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量化金融
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量化金融
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Black–Scholes · European option pricing
Black–Scholes · European option pricing
C=S·N(d1)−K·e^(−rT)·N(d2) · 看涨/看跌 · S100→10.45
Closed-form call & put value from spot, strike, rate, volatility and time
Put–call parity · no-arbitrage relation
Put–call parity · no-arbitrage relation
C−P=S−K·e^(−rT) · 无套利 · 保护性看跌=受托看涨
A fiduciary call and a protective put share one payoff — so they share one price
Binomial option tree · Cox–Ross–Rubinstein
Binomial option tree · Cox–Ross–Rubinstein
CRR u=e^(σ√Δt)、p 风险中性 · 回溯 · 收敛 BS
Price an option by backward induction on a recombining price tree
Sharpe ratio · risk-adjusted return
Sharpe ratio · risk-adjusted return
(Rp−Rf)/σp · 风险调整收益 · 资本市场线斜率
Excess return per unit of risk: S = (Rp − Rf) / σp
Value at Risk · parametric method
Value at Risk · parametric method
参数法 z·σ·V·√t · 95%→1.645/99%→2.326 · 左尾
The loss not exceeded with probability c: VaR = z_c · σ · V · √t
Kelly criterion · optimal bet sizing
Kelly criterion · optimal bet sizing
f*=p−q/b · 最优下注 · 对数增长峰值 · 半凯利
Fraction that maximises long-run growth: f* = p − q/b
Yield curve · term structure of interest rates
Yield curve · term structure of interest rates
期限结构 · 正常/平坦/倒挂 · 10Y−2Y 利差衰退信号
Yield vs maturity — the slope (10Y−2Y spread) classifies the curve
Forward pricing · cost of carry
Forward pricing · cost of carry
持有成本 F=S·e^((r−q)T) · 升水/贴水 · 现货套利
F = S·e^((r + u − q)·T) — no-arbitrage forward/futures price
Dividend discount model · Gordon growth
Dividend discount model · Gordon growth
戈登 P=D1/(r−g) · 内在价值 · g→r 失效
P = D1 / (r − g) — intrinsic value of a constantly-growing dividend
贷款分期偿还 · 等额本息
Loan amortization · fixed-payment loan
月供 M=P·i(1+i)ⁿ/((1+i)ⁿ−1) · 本息分摊 · 余额曲线
Level monthly payment; interest shrinks and principal grows over time
真实收益率 · 费雪方程
Real return · the Fisher equation
费雪 (1+nom)/(1+inf)−1 · 8%/3%→4.85% · 购买力侵蚀
Inflation erodes returns: real ≠ nominal − inflation exactly
蒙特卡洛期权定价
Monte-Carlo option pricing
GBM 路径模拟 · 贴现均值 · 收敛 BS · 标准误∝1/√N
Simulate terminal prices under GBM; converge to the Black–Scholes value