造市者例如城堡證券有獨家的選擇權計價模型嗎?

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Do market makers like citadel have proprietary option pricing model other than black and scholes model?


Yes, market makers like Citadel absolutely use sophisticated proprietary options pricing models that go far beyond the standard Black-Scholes model. The Black-Scholes model, while a foundational academic tool, is far too simplistic for the complexities of modern markets.

The Limitations of the Black-Scholes Model
For a professional market maker, relying solely on the Black-Scholes model would lead to consistent losses. This is because the model makes several unrealistic assumptions:

Constant Volatility: Black-Scholes assumes volatility is a single, constant number for a given stock. In reality, implied volatility is not flat; it varies across different strike prices and maturities, a phenomenon known as the volatility smile or skew.


European-Style Options: The model is only designed for European options, which can be exercised only at expiration. Most options traded in the U.S. are American-style, allowing for early exercise, a feature that has a real value the model ignores.

No Price Jumps: It assumes asset prices change smoothly and continuously. It cannot account for sudden, significant price jumps that can occur due to news events.

Fixed Interest Rates and Dividends: The model assumes constant, known interest rates and dividends, which is rarely the case in practice.

The Sophistication of Proprietary Models
To address these flaws and gain a competitive edge, firms like Citadel invest heavily in developing their own models. These models are a key part of their profitability and function as their intellectual property. They incorporate a range of advanced concepts:

Volatility Surfaces: Instead of a single number, proprietary models use complex mathematical surfaces that map implied volatility across a matrix of strike prices and maturities, reflecting the real-world volatility skew.

Stochastic Volatility Models: These models treat volatility not as a constant, but as a random variable that changes over time, more accurately reflecting market dynamics.

Jump-Diffusion Models: These models incorporate the possibility of sudden, sharp price movements, allowing them to better price options that are sensitive to "tail-risk" events.

Numerical Methods: For American-style options, these models use advanced numerical techniques like binomial trees or Monte Carlo simulations to accurately account for the value of early exercise.

Real-Time Data Integration: A market maker's model is constantly fed with live market data, allowing it to adapt to changing conditions instantly and maintain a competitive edge.

In essence, a market maker's proprietary model is a highly complex, constantly evolving system that moves beyond academic theory to accurately reflect the messy, dynamic, and unpredictable nature of real-time financial markets.

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白話文就是考慮了volatility surface, jump risk, tail risk,, 美式選擇權、以及報價即時性的問題。