Cheat Sheet

Black-Scholes (Call): $C = S_0 N(d_1) - Ke^{-rT}N(d_2)$ where $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$, $d_2 = d_1 - \sigma\sqrt{T}$

Put-Call Parity: $C - P = S_0 - Ke^{-rT}$

Delta Hedging: A delta-neutral portfolio: $\Pi = V - \Delta \cdot S$, rebalanced so $\frac{\partial \Pi}{\partial S} = 0$

Greeks:

$\Delta = \frac{\partial V}{\partial S}$,   $\Gamma = \frac{\partial^2 V}{\partial S^2}$,   $\mathcal{V} = \frac{\partial V}{\partial \sigma}$,   $\Theta = \frac{\partial V}{\partial t}$,   $\rho = \frac{\partial V}{\partial r}$

Geometric Brownian Motion: $dS = \mu S\, dt + \sigma S\, dW$

Ito's Lemma: $df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial S}dS + \frac{1}{2}\frac{\partial^2 f}{\partial S^2}(dS)^2$

Girsanov's Theorem: Under a change of measure $\mathbb{Q}$, $\widetilde{W}_t = W_t + \int_0^t \theta_s\, ds$ is a $\mathbb{Q}$-Brownian motion, where $\theta$ is the market price of risk.

Martingale Property: $\{X_t\}$ is a martingale if $\mathbb{E}[X_t \mid \mathcal{F}_s] = X_s$ for all $s \leq t$.

Bayes' Theorem: $P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}$

Conditional Expectation: $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$  (tower property)

Normal Distribution: $X \sim N(\mu, \sigma^2)$,   MGF: $M_X(t) = \exp\!\left(\mu t + \frac{\sigma^2 t^2}{2}\right)$

Poisson Distribution: $P(X = k) = e^{-\lambda}\frac{\lambda^k}{k!}$,   $\mathbb{E}[X] = \text{Var}(X) = \lambda$

Central Limit Theorem: $\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0,1)$ as $n \to \infty$

Law of Large Numbers: $\bar{X}_n \xrightarrow{p} \mu$ as $n \to \infty$

Covariance: $\text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$

Correlation: $\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$,   $-1 \leq \rho \leq 1$

OLS Estimator: $\hat{\beta} = (X^TX)^{-1}X^TY$

VaR (Parametric, Normal): $\text{VaR}_\alpha = \mu + z_\alpha\, \sigma$

Portfolio Variance: $\sigma_P^2 = \mathbf{w}^T \Sigma\, \mathbf{w}$

Duration: $D = -\frac{1}{P}\frac{dP}{dy}$,   price change $\approx -D \cdot \Delta y \cdot P$

Sharpe Ratio: $SR = \frac{\mathbb{E}[R_p] - R_f}{\sigma_p}$

Kelly Criterion: $f^* = \frac{p(b+1) - 1}{b}$,   where $p$ = win probability, $b$ = odds ratio