Notation and Symbols
Grouped by where each symbol first does real work. Where a letter is used twice, both meanings are listed together and the note says how to tell them apart.
The Loan
| Symbol |
Meaning |
| $L$ |
Original loan principal |
| $i$ |
Monthly interest rate, $w/12$ |
| $N$ |
Number of monthly payments (360 for a 30-year loan) |
| $M$ |
Fixed monthly payment |
| $B_k$, $B_t$ |
Outstanding balance (UPB) after payment $k$, or at time $t$ |
| $w$, $w_t$ |
Mortgage note rate, or the pool's coupon |
The Pool
| Symbol |
Meaning |
| $\text{WAC}$ |
Weighted average coupon â balance-weighted note rate across the pool |
| $\text{WALA}$ |
Weighted average loan age, in months |
| $\text{WAM}$ |
Weighted average maturity â months of contractual term remaining |
| $\text{CPR}$ |
Conditional prepayment rate, annualized |
| $\text{SMM}$ |
Single monthly mortality â the monthly form of CPR |
| $\text{PSA}$ |
Public Securities Association prepayment benchmark |
| $B_j$, $C_j$, $A_j$, $R_j$ |
Current balance, note rate, age (months) and remaining term (months) of loan $j$ in a pool |
Rates and Discounting
| Symbol |
Meaning |
| $r$, $r_t$ |
Short rate or discount rate |
| $\text{SOFR}$ |
Secured Overnight Financing Rate |
| $y(t,T)$ |
Zero rate for maturity $T$ seen at $t$ |
| $P(0,T)$, $P(t,T)$ |
Price at time $0$ (or $t$) of \$1 paid at$T$â the discount factor for maturity$T$ |
| $s$ |
Constant spread added to the discount curve â the quantity an OAS solve returns |
| $\text{OAS}$ |
Option-adjusted spread |
Stochastic Dynamics
| Symbol |
Meaning |
| $W_t, \tilde{W}_t$ |
Brownian motion under $\mathbb{P}$ and $\mathbb{Q}$ |
| $\mathbb{P}$, $\mathbb{Q}$ |
Real-world measure and the risk-neutral (pricing) measure |
| $a$ |
Speed of mean reversion in the short-rate SDE |
| $\theta$, $\theta_t$ |
Long-run level the short rate is pulled toward |
| $\sigma_r$ |
Volatility of the short rate |
| $\rho$ |
Correlation between two driving Brownian motions |
| $\eta_t$ |
Market price of risk (Girsanov drift adjustment) |
| $Q_n$ |
Quadratic variation of a path over an $n$-step partition |
| $\tau_b$ |
First passage time to level $b$ |
Options and Survival
| Symbol |
Meaning |
| $S_0$, $S_t$ |
Spot price of the underlying asset, in the option chapters |
| $S(t)$ |
Survival probability, $\mathbb{P}(T>t)$ â a different quantity |
| $K$ |
Strike, or the refinance threshold that plays its part |
| $T$ |
Maturity, or the horizon of a survival calculation |
| $\lambda_t$ |
Prepayment intensity (continuous-time hazard rate) |
| $\Phi(\cdot)$ |
Standard-normal cumulative distribution |
| $\mathbb{E}^{\mathbb{Q}}[\,\cdot\,]$ |
Expectation under the pricing measure |
The letter $S$ carries two standard meanings that this book cannot avoid, because each is the convention in its own field: $S_t$ is the spot price in option pricing, and $S(t)$ is the survival function in survival analysis. The subscript is the tell â $S_t$ with a subscript is a price, $S(t)$ with an argument is a probability, and the two never appear in the same derivation.
Risk Measures
| Symbol |
Meaning |
| $\text{DV01}$ |
Dollar value of a basis point: $(P_{-} - P_{+})/2$, with $P_{\pm}$ the price after a one-basis-point rise or fall of the whole curve |
| $\text{CV01}$ |
Dollar convexity for a one-basis-point move: the second difference $P_{+} + P_{-} - 2P_0$ |
| $\text{KRD}_k$ |
Key rate duration for tenor bucket $k$: $(P_{-,k} - P_{+,k})/2$ with only knot $k$ bumped |
General conventions used throughout: $\log$ and $\ln$ both mean the natural logarithm; $^\top$ is the transpose; $\mathbb{1}\{\cdot\}$ and $\mathbb{I}_{\{\cdot\}}$ are indicators, 1 when the condition holds and 0 otherwise; $(x)^+ = \max(x, 0)$; $\mathcal{L}$ is a loss function; $\Phi$ and $\phi$ are the standard normal CDF and density; $\mathcal{N}(\mu, \sigma^2)$ is the normal law with mean $\mu$ and variance $\sigma^2$; $\theta$, $\phi$, $\psi$ as subscripts on a network ($f_\theta$, $\mu_\phi$) are its trainable weights, while a bare $\theta$ or $\theta_t$ is the short rate's mean-reversion level.