Quantitative Finance & Risk Analytics

Cost of Carry Model → $$F_t = S_t e^{(r + d)T}$$

Black-Scholes Call Option → $$C = S_0 N(d_1) - X e^{-rT} N(d_2)$$

Black-Scholes Parameters → $$d_1 = \frac{\ln(S_0/X) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$

Digital Call Option → $$\text{Payoff} = \begin{cases} 1, & S_T > X \\ 0, & S_T \le X \end{cases}$$
$$\text{Price (today)}: V_{\text{call}} = e^{-rT} N(d_2)$$

Digital Put Option → $$\text{Payoff} = \begin{cases} 1, & S_T < X \\ 0, & S_T \ge X \end{cases}$$
$$\text{Price (today)}: V_{\text{put}} = e^{-rT} N(-d_2)$$

Geometric Brownian Motion → $$dS_t = \mu S_t dt + \sigma S_t dW_t$$

GBM Discrete Simulation → $$S_{t+\Delta t} = S_t \exp\left[(\mu - \frac{1}{2}\sigma^2)\Delta t + \sigma\sqrt{\Delta t}Z\right]$$

Value at Risk (VaR) → $$VaR = Z_\alpha \sigma \sqrt{h} P$$

Risk-Neutral Pricing → $$V = e^{-rT}\mathbb{E}[\text{Payoff}]$$

Structured Note Pricing → $$P_n = F + \alpha (C_T - C_0)$$

Altman Z-Score → $$Z = 1.2X_1 + 1.4X_2 + 3.3X_3 + 0.6X_4 + 0.999X_5$$

Bond Pricing → $$P = \sum_{t=1}^{T} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^T}$$

Commodity-Linked Bond → $$P_b = P_f + \lambda \Delta C$$

Linear Regression → $$Y = \alpha + \beta X + \varepsilon$$

Expected Value → $$E[X] = \sum x_i p_i$$

Variance & Standard Deviation → $$\text{Var}(X) = E[(X-\mu)^2]$$ $$\sigma = \sqrt{\text{Var}(X)}$$