CME's shift for oil futures, could accelerate ABM adoption.
Technological Advancements: Advancements in computational power and numerical methods can make it easier to implement and calibrate more complex ABM models, potentially making them more accessible for wider use.
Demand for More Realistic Pricing: As financial markets become more complex and interconnected, the demand for more realistic pricing models that accurately capture the dynamics of various asset classes could lead to greater adoption of ABM.
Factors Hindering ABM Adoption:

Inertia and Familiarity with GBM: The historical dominance and widespread familiarity with GBM within the financial industry create inertia against adopting new models like ABM.
Model Complexity and Implementation Costs: ABM models can be more complex to implement and calibrate compared to GBM, potentially leading to higher computational and implementation costs.
Lack of Standardized Approaches: The lack of standardized approaches for implementing and calibrating ABM models across different asset classes could hinder wider adoption.
Limited Empirical Support for Certain Asset Classes: For asset classes where negative prices are highly unlikely or where GBM provides a reasonable approximation, the benefits of adopting ABM might not outweigh the costs and complexities.
In conclusion, the wider adoption of ABM in other asset classes depends on the interplay of various factors. While empirical evidence supporting its superior performance and regulatory changes could accelerate adoption, inertia, model complexity, and the lack of standardized approaches might pose challenges.