什么是期权凸性套利? Q 由于种种原因,期权市场往往会出 现市场交易价格与其理论价格出现差异 的情况,为无风险套利 提供了机会。常 见的无风险套利有平价套利、箱体套 利、凸性套利等。那什么是期权的凸性 套利呢? 凸函数的定义 首先我们来看一个概念,什么叫凸函 数? C 1 C1 C2 L2 C3 K1 K2 K3 K 上 图 即 为 凸 函 数 , 其 中 |L 1 的 斜 率|>|L2的斜率| l|L1的斜率|=(C1-C2)/(K2- K1) lL2的斜率|=(C2-C3)/(K3- K2) (1-λ)C1+ λC3>C2 其 中 λ=(Κ2-Κ1)/(Κ3-Κ1) K1(C2-C3)/(K3- K2） 公式① 假 设 λ=(Κ2-Κ1)/(Κ3-Κ1)，其 中 K1<K2<K3 公式①可简化为:C2<(1-λ)C1+λC3 欧式期权、的凸函数特性 欧式认购期权和认洁期权的价格C是 关于行权价 K的凸函数,所以需满足凸 函数的特性: C1、C2、C3分别表示行权价为K1、 K2 、 K3 的 相 同 类 型 ( 同 为 认 购 或 认 沽）、相同到期日的期权价格 如 果 上 式 不 成 立 , 即 (1- ...

Core Insights - The article discusses the evolution and significance of the binomial option pricing model, which serves as a crucial complement to the Black-Scholes model in the field of option pricing [1][10]. Group 1: Historical Context - The Black-Scholes model revolutionized option pricing in the 1970s, providing a mathematical framework that gained widespread acceptance in both academia and practice [1]. - The limitations of the Black-Scholes model, such as its strict assumptions about market conditions, led to the development of the binomial model by Cox, Ross, and Rubinstein in 1979 [2][10]. Group 2: Binomial Model Fundamentals - The binomial model divides the option's life into multiple discrete time intervals, allowing for a more intuitive representation of asset price movements [2]. - In each time interval, the asset price can either increase or decrease, creating a branching structure similar to a binomial tree [2][3]. - The model operates under the no-arbitrage principle, ensuring that there are no risk-free profit opportunities in the market [2]. Group 3: Pricing Mechanism - The single-period model serves as the foundation for the multi-period binomial model, where option values are calculated recursively from the expiration date back to the present [5]. - The risk-neutral probability is a key concept in the model, simplifying the calculation of expected option values [4][6]. Group 4: Application to American Options - The binomial model is particularly suited for pricing American options, which can be exercised at any time before expiration, by evaluating the option's value at each node [8]. - The model allows for the comparison of holding the option until expiration versus exercising it early, thus accurately reflecting the value of early exercise [8]. Group 5: Limitations and Challenges - Despite its advantages, the binomial model faces challenges such as exponential growth in computational nodes with increasing periods, which can hinder real-time pricing in high-frequency trading [9]. - The model's accuracy is highly dependent on the volatility input; discrepancies between assumed and actual market volatility can lead to significant pricing errors [9]. Group 6: Future Outlook - The binomial model has become a foundational tool in option pricing, addressing the limitations of the Black-Scholes model and adapting to complex derivatives [10]. - Ongoing advancements in algorithms and technology are expected to expand the model's applicability, supporting risk management and valuation across various financial products [10].