期权理论 / Options Theory

本页结构

核心概念

标的资产、贴现与合约结构 Underlying assets, discounting and contracts
看涨看跌平价与无套利推理 Put-call parity and no-arbitrage reasoning
Black-Scholes-Merton 模型与 Greeks Black-Scholes-Merton model and Greeks

学习顺序

期权推理优先从复制组合或无套利出发。 Start option arguments from replication or no-arbitrage.
使用 Black-Scholes 公式前先说明模型假设。 State model assumptions before using Black-Scholes formulas.
把 Greeks 解释成对冲敏感度，而不是只背符号。 Explain Greeks as sensitivities for hedging, not just symbols.

Options Theory is the mathematical framework for valuing derivative securities. At its core, it relies on the principle of no-arbitrage and the concept of risk-neutral valuation.

期权理论是评估衍生证券的数学框架。其核心依赖于无套利的原则和风险中性估值的概念。

Underlying Assets and Discounting

基础资产和贴现

Options are derivatives, meaning their value is derived from an Underlying Asset ( $S$ ), typically a stock, index, or commodity. The Bond ( $B$ ) represents the risk-free rate ( $r$ ), used for discounting future cash flows.

期权是衍生品，这意味着它们的价值源自基础资产（ $S$ ），通常是股票、指数或商品。债券 ( $B$ ) 代表无风险利率 ( $r$ )，用于贴现未来现金流量。

Discount Factor: The present value of one unit of currency received at time $T$ is $e^{-rT}$ .
Vanilla Options: Call Option ( $C$ ): Right to buy the underlying at the Strike Price ( $K$ ) at time $T$ . Payoff: $\max(S_T - K, 0)$ . Put Option ( $P$ ): Right to sell the underlying at the Strike Price ( $K$ ) at time $T$ . Payoff: $\max(K - S_T, 0)$ .

折扣系数：在 $T$ 时间收到的一单位货币的现值为 $e^{-rT}$ 。
普通期权：看涨期权 ( $C$ )：在时间 $T$ 以执行价格 ( $K$ ) 购买标的资产的权利。回报： $\max(S_T - K, 0)$ 。 看跌期权 ( $P$ )：在 $T$ 时间以执行价格 ( $K$ ) 出售标的资产的权利。回报： $\max(K - S_T, 0)$ 。

Put-Call Parity

看跌期权平价

Put-Call Parity is a fundamental no-arbitrage relationship between the prices of a European call option, a European put option, the underlying stock, and a zero-coupon bond.

看跌期权平价是欧式看涨期权、欧式看跌期权、标的股票和零息债券价格之间基本的无套利关系。

This equation states that a portfolio consisting of a long call and a zero-coupon bond with face value $K$ (left side) must have the same value as a portfolio consisting of a long put and a long share of the stock (right side). Any deviation from this parity implies an arbitrage opportunity.

该方程表明，由多头看涨期权和面值为 $K$ 的零息债券（左侧）组成的投资组合必须与由多头看跌期权和多头股票组成的投资组合（右侧）具有相同的价值。任何偏离该平价的行为都意味着套利机会。

The BSM model provides a closed-form solution for pricing European options under several key assumptions, most notably that the underlying asset price follows a Geometric Brownian Motion (GBM).

BSM 模型提供了在几个关键假设下对欧式期权进行定价的封闭式解决方案，最值得注意的是基础资产价格遵循几何布朗运动 (GBM)。

The Black-Scholes Partial Differential Equation (PDE)

Black-Scholes 偏微分方程 (PDE)

The BSM PDE is a second-order parabolic PDE that must be satisfied by the price of any derivative $V(S, t)$ that is a function of the underlying asset price $S$ and time $t$ , assuming no arbitrage.

BSM PDE 是二阶抛物线 PDE，必须由任何衍生品 $V(S, t)$ 的价格满足，该价格是标的资产价格 $S$ 和时间 $t$ 的函数（假设没有套利）。

Interpretation: The equation represents the idea that a portfolio consisting of the derivative and a dynamically adjusted position in the underlying asset (the Delta-Hedge) must earn the risk-free rate $r$ .

解释：该方程表示由衍生品和标的资产动态调整头寸（Delta-Hedge）组成的投资组合必须赚取无风险利率 $r$ 。

The BSM Pricing Formula (European Call)

BSM 定价公式（欧洲看涨期权）

The solution to the PDE, with the call option payoff as the boundary condition, is:

以看涨期权收益为边界条件的偏微分方程的解为：

where:

在哪里：

$N(\cdot)$ : Cumulative distribution function of the standard normal distribution.
Interpretation: $S N(d_1)$ is the expected present value of receiving the stock, and $K e^{-r(T-t)} N(d_2)$ is the expected present value of paying the strike price, both under the risk-neutral measure $\mathbb{Q}$ .

$N(\cdot)$ : Cumulative distribution function of the standard normal distribution.
解释： $S N(d_1)$ 是接收股票的预期现值， $K e^{-r(T-t)} N(d_2)$ 是支付执行价格的预期现值，两者均根据风险中性措施 $\mathbb{Q}$ 进行。

The Greeks are the partial derivatives of the option price with respect to various input parameters. They are essential for understanding the sensitivity of an option's price and for constructing hedging strategies.

Greeks 是期权价格相对于各项输入参数的偏导数。它们用于刻画期权价格敏感度，也是构建对冲策略的核心风险指标。

Greek	Formula (Partial Derivative)	Interpretation	Hedging Application
Delta ( $\Delta$ )	$\frac{\partial V}{\partial S}$	Change in option price for a one-unit change in the underlying price.	Primary Hedge: Used to create a delta-neutral portfolio (a portfolio whose value does not change with small movements in the underlying price).
Gamma ( $\Gamma$ )	$\frac{\partial^2 V}{\partial S^2}$	Change in Delta for a one-unit change in the underlying price.	Delta-Hedge Stability: Measures the effectiveness of the delta hedge. High Gamma means the hedge must be rebalanced frequently.
Theta ( $\Theta$ )	$\frac{\partial V}{\partial t}$	Change in option price for a one-unit change in time (time decay).	Time Risk: Measures the cost of holding the option over time. Typically negative for long options.
Vega ( $\mathcal{V}$ )	$\frac{\partial V}{\partial \sigma}$	Change in option price for a one-unit change in volatility ( $\sigma$ ).	Volatility Risk: Used to hedge against changes in the market's implied volatility.
Rho ( $\rho$ )	$\frac{\partial V}{\partial r}$	Change in option price for a one-unit change in the risk-free rate ( $r$ ).	Interest Rate Risk: Less critical than other Greeks but relevant for long-dated options.

Greek	公式（偏导数）	解释	对冲应用
Delta ( $\Delta$ )	$\frac{\partial V}{\partial S}$	标的价格每变化一单位时期权价格的变化。	主要对冲：用于构建 Delta 中性组合，即组合价值对标的小幅价格变化近似不敏感。
Gamma ( $\Gamma$ )	$\frac{\partial^2 V}{\partial S^2}$	标的价格每变化一单位时 Delta 的变化。	Delta 对冲稳定性：衡量 Delta 对冲随标的价格变化失效的速度。Gamma 较高时需要更频繁地再平衡。
Theta ( $\Theta$ )	$\frac{\partial V}{\partial t}$	时间变化一单位时期权价格的变化，也称时间衰减。	时间风险：衡量持有期权随时间流逝产生的价值变化，通常对期权多头为负。
Vega ( $\mathcal{V}$ )	$\frac{\partial V}{\partial \sigma}$	波动率 $\sigma$ 变化一单位时期权价格的变化。	波动率风险：用于管理和对冲市场隐含波动率变化。
Rho ( $\rho$ )	$\frac{\partial V}{\partial r}$	无风险利率 $r$ 变化一单位时期权价格的变化。	利率风险：通常不如其他 Greeks 关键，但对长期期权更重要。

Implied Volatility and the Volatility Smile

隐含波动率和波动率微笑

Implied Volatility ( $\sigma_{implied}$ ): The value of $\sigma$ that, when plugged into the BSM formula, yields the current market price of the option. It is a forward-looking measure of the market's expectation of future volatility.
Volatility Smile/Skew: The empirical observation that implied volatility is not constant across different strike prices and maturities, contradicting the BSM assumption of constant volatility. This phenomenon is a key area of research and modeling in quantitative finance (e.g., Stochastic Volatility Models).

隐含波动率 ( $\sigma_{implied}$ )： $\sigma$ 的值，当插入 BSM 公式时，产生期权的当前市场价格。它是市场对未来波动预期的前瞻性衡量指标。
波动率微笑/偏斜：经验观察表明隐含波动率在不同的执行价格和期限内不是恒定的，这与 BSM 恒定波动率的假设相矛盾。这种现象是定量金融研究和建模的关键领域（例如随机波动模型）。

Risk-Neutral Valuation

风险中性估值

The BSM model is derived under the Risk-Neutral Measure ( $\mathbb{Q}$ ).

BSM 模型是根据 风险中性衡量标准 ( $\mathbb{Q}$ ) 得出的。

Principle: In a complete and arbitrage-free market, the price of any derivative is the discounted expected value of its future payoff, where the expectation is taken under a measure where all assets grow at the risk-free rate $r$ .
Relevance: This concept simplifies pricing by allowing us to ignore the true market risk premium and focus only on the probability distribution of the underlying asset under the risk-neutral world. The drift of the underlying asset price process is set to $r$ instead of the true expected return $\mu$ .

原理：在一个完整且无套利的市场中，任何衍生品的价格都是其未来收益的贴现预期值，其中预期是在所有资产以无风险利率 $r$ 增长的情况下得出的。
相关性：这个概念允许我们忽略真实的市场风险溢价，只关注风险中性世界下标的资产的概率分布，从而简化了定价。标的资产价格过程的漂移设置为 $r$ ，而不是真实的预期收益 $\mu$ 。

补充讲解

无套利是第一原则

No-arbitrage comes first

期权定价从复制和无套利出发，而不是预测股票期望收益。这也是风险中性定价替换真实漂移率的原因。

Option pricing starts from replication and no-arbitrage, not from forecasting the expected stock return. This is why risk-neutral pricing replaces the physical drift.

Greeks 是局部近似

Greeks are local approximations

Delta、gamma、vega、theta 和 rho 描述局部敏感度。它们是有用的风险坐标，但会随标的、波动率和时间变化而更新。

Delta, gamma, vega, theta, and rho describe local sensitivities. They are useful risk coordinates, but must be refreshed as spot, volatility, and time change.

波动率曲面是市场证据

The volatility surface is evidence

波动率微笑或偏斜说明真实市场不满足常数波动率假设。BSM 应作为基准模型，而不是完整市场模型。

A volatility smile or skew shows that real markets violate constant-volatility assumptions. Treat BSM as a baseline, not the full model.

期权理论