CQF Comprehensive Cheatsheet - Fixed Income/Interest Rates

BOND FUNDAMENTALS

Basic Bond Terminology

Face Value/Par ($F$): Principal amount (typically $100 or $1000)

Coupon Rate ($c$): Annual interest rate on face value

Coupon Payment: $C = c \times F$ (often paid semi-annually)

Maturity ($T$): Time until principal repayment

Yield to Maturity (YTM) ($y$): IRR of bond cash flows

Bond Pricing

Discrete compounding (annual):

$$P = \sum_{i=1}^n \frac{C}{(1+y)^{t_i}} + \frac{F}{(1+y)^T}$$

Semi-annual compounding (US convention):

$$P = \sum_{i=1}^{2n} \frac{C/2}{(1+y/2)^i} + \frac{F}{(1+y/2)^{2n}}$$

Continuous compounding:

$$P = \sum_{i=1}^n C e^{-y t_i} + F e^{-yT}$$

Example: 5-year bond, 6% coupon (annual), face $100, YTM = 5%
$P = \frac{6}{1.05} + \frac{6}{1.05^2} + \frac{6}{1.05^3} + \frac{6}{1.05^4} + \frac{106}{1.05^5}$
$= 5.71 + 5.44 + 5.18 + 4.94 + 83.06 = \boxed{\$104.33}$
Bond trades at premium (coupon > YTM)

Bond Price-Yield Relationship

$y < c$: Bond at premium ($P > F$)
$y = c$: Bond at par ($P = F$)
$y > c$: Bond at discount ($P < F$)

Convexity: Price-yield relationship is convex (good for bondholders)

Zero-Coupon Bonds

Price:

$$P(t,T) = e^{-r(t,T)(T-t)}$$

or discrete:

$$P(t,T) = \frac{F}{(1+y)^{T-t}}$$

Spot rate $r(t,T)$: Zero-coupon yield for maturity $T$

Example: 10-year zero, face $100, $y = 4\%$
$P = \frac{100}{1.04^{10}} = \boxed{\$67.56}$

Accrued Interest

Clean Price: Quoted price (without accrued)

Dirty Price (Full Price):

$$\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}$$

Accrued Interest:

$$AI = C \times \frac{\text{Days since last coupon}}{\text{Days in coupon period}}$$

Day Count Conventions:

Actual/Actual: US Treasuries
30/360: US Corporate bonds
Actual/360: Money markets
Actual/365: UK Gilts

YIELD CURVE CONSTRUCTION

Spot Rates and Forward Rates

Spot Rate $r(0,T)$: Zero-coupon rate from now (0) to time $T$

Forward Rate $f(t,T_1,T_2)$: Rate for period $[T_1, T_2]$ locked in at $t$

Relationship (discrete):

$$(1+r_2)^{T_2} = (1+r_1)^{T_1}(1+f)^{T_2-T_1}$$

Solving for forward:

$$f(0,T_1,T_2) = \frac{(1+r_2)^{T_2}}{(1+r_1)^{T_1}}^{\frac{1}{T_2-T_1}} - 1$$

Continuous compounding:

$$f(0,T_1,T_2) = \frac{r_2 T_2 - r_1 T_1}{T_2 - T_1}$$

Instantaneous forward rate:

$$f(t,T) = -\frac{\partial \ln P(t,T)}{\partial T} = r(t,T) + (T-t)\frac{\partial r(t,T)}{\partial T}$$

Example: 1Y spot = 3%, 2Y spot = 4%
1Y forward 1Y ahead:
$f = \frac{1.04^2}{1.03^1} - 1 = \frac{1.0816}{1.03} - 1 = 0.0501$ or 5.01%

Bootstrapping the Curve

Method: Extract zero rates from coupon bond prices

Steps:

Start with shortest maturity (use directly if zero)
For coupon bond, use previously solved zeros for earlier cash flows
Solve for the new zero rate
Move to next maturity

Example:
Given:

6M T-bill: Price = $97.50, Face = $100 → $r_{0.5} = 5.13\%$
1Y bond: Coupon = 5%, Price = $98.50

Solve for $r_1$:
$98.50 = \frac{2.5}{e^{0.0513 \times 0.5}} + \frac{102.5}{e^{r_1 \times 1}}$
$98.50 = 2.44 + 102.5e^{-r_1}$
$96.06 = 102.5e^{-r_1}$
$e^{-r_1} = 0.9372$
$r_1 = 0.0648$ or 6.48%

Interpolation Methods

Linear (on zero rates):

$$r(T) = r_1 + \frac{T - T_1}{T_2 - T_1}(r_2 - r_1)$$

Linear (on log discount factors):

$$\ln P(0,T) = \ln P(0,T_1) + \frac{T - T_1}{T_2 - T_1}[\ln P(0,T_2) - \ln P(0,T_1)]$$

Cubic Spline: Smooth curve through points

Nelson-Siegel:

$$r(T) = \beta_0 + \beta_1\frac{1-e^{-T/\tau}}{T/\tau} + \beta_2\left(\frac{1-e^{-T/\tau}}{T/\tau} - e^{-T/\tau}\right)$$

$\beta_0$: Long-term level
$\beta_1$: Short-term component
$\beta_2$: Medium-term component (hump)
$\tau$: Decay parameter

Svensson Extension: Adds second hump term

Further Reading: Wikipedia: Yield Curve | Investopedia: Bootstrapping | BIS: Nelson-Siegel Model

DURATION AND CONVEXITY

Macaulay Duration

Definition: Weighted average time to receive cash flows

$$D_{Mac} = \frac{1}{P}\sum_{i=1}^n t_i \frac{CF_i}{(1+y)^{t_i}}$$

where $CF_i$ includes both coupons and principal

Properties:

Measured in years
Zero-coupon bond: $D = T$
Lower coupon → higher duration
Higher yield → lower duration

Modified Duration

Definition: Price sensitivity to yield changes

$$D_{Mod} = \frac{D_{Mac}}{1+y}$$

Price approximation:

$$\frac{dP}{P} \approx -D_{Mod} \cdot dy$$

or:

$$\Delta P \approx -D_{Mod} \times P \times \Delta y$$

Example: Bond price = $105, $D_{Mac} = 7.5$ years, $y = 5\%$
$D_{Mod} = \frac{7.5}{1.05} = 7.14$

If $y$ increases from 5% to 5.5% ($\Delta y = 0.005$):
$\Delta P \approx -7.14 \times 105 \times 0.005 = -\$3.75$
New price ≈ $105 - $3.75 = $101.25

Dollar Duration (DV01)

Definition: Dollar change for 1 bp (0.01%) yield change

$$\text{DV01} = -\frac{\partial P}{\partial y} \times 0.0001 = D_{Mod} \times P \times 0.0001$$

Example: $P = \$10M$, $D_{Mod} = 7.14$
$\text{DV01} = 7.14 \times 10,000,000 \times 0.0001 = \boxed{\$7,140}$
1 bp increase → lose $7,140

Convexity

Definition: Second-order price sensitivity

$$C = \frac{1}{P}\sum_{i=1}^n t_i(t_i+1)\frac{CF_i}{(1+y)^{t_i+2}}$$

Better price approximation:

$$\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y + \frac{1}{2}C \cdot (\Delta y)^2$$

Why convexity matters:

Duration alone underestimates price gain when yields fall
Duration alone overestimates price loss when yields rise
Convexity is always positive for option-free bonds
Higher convexity = better (more upside, less downside)

Example: $P = \$100$, $D_{Mod} = 7$, $C = 60$, $\Delta y = -1\%$

Duration only:
$\Delta P = -7 \times 100 \times (-0.01) = \$7.00$

With convexity:
$\Delta P = -7 \times 100 \times (-0.01) + 0.5 \times 60 \times 100 \times 0.01^2$
$= 7.00 + 0.30 = \boxed{\$7.30}$

Convexity adds $0.30 (more gain than duration predicts)

Effective Duration

For bonds with embedded options:

$$D_{Eff} = \frac{P_{-\Delta y} - P_{+\Delta y}}{2 \times P_0 \times \Delta y}$$

where $P_{-\Delta y}$ and $P_{+\Delta y}$ are prices when yield shifts by $\pm \Delta y$

Further Reading: Wikipedia: Duration | Investopedia: Convexity | CFI: Duration Explained

INTEREST RATE SWAPS

Plain Vanilla IRS

Definition: Exchange fixed rate for floating rate

Fixed leg: Pay/receive fixed rate $R$
Floating leg: Receive/pay LIBOR (or SOFR)
Notional: $N$ (not exchanged)
Tenor: Swap maturity (e.g., 5Y, 10Y)

Fixed payer = Short bond + Long FRN

Fixed receiver = Long bond + Short FRN

Swap Rate Derivation

Fair swap rate $R$: Rate that makes initial swap value = 0

Fixed leg PV:

$$PV_{fixed} = R \times N \times \sum_{i=1}^n \delta_i P(0,T_i)$$

where $\delta_i$ = accrual period (e.g., 0.5 for semi-annual)

Floating leg PV:

$$PV_{float} = N \times [1 - P(0,T_n)]$$

Key insight: Floating leg worth par at reset dates

Swap rate (sets $PV_{fixed} = PV_{float}$):

$$R = \frac{1 - P(0,T_n)}{\sum_{i=1}^n \delta_i P(0,T_i)}$$

The denominator is the annuity factor or PV01

Example: 2-year swap, semi-annual payments
Discount factors: $P(0,0.5)=0.975, P(0,1)=0.950, P(0,1.5)=0.925, P(0,2)=0.900$

Annuity = $0.5(0.975 + 0.950 + 0.925 + 0.900) = 1.875$
$R = \frac{1 - 0.900}{1.875} = \frac{0.100}{1.875} = 0.0533$ or 5.33%

Swap Valuation (After Inception)

Value to fixed payer:

$$V_{swap} = N \times \sum_{i=1}^n \delta_i P(0,T_i)[F(0,T_{i-1},T_i) - R]$$

where $F(0,T_{i-1},T_i)$ = forward rate for period $[T_{i-1}, T_i]$

Alternatively:

$$V_{swap} = PV_{float} - PV_{fixed}$$

If rates rise → fixed payer gains (paying below market)

If rates fall → fixed receiver gains

Swap DV01

Definition: Change in swap value per 1 bp parallel shift

$$\text{DV01} = N \times 0.0001 \times \sum_{i=1}^n \delta_i P(0,T_i)$$

Approximately: $\text{DV01} \approx N \times D_{Mod} \times 0.0001$

Basis Swaps

Definition: Exchange one floating rate for another

Examples:

3M LIBOR vs 6M LIBOR
LIBOR vs SOFR (post-2021)
EURIBOR vs EONIA

Basis spread: Adjustment to make swap fair

Further Reading: Wikipedia: Interest Rate Swaps | Investopedia: IRS Explained | CME: Introduction to IRS

CAPS, FLOORS, AND SWAPTIONS

Interest Rate Caps

Cap: Portfolio of caplets (call options on interest rate)

Caplet Payoff (at $T_i$):

$$\text{Payoff} = N \times \delta \times \max(L(T_{i-1}) - K, 0)$$

where $L(T_{i-1})$ = LIBOR set at $T_{i-1}$, $K$ = strike rate

Caplet Pricing (Black's formula):

$$\text{Caplet} = N \times \delta \times P(0,T_i) \times [F \cdot N(d_1) - K \cdot N(d_2)]$$ $$d_1 = \frac{\ln(F/K) + \sigma^2 T_{i-1}/2}{\sigma\sqrt{T_{i-1}}}, \quad d_2 = d_1 - \sigma\sqrt{T_{i-1}}$$

where $F = F(0,T_{i-1},T_i)$ = forward LIBOR, $\sigma$ = volatility

Cap value:

$$\text{Cap} = \sum_{i=1}^n \text{Caplet}_i$$

Example: Single caplet, $N = \$1M$, $\delta = 0.5$, $K = 5\%$
$F = 5.5\%$, $\sigma = 20\%$, $T = 1$, $P(0,1.5) = 0.93$

$d_1 = \frac{\ln(0.055/0.05) + 0.02 \times 1}{0.20} = 0.5866$
$d_2 = 0.5866 - 0.20 = 0.3866$
$N(d_1) = 0.7213, N(d_2) = 0.6505$

Value = $1M \times 0.5 \times 0.93 \times [0.055 \times 0.7213 - 0.05 \times 0.6505]$
= $465,000 \times [0.0397 - 0.0325] = \boxed{\$3,348}$

Interest Rate Floors

Floor: Portfolio of floorlets (put options on interest rate)

Floorlet Payoff:

$$\text{Payoff} = N \times \delta \times \max(K - L(T_{i-1}), 0)$$

Pricing: Similar to caplet, using Black's formula for puts

Put-Call Parity for Caps/Floors

$$\text{Cap} - \text{Floor} = \text{Swap}$$

More precisely:

$$\text{Cap}(K) - \text{Floor}(K) = \text{Swap}_{\text{floating}} - \text{Swap}_{\text{fixed at K}}$$

Swaptions

Definition: Option to enter into swap

Payer swaption: Right to pay fixed in swap
Receiver swaption: Right to receive fixed in swap

Notation: $m \times n$ swaption = option expires in $m$ years, swap lasts $n$ years

Example: 2×5 payer swaption = right in 2Y to enter 5Y swap paying fixed

Swaption Payoff (payer, at expiry $T_0$):

$$\text{Payoff} = \max\left(0, N \times \sum_{i=1}^n \delta_i P(T_0, T_i)[R_{swap}(T_0) - K]\right)$$

where $R_{swap}(T_0)$ = swap rate at option expiry, $K$ = strike

Black's Formula for Swaptions:

$$V = N \times A(0,T_0,T_n) \times [R_0 N(d_1) - K N(d_2)]$$ $$d_1 = \frac{\ln(R_0/K) + \sigma^2 T_0/2}{\sigma\sqrt{T_0}}, \quad d_2 = d_1 - \sigma\sqrt{T_0}$$

where:

$A(0,T_0,T_n) = \sum_{i=1}^n \delta_i P(0,T_i)$ = annuity factor
$R_0$ = forward swap rate
$\sigma$ = swaption volatility

Payer swaption = Call on swap rate

Receiver swaption = Put on swap rate

Example: 1×2 payer swaption, strike = 5%, forward swap rate = 5.5%
$\sigma = 25\%$, $T_0 = 1$, annuity = 1.85, notional = $10M

$d_1 = \frac{\ln(0.055/0.05) + 0.03125}{0.25} = 0.5116$
$d_2 = 0.2616$

$V = 10M \times 1.85 \times [0.055 \times 0.6953 - 0.05 \times 0.6032]$
= $18.5M × 0.0082 = $151,700

Further Reading: Wikipedia: Caps and Floors | Wikipedia: Swaptions | Investopedia: Swaptions

SHORT RATE MODELS

Vasicek Model

Short rate dynamics:

$$dr_t = \kappa(\theta - r_t)dt + \sigma dW_t$$

$\kappa$: Mean reversion speed
$\theta$: Long-term mean
$\sigma$: Volatility

Properties:

Gaussian (can go negative)
Mean-reverting
Affine term structure

Bond pricing formula:

$$P(t,T) = A(t,T) e^{-B(t,T)r_t}$$

where:

$$B(t,T) = \frac{1-e^{-\kappa(T-t)}}{\kappa}$$ $$\ln A(t,T) = \left(\theta - \frac{\sigma^2}{2\kappa^2}\right)[B(t,T) - (T-t)] - \frac{\sigma^2 B(t,T)^2}{4\kappa}$$

Example: $r_0 = 3\%$, $\kappa = 0.5$, $\theta = 5\%$, $\sigma = 1\%$, $T = 2$
$B(0,2) = \frac{1-e^{-1}}{0.5} = 1.2642$
$\ln A(0,2) = ...$ (calculation omitted)
$P(0,2) = A \times e^{-1.2642 \times 0.03} = \boxed{0.9235}$

Cox-Ingersoll-Ross (CIR) Model

Dynamics:

$$dr_t = \kappa(\theta - r_t)dt + \sigma\sqrt{r_t} dW_t$$

Key difference: Square-root diffusion ensures positive rates

Feller Condition (no boundary at zero):

$$2\kappa\theta \geq \sigma^2$$

Distribution: Non-central chi-squared

Bond pricing: Also affine, but more complex formulas

$$P(t,T) = A(t,T) e^{-B(t,T)r_t}$$

where $A(t,T)$ and $B(t,T)$ are known but lengthy expressions

Hull-White Model (Extended Vasicek)

Dynamics:

$$dr_t = [\theta(t) - \kappa r_t]dt + \sigma dW_t$$

Key feature: Time-dependent $\theta(t)$ allows calibration to initial yield curve

Calibration:

$$\theta(t) = \frac{\partial f(0,t)}{\partial t} + \kappa f(0,t) + \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})$$

where $f(0,t)$ = market instantaneous forward rate

Advantages:

Fits any initial term structure
Tractable (affine structure)
Widely used in practice

Black-Derman-Toy (BDT) Model

Dynamics (lognormal):

$$d\ln r_t = [\theta(t) - \frac{\sigma'(t)}{2}]dt + \sigma(t) dW_t$$

Or equivalently:

$$\frac{dr_t}{r_t} = \theta(t)dt + \sigma(t)dW_t$$

Properties:

Lognormal rates (always positive)
Time-dependent volatility $\sigma(t)$
Calibrated to term structure and volatility structure
Implemented on binomial/trinomial tree

Calibration: Fit $\theta(t)$ to yield curve, $\sigma(t)$ to cap/swaption vols

Model Comparison

Model	Rates	Distribution	Calibration	Use Case
Vasicek	Can be negative	Gaussian	Simple	Theory, ALM
CIR	Positive (Feller)	Chi-squared	Simple	Credit, rates
Hull-White	Can be negative	Gaussian	Yield curve	Derivatives pricing
BDT	Positive	Lognormal	Yield + vol	Exotic options
HJM	Flexible	Any	Full curve	Complex derivatives
LMM	Positive	Lognormal	Caps/swaptions	Market standard

Further Reading: Wikipedia: Vasicek Model | Wikipedia: CIR Model | Wikipedia: Hull-White Model

HEATH-JARROW-MORTON (HJM) FRAMEWORK

HJM Setup

Key Idea: Model entire forward rate curve, not just short rate

Forward rate dynamics (under $\mathbb{P}$):

$$df(t,T) = \alpha(t,T)dt + \sigma(t,T)dW_t$$

where $f(t,T)$ = instantaneous forward rate for maturity $T$

No-Arbitrage Condition (HJM Drift Condition)

Under risk-neutral measure $\mathbb{Q}$:

$$\alpha(t,T) = \sigma(t,T) \int_t^T \sigma(t,u)du$$

Interpretation: Drift completely determined by volatility structure

Only need to specify $\sigma(t,T)$; drift is then given by no-arbitrage

Short Rate from Forward Rates

$$r_t = f(t,t)$$

Bond price:

$$P(t,T) = \exp\left(-\int_t^T f(t,u)du\right)$$

Special Cases

Constant volatility $\sigma(t,T) = \sigma$:

Leads to Ho-Lee model

Exponential volatility $\sigma(t,T) = \sigma e^{-\kappa(T-t)}$:

Leads to Hull-White model

Multi-factor HJM:

$$df(t,T) = \alpha(t,T)dt + \sum_{i=1}^n \sigma_i(t,T)dW_t^i$$

HJM Challenges:

High-dimensional (infinite-dimensional in theory)
Requires numerical methods (Monte Carlo, PDE)
Volatility structure must be specified carefully
Computationally intensive

Further Reading: Wikipedia: HJM Framework | QuantStart: HJM Framework

LIBOR MARKET MODEL (LMM/BGM)

LMM Framework

Idea: Model observable forward LIBORs directly

Forward LIBOR $L_k(t)$ for period $[T_k, T_{k+1}]$:

$$L_k(t) = \frac{1}{\delta_k}\left[\frac{P(t,T_k)}{P(t,T_{k+1})} - 1\right]$$

where $\delta_k = T_{k+1} - T_k$ (typically 3M or 6M)

LMM Dynamics

Under $T_{k+1}$-forward measure $\mathbb{Q}^{k+1}$:

$$\frac{dL_k(t)}{L_k(t)} = \sigma_k(t) dW_t^{k+1}$$

$L_k(t)$ is a martingale under its "natural" measure $\mathbb{Q}^{k+1}$

Under terminal measure $\mathbb{Q}^N$:

$$\frac{dL_k(t)}{L_k(t)} = -\sum_{j=k+1}^{N-1} \frac{\delta_j \rho_{kj} \sigma_j(t) L_j(t)}{1 + \delta_j L_j(t)} dt + \sigma_k(t) dW_t^N$$

where $\rho_{kj}$ = correlation between $L_k$ and $L_j$

Caplet Pricing in LMM

Black's formula (exact in LMM):

$$\text{Caplet}_k = N \delta_k P(0,T_{k+1})[L_k(0)N(d_1) - K N(d_2)]$$ $$d_1 = \frac{\ln(L_k(0)/K) + \frac{1}{2}\bar{\sigma}_k^2 T_k}{\bar{\sigma}_k\sqrt{T_k}}$$

where $\bar{\sigma}_k^2 = \frac{1}{T_k}\int_0^{T_k} \sigma_k^2(t)dt$

Calibration: Choose $\sigma_k(t)$ to match market cap volatilities

Swaption Approximations

Problem: Swap rate is not lognormal in LMM

Solutions:

Monte Carlo: Simulate LIBOR paths, calculate payoff
Approximations: "Frozen drift" or other moment-matching

Volatility Structure

Separable form:

$$\sigma_k(t) = \phi_k g(t), \quad t \leq T_k$$

$\phi_k$: Terminal volatility (calibrated to caps)
$g(t)$: Time function (calibrated to swaptions)

Common choices for $g(t)$:

Constant: $g(t) = 1$
Exponential: $g(t) = e^{-\kappa(T_k - t)}$
Piecewise constant

Correlation Structure

Common specifications:

Exponential:

$$\rho_{ij} = e^{-\beta|T_i - T_j|}$$

Parametric:

$$\rho_{ij} = \rho_{\infty} + (1-\rho_{\infty})e^{-\beta|i-j|}$$

Further Reading: Wikipedia: LIBOR Market Model | Risk.net: LMM Definition

INTEREST RATE DERIVATIVES STRATEGIES

Curve Steepeners/Flatteners

Steepener: Profit if curve steepens (long end - short end increases)

Strategy: Receive fixed 10Y, pay fixed 2Y

Flattener: Profit if curve flattens

Strategy: Pay fixed 10Y, receive fixed 2Y

Example: 2Y rate = 2%, 10Y rate = 3%, spread = 100 bps
Enter 2Y-10Y steepener (notional adjusted for DV01 neutrality)

If spread → 120 bps, profit
If spread → 80 bps, loss

Butterfly Trades

Setup: Combine three maturities (short, medium, long)

Long butterfly: Profit if middle matures outperforms

Example: Long 5Y, short 2Y and 10Y (DV01-weighted)

Convexity Trade

Long convexity: Buy bonds, profit from large yield moves

Short convexity: Sell options, collect premium, lose if large moves

Basis Trading

LIBOR-OIS spread: Reflects bank credit risk

Trade: If spread widens, banks perceived as riskier

Swap spread: Swap rate - Treasury yield

Trade: Relative value between swaps and government bonds

LIBOR TRANSITION TO SOFR

Background

LIBOR scandal (2012): Banks manipulating submissions

LIBOR cessation: Discontinued after 2021 (USD LIBOR by June 2023)

Replacement rates:

SOFR (USD): Secured Overnight Financing Rate
SONIA (GBP): Sterling Overnight Index Average
€STR (EUR): Euro Short-Term Rate
TONAR (JPY): Tokyo Overnight Average Rate

SOFR vs LIBOR

Feature	LIBOR	SOFR
Type	Unsecured	Secured (repo)
Tenor	Multiple (1M, 3M, 6M, 12M)	Overnight only
Credit risk	Includes bank credit	Risk-free (repo collateral)
Transaction-based	No (panel submissions)	Yes (~$1T daily volume)
Volatility	Lower	Higher (overnight rate)

Conversion Issues

Spread Adjustment: SOFR ≈ LIBOR - credit spread

ARRC recommendation: Add credit spread adjustment when converting

Example: 3M LIBOR + spread → SOFR + spread + adjustment

Term SOFR: Constructed from SOFR derivatives (forward-looking)

KEY FORMULAS SUMMARY

Concept	Formula
Bond Price	$\sum C e^{-yt_i} + F e^{-yT}$
Forward Rate (cont.)	$f = \frac{r_2T_2 - r_1T_1}{T_2-T_1}$
Instantaneous Forward	$f(t,T) = -\frac{\partial \ln P(t,T)}{\partial T}$
Modified Duration	$D_{Mod} = \frac{D_{Mac}}{1+y}$
Price Change	$\frac{\Delta P}{P} \approx -D_{Mod}\Delta y + \frac{1}{2}C(\Delta y)^2$
DV01	$D_{Mod} \times P \times 0.0001$
Swap Rate	$R = \frac{1-P(0,T_n)}{\sum \delta_i P(0,T_i)}$
Caplet (Black)	$N\delta P(0,T)[FN(d_1) - KN(d_2)]$
Vasicek	$dr = \kappa(\theta-r)dt + \sigma dW$
CIR	$dr = \kappa(\theta-r)dt + \sigma\sqrt{r}dW$
HJM Drift	$\alpha(t,T) = \sigma(t,T)\int_t^T\sigma(t,u)du$
LMM (forward measure)	$\frac{dL_k}{L_k} = \sigma_k dW^{k+1}$

COMMON MISTAKES & TIPS

Common Mistakes:

Confusing duration types: Macaulay ≠ Modified ≠ Effective
DV01 sign: Price falls when yield rises (negative relationship)
Swap direction: Fixed payer = short bond (loses when rates fall)
Convexity sign: Always positive for option-free bonds
Forward rates vs spot rates: Forward can exceed spot in upward-sloping curve
Day count conventions: Critical for precise calculations
Clean vs dirty price: Must add accrued interest
Swaption terminology: "Payer" means paying fixed
Cap/floor parity: Must be at same strike
LMM measure: Each LIBOR martingale under different forward measure

Quick Interview Tips:

Duration intuition: Higher coupon → lower duration (cash sooner)
Convexity benefit: Asymmetric - more gain up than loss down
Swap fixed payer: Wants rates to rise (paying below market)
Why caps like calls: Benefit when rate (underlying) rises above strike
Swaption = option on swap: Not option on bond
HJM vs LMM: HJM models all forwards, LMM models discrete LIBORs
Vasicek negative rates: Can be feature (Europe) or bug
CIR always positive: If Feller condition holds
Hull-White advantage: Fits initial curve exactly
LIBOR → SOFR: SOFR lower (no credit spread), more volatile (overnight)
Curve steepener: Receive long, pay short
Basis risk: Different floating indices (3M vs 6M LIBOR)

PRACTICAL EXAMPLE: COMPLETE SWAP PRICING

Real-World Interest Rate Swap Pricing:

Market Data:

Pricing date: Today
Swap: 3-year, semi-annual, USD
Notional: $10M
Day count: Actual/360
Discount curve (zero rates, continuous):

0.5Y: 2.00%	1.0Y: 2.20%	1.5Y: 2.40%
2.0Y: 2.55%	2.5Y: 2.68%	3.0Y: 2.80%

Task 1: Calculate discount factors

$P(0,0.5) = e^{-0.02 \times 0.5} = 0.9900$
$P(0,1.0) = e^{-0.022 \times 1.0} = 0.9782$
$P(0,1.5) = e^{-0.024 \times 1.5} = 0.9646$
$P(0,2.0) = e^{-0.0255 \times 2.0} = 0.9503$
$P(0,2.5) = e^{-0.0268 \times 2.5} = 0.9350$
$P(0,3.0) = e^{-0.028 \times 3.0} = 0.9192$

Task 2: Calculate forward rates

For semi-annual period [0.5, 1.0]:
$F(0,0.5,1.0) = \frac{P(0,0.5) - P(0,1.0)}{0.5 \times P(0,1.0)} = \frac{0.9900 - 0.9782}{0.5 \times 0.9782} = 0.0241$ or 2.41%

Similarly:
$F(0,1.0,1.5) = 2.62\%$, $F(0,1.5,2.0) = 2.79\%$
$F(0,2.0,2.5) = 2.94\%$, $F(0,2.5,3.0) = 3.09\%$

Task 3: Calculate fair swap rate

Annuity factor:
$A = 0.5 \times (0.9900 + 0.9782 + 0.9646 + 0.9503 + 0.9350 + 0.9192)$
$= 0.5 \times 5.7373 = 2.8687$

Swap rate:
$R = \frac{1 - P(0,3.0)}{A} = \frac{1 - 0.9192}{2.8687} = \frac{0.0808}{2.8687} = 0.02817$ or 2.817%

Task 4: Value existing swap (contract rate = 2.5%)

Fixed leg PV (receiving fixed at 2.5%):
$PV_{fixed} = 0.025 \times 10M \times 2.8687 = \$716,175$

Floating leg PV:
$PV_{float} = 10M \times (1 - 0.9192) = \$808,000$

Value to fixed receiver:
$V = \$716,175 - \$808,000 = -\$91,825$

Fixed receiver has loss of $91,825 (receiving 2.5% when market is 2.817%)

Task 5: Calculate swap DV01

$\text{DV01} = 10M \times 0.0001 \times 2.8687 = \boxed{\$2,869}$

If rates rise 1 bp across curve, fixed payer gains ~$2,869

Task 6: Hedge with 3Y bond

Need bond position with same DV01:
If 3Y bond has $D_{Mod} = 2.85$, price = $98:
Bond notional = $\frac{\$2,869}{2.85 \times 98 \times 0.0001} = \boxed{\$10.26M}$

To hedge fixed receiver position: short $10.26M of 3Y bonds

Concept	Formula
Bond Price	\(\sum C e^{-yt_i} + F e^{-yT}\)
Forward Rate (cont.)	\(f = \frac{r_2T_2 - r_1T_1}{T_2-T_1}\)
Instantaneous Forward	\(f(t,T) = -\frac{\partial \ln P(t,T)}{\partial T}\)
Modified Duration	\(D_{Mod} = \frac{D_{Mac}}{1+y}\)
Price Change	\(\frac{\Delta P}{P} \approx -D_{Mod}\Delta y + \frac{1}{2}C(\Delta y)^2\)
DV01	\(D_{Mod} \times P \times 0.0001\)
Swap Rate	\(R = \frac{1-P(0,T_n)}{\sum \delta_i P(0,T_i)}\)
Caplet (Black)	\(N\delta P(0,T)[FN(d_1) - KN(d_2)]\)
Vasicek	\(dr = \kappa(\theta-r)dt + \sigma dW\)
CIR	\(dr = \kappa(\theta-r)dt + \sigma\sqrt{r}dW\)
HJM Drift	\(\alpha(t,T) = \sigma(t,T)\int_t^T\sigma(t,u)du\)
LMM (forward measure)	\(\frac{dL_k}{L_k} = \sigma_k dW^{k+1}\)