CQF Quantitative Finance Cheatsheet

CQF CORE SUMMARY

Stochastic Foundations

Wiener Process $W_t$

$W_0 = 0$, independent increments, $W_t - W_s \sim \mathcal{N}(0, t-s)$
$\mathbb{E}[W_t] = 0$, $\text{Var}(W_t) = t$, $[W, W]_t = t$

Itô's Lemma

For $dX = \mu dt + \sigma dW$ and $f(X,t)$:

$$df = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial X} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial X^2}\right)dt + \sigma\frac{\partial f}{\partial X}dW$$

Key SDEs

GBM: $dS = \mu S dt + \sigma S dW$

Solution: $S_t = S_0 \exp\left[(\mu - \frac{\sigma^2}{2})t + \sigma W_t\right]$

Ornstein-Uhlenbeck: $dX = \kappa(\theta - X)dt + \sigma dW$

CIR: $dr = \kappa(\theta - r)dt + \sigma\sqrt{r}dW$

Feller condition: $2\kappa\theta \geq \sigma^2$ ensures $r > 0$

Black-Scholes Framework

BS PDE

$$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV$$

European Options

Call: $C = S_0N(d_1) - Ke^{-rT}N(d_2)$

Put: $P = Ke^{-rT}N(-d_2) - S_0N(-d_1)$

$$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$

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

Greeks

$\Delta = \partial V/\partial S$: Call $N(d_1)$, Put $N(d_1)-1$
$\Gamma = \partial^2 V/\partial S^2 = N'(d_1)/(S\sigma\sqrt{T})$
$\mathcal{V} = \partial V/\partial \sigma = S\sqrt{T}N'(d_1)$
$\Theta = \partial V/\partial t$, $\rho = \partial V/\partial r$

Risk Management

VaR (Value at Risk)

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

CVaR/ES: $\text{CVaR}_\alpha = \mathbb{E}[L | L \geq \text{VaR}_\alpha]$

GARCH(1,1)

$$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$$

Long-run variance: $\bar{\sigma}^2 = \omega/(1 - \alpha - \beta)$

Interest Rate Models

Vasicek

$dr = a(b - r)dt + \sigma dW$

Bond: $P(t,T) = A(t,T)e^{-B(t,T)r_t}$ where $B = (1-e^{-a\tau})/a$

Hull-White

$dr = [\theta(t) - ar]dt + \sigma dW$

Calibrates to initial yield curve

CREDIT RISK - OVERVIEW

Credit Risk Summary Table

Concept	Formula / Definition
Default Probability	$PD = \mathbb{P}(\tau \leq T)$
Survival Probability	$Q(t,T) = \mathbb{P}(\tau > T \| \tau > t)$
Hazard Rate	$\lambda(t) = -\frac{d\ln Q(0,t)}{dt}$
Recovery Rate	$R$ = fraction recovered at default
Loss Given Default	$LGD = 1 - R$
Credit Spread	$s = (1-R)\lambda$ (approx)
Expected Loss	$EL = PD \times LGD \times EAD$

Hazard Rate Framework

Intensity-Based Default

Default time $\tau$ is first jump of a Poisson process with intensity $\lambda(t)$:

$$\mathbb{P}(\tau > T) = Q(0,T) = \exp\left(-\int_0^T \lambda(s)ds\right)$$

Constant hazard rate $\lambda$:

$$Q(0,T) = e^{-\lambda T}$$

Conditional default probability in $[t, t+dt]$:

$$\mathbb{P}(\tau \in [t, t+dt] | \tau > t) = \lambda(t)dt$$

Term Structure of Hazard Rates

Piecewise constant: Given hazard rates $\lambda_1, \lambda_2, ...$ for periods $[0,T_1], [T_1,T_2], ...$:

$$Q(0,T_n) = \exp\left(-\sum_{i=1}^{n} \lambda_i (T_i - T_{i-1})\right)$$

Example - Survival Probability
Given: $\lambda_1 = 1\%$ for [0,1], $\lambda_2 = 1.5\%$ for [1,3], $\lambda_3 = 2\%$ for [3,5]
$Q(0,5) = e^{-0.01 \times 1} \times e^{-0.015 \times 2} \times e^{-0.02 \times 2}$
$Q(0,5) = e^{-0.01 - 0.03 - 0.04} = e^{-0.08} = \boxed{0.9231}$

Relationship to Credit Spreads

$$\lambda \approx \frac{s}{1-R}$$

Example - Hazard Rate from Spread
Credit spread $s = 200$ bps = 2%, Recovery $R = 40\%$
$\lambda = \frac{0.02}{1 - 0.40} = \frac{0.02}{0.60} = \boxed{3.33\%}$ per annum

Further Reading: Wikipedia: Credit Risk | Investopedia: Hazard Rate | Risk.net: Hazard Rate Definition

Structural Models

Merton Model (1974)

Firm value $V_t$ follows GBM. Equity = call option on firm assets:

$$E_t = V_t N(d_1) - De^{-r(T-t)}N(d_2)$$ $$d_1 = \frac{\ln(V_t/D) + (r + \sigma_V^2/2)(T-t)}{\sigma_V\sqrt{T-t}}$$

Distance to Default:

$$DD = \frac{\ln(V_t/D) + (\mu - \sigma_V^2/2)(T-t)}{\sigma_V\sqrt{T-t}}$$

Default Probability: $PD = N(-DD)$

Further Reading: Wikipedia: Merton Model | Investopedia: Merton Model | CFI: Distance to Default

RISKY BOND PRICING

Fundamental Pricing Framework

Risk-Free vs Risky Bond

Risk-free zero-coupon bond:

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

Risky zero-coupon bond with default risk:

$$P^{Risky}(t,T) = e^{-r(T-t)} \times \left[Q(t,T) + (1-Q(t,T)) \times R\right]$$

Alternatively, using risk-adjusted discount rate:

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

where $s$ is the credit spread.

Key Pricing Equations

Model	Risky Bond Price
Constant $\lambda$, $R=0$	$P = e^{-(r+\lambda)T}$
Constant $\lambda$, $R>0$	$P = e^{-rT}[e^{-\lambda T} + R(1-e^{-\lambda T})]$
Time-varying $\lambda(t)$	$P = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\mathbf{1}_{\tau>T} + R\mathbf{1}_{\tau\leq T}]$

Zero Recovery Model

Simplest case: bondholder receives nothing if default occurs.

Risky Zero Price (R=0): $$\bar{P}(t,T) = e^{-r(T-t)} \times Q(t,T) = e^{-(r+\lambda)(T-t)}$$

Example - Zero Recovery Pricing
Given: Face = $100, $r = 5\%$, $\lambda = 2\%$, $T = 3$ years

Risk-free price: $P^{RF} = 100 \times e^{-0.05 \times 3} = 100 \times 0.8607 = \$86.07$
Survival prob: $Q(0,3) = e^{-0.02 \times 3} = e^{-0.06} = 0.9418$
Risky price: $P^{Risky} = 86.07 \times 0.9418 = \boxed{\$81.06}$

Alternative: $P^{Risky} = 100 \times e^{-(0.05+0.02) \times 3} = 100 \times e^{-0.21} = \boxed{\$81.06}$ ✓
Credit spread: $s = \lambda = 200$ bps

Non-Zero Recovery Model

Bondholder recovers fraction $R$ of face value at default.

Risky Zero Price (R > 0): $$\bar{P}(t,T) = e^{-r(T-t)} \times \left[Q(t,T) + R \times (1-Q(t,T))\right]$$

Equivalently:

$$\bar{P}(t,T) = e^{-r(T-t)} \times \left[R + (1-R)Q(t,T)\right]$$

Example - Non-Zero Recovery
Given: Face = $100, $r = 4\%$, $\lambda = 3\%$, $R = 40\%$, $T = 5$ years

Survival probability: $Q = e^{-0.03 \times 5} = e^{-0.15} = 0.8607$
Default probability: $1 - Q = 0.1393$
Risk-free discount: $e^{-0.04 \times 5} = e^{-0.20} = 0.8187$

Risky bond price:
$P = 100 \times 0.8187 \times [0.8607 + 0.40 \times 0.1393]$
$P = 81.87 \times [0.8607 + 0.0557]$
$P = 81.87 \times 0.9164 = \boxed{\$75.02}$

Credit spread: $s = -\frac{1}{T}\ln(P/100) - r = -\frac{1}{5}\ln(0.7502) - 0.04$
$s = 0.0575 - 0.04 = 1.75\%$ = \boxed{175 \text{ bps}}

Risky Coupon Bond Pricing

General Formula

For a bond paying coupons $c$ at times $T_1, T_2, ..., T_n$ with face $F$ at $T_n$:

$$P = \sum_{i=1}^{n} c \cdot D(0,T_i) \cdot Q(0,T_i) + F \cdot D(0,T_n) \cdot Q(0,T_n) + R \cdot F \cdot \sum_{i=1}^{n} D(0,T_i) \cdot [Q(0,T_{i-1}) - Q(0,T_i)]$$

where $D(0,T_i) = e^{-rT_i}$ is the risk-free discount factor.

Simplified Formula (No Recovery on Accrued)

$$P = \sum_{i=1}^{n} c \cdot D(0,T_i) \cdot Q(0,T_i) + F \cdot D(0,T_n) \cdot \left[R + (1-R)Q(0,T_n)\right]$$

Example - Risky Coupon Bond
Given: Face = $100, Annual coupon = $5 (5%), $r = 3\%$, $\lambda = 2\%$, $R = 40\%$, T = 3 years

Discount factors:
$D_1 = e^{-0.03} = 0.9704$, $D_2 = e^{-0.06} = 0.9418$, $D_3 = e^{-0.09} = 0.9139$

Survival probabilities:
$Q_1 = e^{-0.02} = 0.9802$, $Q_2 = e^{-0.04} = 0.9608$, $Q_3 = e^{-0.06} = 0.9418$

Coupon PVs:
Year 1: $5 \times 0.9704 \times 0.9802 = \$4.756$
Year 2: $5 \times 0.9418 \times 0.9608 = \$4.525$
Year 3: $5 \times 0.9139 \times 0.9418 = \$4.305$
Total coupon PV = \$13.586

Principal PV:
$100 \times 0.9139 \times [0.40 + 0.60 \times 0.9418] = 91.39 \times [0.40 + 0.5651]$
$= 91.39 \times 0.9651 = \$88.20$

Total bond price = 13.586 + 88.20 = \boxed{\$101.79}

Credit Spread Relationships

Spread vs Hazard Rate

$$s \approx (1-R)\lambda$$

Exact relationship for risky zero:

$$e^{-sT} = Q(0,T) + R(1-Q(0,T))$$

Z-Spread

Constant spread $z$ added to risk-free curve that prices the bond:

$$P_{market} = \sum_{i} CF_i \cdot e^{-(r_i + z)T_i}$$

Asset Swap Spread

Spread over LIBOR paid/received to convert fixed bond to floating.

OAS (Option-Adjusted Spread)

Spread after removing value of embedded options.

Example - Implied Hazard Rate
Given: 5-year risky zero trades at $72, Risk-free rate = 4%, Recovery = 40%

Risk-free 5Y zero: $P^{RF} = 100 \times e^{-0.04 \times 5} = \$81.87$
Risky/Risk-free ratio: $72/81.87 = 0.8794$

From $0.8794 = R + (1-R)Q = 0.40 + 0.60 \times Q$:
$Q = (0.8794 - 0.40)/0.60 = 0.7990$

$Q = e^{-\lambda \times 5} = 0.7990$
$\lambda = -\ln(0.7990)/5 = 0.2244/5 = \boxed{4.49\%}$

Credit spread: $s = (1-0.40) \times 0.0449 = \boxed{269 \text{ bps}}$

Further Reading: Wikipedia: Credit Spread | Investopedia: Credit Spread | Risk.net: Z-Spread

CDS PRICING

CDS Structure

Key Components

Protection Buyer: Pays periodic premium (spread $s$)
Protection Seller: Pays $(1-R)$ × Notional at default
Credit Event: Triggers protection payment
Notional: $N$ (reference amount)

Cash Flows

Leg	Payment	Timing
Premium Leg	$s \times N \times \Delta$ per period	Until default or maturity
Accrued Premium	Prorated premium to default date	At default
Protection Leg	$(1-R) \times N$	At default

CDS Pricing Formulas

Premium Leg (PV of Spread Payments)

$$PV_{Premium} = s \times N \times \sum_{i=1}^{n} \Delta_i \cdot D(0,T_i) \cdot Q(0,T_i)$$

where $\Delta_i$ is the day count fraction for period $i$.

Protection Leg (PV of Contingent Payment)

$$PV_{Protection} = (1-R) \times N \times \sum_{i=1}^{n} D(0,T_i) \cdot [Q(0,T_{i-1}) - Q(0,T_i)]$$

Continuous approximation:

$$PV_{Protection} = (1-R) \times N \times \int_0^T D(0,t) \cdot Q(0,t) \cdot \lambda(t) dt$$

Risky PV01 (RPV01 or Risky Duration)

$$RPV01 = \sum_{i=1}^{n} \Delta_i \cdot D(0,T_i) \cdot Q(0,T_i)$$

This is the present value of 1 bp paid until default or maturity.

Par CDS Spread

At inception, CDS has zero value: $PV_{Premium} = PV_{Protection}$

Par Spread: $$s_{par} = \frac{(1-R) \times \sum_{i=1}^{n} D(0,T_i) \cdot [Q(0,T_{i-1}) - Q(0,T_i)]}{\sum_{i=1}^{n} \Delta_i \cdot D(0,T_i) \cdot Q(0,T_i)}$$

Simplified (constant $\lambda$, continuous):

$$s_{par} \approx (1-R) \times \lambda$$

CDS Valuation Examples

Example 1 - Par CDS Spread Calculation
Given: 3-year CDS, Notional = $10M, $r = 3\%$, $\lambda = 2\%$, $R = 40\%$, Quarterly payments

For simplicity, use annual payments:

Discount factors: $D_1 = 0.9704$, $D_2 = 0.9418$, $D_3 = 0.9139$
Survival probs: $Q_1 = 0.9802$, $Q_2 = 0.9608$, $Q_3 = 0.9418$

RPV01 = $1 \times D_1 Q_1 + 1 \times D_2 Q_2 + 1 \times D_3 Q_3$
= $0.9704 \times 0.9802 + 0.9418 \times 0.9608 + 0.9139 \times 0.9418$
= $0.9512 + 0.9049 + 0.8607 = 2.7168$

Protection PV per $1:
Year 1: $D_1 \times [Q_0 - Q_1] = 0.9704 \times [1 - 0.9802] = 0.9704 \times 0.0198 = 0.01921$
Year 2: $D_2 \times [Q_1 - Q_2] = 0.9418 \times [0.9802 - 0.9608] = 0.9418 \times 0.0194 = 0.01827$
Year 3: $D_3 \times [Q_2 - Q_3] = 0.9139 \times [0.9608 - 0.9418] = 0.9139 \times 0.0190 = 0.01736$
Total = 0.05484

Par spread: $s = \frac{(1-0.40) \times 0.05484}{2.7168} = \frac{0.03290}{2.7168} = 0.01211 = \boxed{121 \text{ bps}}$

Check: $s \approx (1-R)\lambda = 0.60 \times 0.02 = 1.20\%$ ✓

Further Reading: Wikipedia: Credit Default Swap | Investopedia: CDS Explained | CME: Introduction to CDS

Example 2 - CDS Mark-to-Market
Existing CDS: 5Y CDS bought 2 years ago at $s_0 = 150$ bps
Current: 3Y CDS par spread = 200 bps, $r = 4\%$, $\lambda = 3.33\%$, $R = 40\%$
Notional = $10M

CDS MTM = (Current spread - Contract spread) × RPV01 × Notional

Calculate RPV01 (3Y remaining, annual):
$D_1 = 0.9608$, $D_2 = 0.9231$, $D_3 = 0.8869$
$Q_1 = e^{-0.0333} = 0.9672$, $Q_2 = 0.9357$, $Q_3 = 0.9053$
RPV01 = $0.9608 \times 0.9672 + 0.9231 \times 0.9357 + 0.8869 \times 0.9053$
= $0.9293 + 0.8638 + 0.8029 = 2.596$

MTM (protection buyer):
MTM = (0.0200 - 0.0150) × 2.596 × $10M
MTM = 0.0050 × 2.596 × $10M = \boxed{\$129,800} gain

Bootstrapping Hazard Rates from CDS

Procedure

Start with shortest maturity CDS spread
Assume constant hazard rate for first period
Solve for $\lambda_1$ such that par spread is matched
Move to next maturity, solve for $\lambda_2$, etc.

Example - Bootstrap from CDS Spreads
Given: 1Y CDS = 100 bps, 2Y CDS = 120 bps, 3Y CDS = 150 bps
$r = 5\%$, $R = 40\%$, annual payments

Step 1: Solve for $\lambda_1$
Using $s_1 \approx (1-R)\lambda_1$:
$\lambda_1 = 0.01/0.60 = 1.67\%$
$Q_1 = e^{-0.0167} = 0.9834$

Step 2: Solve for $\lambda_2$
For 2Y CDS with spread 120 bps:
Premium PV = $0.0120 \times [D_1 Q_1 + D_2 Q_2]$
Protection PV = $0.60 \times [D_1(1-Q_1) + D_2(Q_1-Q_2)]$
With $Q_2 = Q_1 \times e^{-\lambda_2}$, solve iteratively...
Result: $\lambda_2 \approx 2.0\%$, $Q_2 = 0.9834 \times e^{-0.020} = 0.9640$

Step 3: Solve for $\lambda_3$
Similar procedure gives $\lambda_3 \approx 3.0\%$

CDS Greeks and Sensitivities

Credit DV01 (CS01)

Change in CDS value for 1 bp change in spread:

$$CS01 = \frac{\partial V_{CDS}}{\partial s} \approx RPV01 \times N$$

Example - CS01
Long protection on $10M notional, RPV01 = 4.2 years
CS01 = 4.2 × $10M × 0.0001 = \boxed{\$4,200} per bp

IR01

Sensitivity to 1 bp parallel shift in rates:

$$IR01 = \frac{\partial V_{CDS}}{\partial r} \times 0.0001$$

Recovery Sensitivity

$$\frac{\partial V_{CDS}}{\partial R} \approx -\frac{PV_{Protection}}{1-R}$$

Upfront vs Running CDS

Standard CDS Conventions

Post-2009 "Big Bang": CDS trade with standard coupons (100 or 500 bps) plus upfront:

$$\text{Upfront} = (s_{par} - s_{coupon}) \times RPV01 \times N$$

Example - Upfront Calculation
Par spread = 350 bps, Standard coupon = 500 bps, RPV01 = 4.5, Notional = $10M

Protection buyer receives upfront:
Upfront = (0.0350 - 0.0500) × 4.5 × $10M = -0.0150 × 4.5 × $10M
= \boxed{-\$675,000} (i.e., pays $675,000 to seller)

Since par spread < coupon, buyer "overpays" premium so compensated upfront

CDS vs Bond Basis

CDS-Bond Basis

$$\text{Basis} = \text{CDS Spread} - \text{Bond Z-Spread}$$

Positive basis: CDS spread > Bond spread (basis trade: short CDS, long bond)

Negative basis: CDS spread < Bond spread (basis trade: long CDS, short bond)

Drivers of Basis

Funding costs (repo, liquidity)
Counterparty risk in CDS
CTD option in physical settlement
Delivery squeeze risk

Further Reading: Risk.net: CDS-Bond Basis | Investopedia: Basis Trading

Quick Reference Formulas

Formula	Expression
Survival Prob (const λ)	$Q(T) = e^{-\lambda T}$
Hazard from Spread	$\lambda = s/(1-R)$
Par CDS Spread (approx)	$s \approx (1-R)\lambda$
Risky Zero (R=0)	$P = e^{-(r+\lambda)T}$
Risky Zero (R>0)	$P = e^{-rT}[R + (1-R)e^{-\lambda T}]$
CDS MTM	$(s_{mkt} - s_0) \times RPV01 \times N$
Upfront	$(s_{par} - s_{cpn}) \times RPV01 \times N$

Interview Tips - Credit

$s \approx (1-R)\lambda$ is the most important approximation
RPV01 ≈ sum of survival-weighted discount factors
Protection buyer benefits from spread widening
CDS vs bond basis driven by funding and liquidity
Upfront = (par spread - coupon) × RPV01 × Notional
CS01 ≈ RPV01 × Notional × 1 bp

Common Mistakes

Forgetting recovery rate in hazard rate conversion
Confusing survival probability with default probability
Wrong sign on upfront: buyer pays if par < coupon
Not discounting default payments to appropriate time
Mixing annualized spread with period spread

Model	Risky Bond Price
Constant \(\lambda\), \(R=0\)	\(P = e^{-(r+\lambda)T}\)
Constant \(\lambda\), \(R>0\)	\(P = e^{-rT}[e^{-\lambda T} + R(1-e^{-\lambda T})]\)
Time-varying \(\lambda(t)\)	\(P = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\mathbf{1}_{\tau>T} + R\mathbf{1}_{\tau\leq T}]\)

Concept	Formula / Definition
Default Probability	\(PD = \mathbb{P}(\tau \leq T)\)
Survival Probability	\(Q(t,T) = \mathbb{P}(\tau > T \| \tau > t)\)
Hazard Rate	\(\lambda(t) = -\frac{d\ln Q(0,t)}{dt}\)
Recovery Rate	\(R\) = fraction recovered at default
Loss Given Default	\(LGD = 1 - R\)
Credit Spread	\(s = (1-R)\lambda\) (approx)
Expected Loss	\(EL = PD \times LGD \times EAD\)

Leg	Payment	Timing
Premium Leg	\(s \times N \times \Delta\) per period	Until default or maturity
Accrued Premium	Prorated premium to default date	At default
Protection Leg	\((1-R) \times N\)	At default

Formula	Expression
Survival Prob (const λ)	\(Q(T) = e^{-\lambda T}\)
Hazard from Spread	\(\lambda = s/(1-R)\)
Par CDS Spread (approx)	\(s \approx (1-R)\lambda\)
Risky Zero (R=0)	\(P = e^{-(r+\lambda)T}\)
Risky Zero (R>0)	\(P = e^{-rT}[R + (1-R)e^{-\lambda T}]\)
CDS MTM	\((s_{mkt} - s_0) \times RPV01 \times N\)
Upfront	\((s_{par} - s_{cpn}) \times RPV01 \times N\)