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Greeks：Vega（ν\nu）与隐含波动率

Vega

专题
Finance / 金融
难度
L4

题目详情

A. 解释隐含波动率与波动率微笑，对 Black–Scholes 有何含义？

B. 若 σ\sigma 固定为 30% 或随机且均值 30%，哪种情况下看涨更贵？

C. 若已知所有行权价的看涨价格曲线，能否恢复到期时的风险中性密度？

v=∂f∂σv = \frac{\partial f}{\partial \sigma}

For a European call or put on a stock with dividend yield yy: >v>=>S e−yτ τ N′(d1).>> v > = > S\,e^{-y\tau}\,\sqrt{\tau}\,N'(d_1). >

  • ATM has highest vega.
  • Longer maturity => higher vega.

A. Explain implied volatility and vol smile. Implication for Black-Scholes?

B. If σ\sigma is either a fixed 30% or random with mean 30%, which call is more expensive?

C. Knowing call prices for all strikes, can we recover the risk-neutral PDF at TT?

解析

A. 隐含波动率：使 BS 价格等于市场价格的 σ\sigma。波动率微笑：隐含波动率随行权价/期限变化，说明“常数波动率”的 BS 假设不完备。

B. 期权价格通常对 σ\sigma 是凸的，因此

E[c(σ)]≥c(E[σ]),\mathbb{E}[c(\sigma)]\ge c(\mathbb{E}[\sigma]),

随机波动率通常更贵。

C. 可以。

fST(K)=erτ ∂2c∂K2.f_{S_T}(K)=e^{r\tau}\,\frac{\partial^2 c}{\partial K^2}.

Original Explanation

AnswerA:

  • Implied volatility = σ\sigma solving BS price = market price.
  • Vol smile = the pattern that implied vol varies with strike (or maturity).
  • In reality, volatility isn’t constant => The simple lognormal BS assumption is incomplete.

AnswerB:
Option payoff is generally convex in σ\sigma, so  E[c(σ)]≥c(E[σ]).\,E[c(\sigma)] \ge c(E[\sigma]). The random-vol case is usually more expensive, except possibly near certain boundary conditions (ATM near expiry).

AnswerC:
Yes: fST(K)=e rτ ∂2c∂K2.f_{S_T}(K) = e^{\,r\tau}\,\frac{\partial^2 c}{\partial K^2}.