Master Formula Sheet
Every formula the curriculum adds, organized by phase and rendered locally. Rebuild this from memory weekly (Phase 0.5) - the gaps are your study list. Printable: use your browser’s print.
Phase 0 - Learning How to Learn Quant Material
Working-memory model
\[\text{free slots} = \max\!\big(0,\; 4 - \lceil \text{substeps}/\text{chunk size}\rceil\big)\]
A toy heuristic: bigger chunks leave more room for higher-order reasoning. §0.1
Forgetting curve
\[R(t)=e^{-t/S}\]
Retrievability decays with time; stability S rises after each spaced recall. §0.2
SM-2 interval update
\[I_{n+1}=I_n\cdot \mathrm{EF},\quad \mathrm{EF}\ge 1.3\]
Successful recalls expand the interval; a lapse resets to the minimum. §0.2
Difficulty target
\[\text{advance when success rate} \gtrsim 80\%\]
Keep tasks in the zone of proximal difficulty; too easy = no growth, too hard = no completion. §0.3
Evidence, not authority
\[\text{trust} \sim \text{(quality} \times \text{convergence of evidence)},\ \ \text{not prestige}\]
A heuristic: weight replicated, converging, high-quality evidence; discount authority and single positive results. §0.4
Derivative (definition)
\[f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\]
Your first formula-sheet entry; make bidirectional cards for it. §0.5
Linearity of expectation
\[\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]\]
Always holds; contrast with variance, which needs the covariance term. §0.5
Phase 1 - Mathematical Language & Proof
De Morgan for sets
\[(A\cup B)^c=A^c\cap B^c,\quad (A\cap B)^c=A^c\cup B^c\]
Complement swaps union and intersection; the same law reappears for logic and for events. §1.1
Sum of squares
\[\sum_{i=1}^n i^2=\tfrac{n(n+1)(2n+1)}{6}\]
A standard closed form, proved by induction and reused in variance calculations. §1.4
Epsilon-N limit
\[a_n\to L\iff\forall\varepsilon\gt 0\,\exists N:\,n\ge N\Rightarrow|a_n-L|\lt \varepsilon\]
The template that becomes epsilon-delta continuity and stochastic convergence later. §1.5
Phase 2 - Single-Variable Calculus
Epsilon-delta limit
\[\lim_{x\to c}f(x)=L\iff\forall\varepsilon\gt 0\,\exists\delta\gt 0:\,0\lt |x-c|\lt \delta\Rightarrow|f(x)-L|\lt \varepsilon\]
The template that the derivative and continuity both specialize. §2.1
Continuous compounding
\[\frac{dA}{dt}=rA\ \Rightarrow\ A(t)=A_0e^{rt}\]
Exponential growth is the unique law whose rate is proportional to the level. §2.2
Chain rule
\[(f\circ g)'(x)=f'(g(x))\,g'(x)\]
The composition rule that becomes Itô’s lemma in stochastic calculus. §2.3
Taylor polynomial
\[P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k\]
The local polynomial model underlying linearization, duration/convexity, and Itô. §2.4
Duration-convexity
\[\frac{\Delta P}{P}\approx -D\,\Delta y+\tfrac12 C\,\Delta y^2\]
A second-order Taylor expansion of bond price in yield. §2.4
Profit optimum
\[\pi'(q)=R'(q)-C'(q)=0\ \Rightarrow\ R'(q)=C'(q)\]
Marginal revenue equals marginal cost - Fermat’s theorem in economics. §2.5
PV of continuous stream
\[\mathrm{PV}=\int_0^T c(t)e^{-rt}\,dt\]
Discount each instant’s cash flow and accumulate. §2.6
Geometric series
\[\sum_{n=0}^{\infty} ar^n=\frac{a}{1-r},\quad |r|\lt 1\]
The convergent infinite sum behind perpetuities and discounting. §2.7
Annuity present value
\[\mathrm{PV}=C\cdot\frac{1-(1+i)^{-n}}{i}\]
A finite geometric sum of discounted level payments. §2.7
Phase 3 - Multivariable Calculus
Dot product & angle
\[x\cdot y=\lVert x\rVert\,\lVert y\rVert\cos\theta\]
Length, angle, and orthogonality in one identity. §3.1
Gradient of a quadratic
\[\nabla\big(\tfrac12 x^\top A x+b^\top x\big)=Ax+b,\quad H=A\]
The workhorse identity for optimization. §3.2
Change of variables
\[\iint_R f\,dx\,dy=\iint_{T^{-1}(R)} (f\circ T)\,|\det J_T|\,du\,dv\]
Jacobian determinant = local area-scaling factor. §3.3
2-D second-order test
\[D=f_{xx}f_{yy}-f_{xy}^2:\ D\gt 0,f_{xx}\gt 0\Rightarrow\text{min};\ D\lt 0\Rightarrow\text{saddle}\]
Definiteness via a determinant shortcut. §3.4
Min-variance portfolio
\[w^*=\frac{\Sigma^{-1}\mathbf 1}{\mathbf 1^\top\Sigma^{-1}\mathbf 1}\]
Lagrange applied to a quadratic risk objective with a budget constraint. §3.5
Phase 4 - Linear Algebra
Independence via rank
\[v_1,\dots,v_k\ \text{independent}\iff \rank[v_1\,\cdots\,v_k]=k\]
Redundancy is exactly rank deficiency. §4.1
Trace & determinant
\[\tr A=\sum_i\lambda_i,\quad \det A=\prod_i\lambda_i\]
Two invariants that verify any eigenvalue computation. §4.3
Function of a matrix
\[f(A)=P\,f(D)\,P^{-1},\quad A^k=PD^kP^{-1}\]
Diagonalize once, then act on the diagonal. §4.4
Normal equations / projection
\[\hat x=(A^\top A)^{-1}A^\top b,\quad P=A(A^\top A)^{-1}A^\top\]
Least squares = orthogonal projection onto the column space. §4.5
Spectral decomposition
\[A=Q\Lambda Q^\top,\quad x^\top Ax=\sum_i\lambda_i y_i^2,\ y=Q^\top x\]
Symmetric = orthogonal stretch along principal axes. §4.6
SVD & PCA
\[A=U\Sigma V^\top,\quad \Sigma_{\text{cov}}=Q\Lambda Q^\top,\ \text{var frac}=\lambda_i/\textstyle\sum_j\lambda_j\]
Singular values, principal components, and explained variance. §4.7
Phase 7 - Probability Theory
Law / pushforward
\[\mu_X(B)=\Prob(X\in B),\quad F_X(x)=\Prob(X\le x)\]
The distribution is the measure X induces on the line. §7.1
Chebyshev
\[\Prob(|X-\mu|\ge k\sigma)\le 1/k^2\]
Distribution-free spread bound from the variance. §7.2
Jensen
\[\varphi\ \text{convex}\Rightarrow \varphi(\E X)\le \E[\varphi(X)]\]
Curvature makes averaging inputs an underestimate. §7.2
Bayes
\[\Prob(B_j\mid A)=\frac{\Prob(A\mid B_j)\Prob(B_j)}{\sum_i\Prob(A\mid B_i)\Prob(B_i)}\]
Invert the direction of conditioning. §7.3
Tower property
\[\mathcal{H}\subseteq\mathcal{G}\Rightarrow \E[\E[X\mid\mathcal{G}]\mid\mathcal{H}]=\E[X\mid\mathcal{H}]\]
Forecasts collapse to the coarser information. §7.3
Gaussian MGF
\[X\sim\Normal(\mu,\sigma^2)\Rightarrow M_X(t)=e^{\mu t+\sigma^2 t^2/2}\]
The quadratic exponent is the Gaussian signature. §7.4
Standard error of the mean
\[\mathrm{sd}(\bar X_n)=\sigma/\sqrt n\]
Quadruple the sample to halve the error. §7.5
CLT
\[\frac{\sum X_i-n\mu}{\sigma\sqrt n}\Rightarrow\Normal(0,1)\]
Universal Gaussian fluctuation at the \(\sqrt n\) scale. §7.6
Convergence hierarchy
\[\text{a.s.},\,L^p\ \Rightarrow\ \xrightarrow{p}\ \Rightarrow\ \text{in dist.}\]
Only these arrows hold in general. §7.6
Martingale property
\[\E[M_{n+1}\mid\F_n]=M_n\]
Fair game: today is the best forecast of tomorrow. §7.7
Optional stopping
\[\E[M_\tau]=\E[M_0]\ \ (\text{under regularity})\]
Non-anticipating strategies cannot beat a fair game. §7.7
Phase 8 - Stochastic Processes & Brownian Motion
Stationarity
\[\pi P=\pi,\ \textstyle\sum_i\pi_i=1\]
Long-run distribution as a left eigenvector. §8.1
Detailed balance
\[\pi_i p_{ij}=\pi_j p_{ji}\Rightarrow \pi\text{ stationary}\]
Reversibility, the basis of MCMC. §8.1
Diffusive scaling
\[\Var(S_n)=n,\ \mathrm{sd}(S_n)=\sqrt n\]
Distance grows like the square root of time. §8.2
Martingale transform
\[(H\cdot S)_n=\sum_{k\le n}H_k\,\xi_k,\ \E[(H\cdot S)_n]=0\]
Predictable strategies preserve fairness. §8.2
BM moments
\[\E[B_t]=0,\ \Var(B_t)=t,\ \Cov(B_s,B_t)=\min(s,t)\]
The defining second-order structure. §8.3
Increment law
\[B_t-B_s\sim\Normal(0,t-s),\ \text{independent of }\F_s\]
Independent stationary Gaussian increments. §8.3
Brownian QV
\[[B]_t=t,\qquad (dB_t)^2=dt\]
The surviving second-order term; the engine of Ito’s lemma. §8.4
Scaled QV
\[X_t=\sigma B_t\Rightarrow [X]_t=\sigma^2 t\]
Realized variance measures accumulated \(\sigma^2\). §8.4
Girsanov density
\[Z_t=\exp(-\theta B_t-\tfrac12\theta^2 t),\ \tilde B_t=B_t+\theta t\]
Exponential martingale that removes drift. §8.5
Phase 11 - Statistics & Statistical Learning
MSE decomposition
\[\text{MSE}(\hat\theta)=\text{bias}(\hat\theta)^2+\Var(\hat\theta)\]
The master identity behind every model-complexity tradeoff. §11.1
Prediction error
\[\E[(Y-\hat f)^2]=\sigma^2+\text{bias}^2+\Var(\hat f)\]
Adds the irreducible-noise floor to the estimator decomposition. §11.1
OLS estimator
\[\hat\beta=(X^\top X)^{-1}X^\top y\]
Solution of the normal equations; solve, don't invert, when collinear. §11.2
Coefficient covariance
\[\Var(\hat\beta)=\sigma^2(X^\top X)^{-1}\]
Basis of standard errors; explodes under collinearity. §11.2
Optimism
\[\E[\text{err}_{\text{test}}]-\E[\text{err}_{\text{train}}]=\tfrac{2}{n}\sigma^2\,\text{df}\]
Grows with effective degrees of freedom; the reason training error lies. §11.3
BIC
\[\text{BIC}=-2\ell+p\log n\]
Heavier penalty than AIC; consistent for the true model, favors parsimony. §11.3
Ridge estimator
\[\hat\beta_{\text{ridge}}=(X^\top X+\lambda I)^{-1}X^\top y\]
Closed form; exists even when \(X^\top X\) is singular. §11.4
Soft-thresholding
\[S_\lambda(z)=\text{sign}(z)(|z|-\lambda)_+\]
Lasso solution under orthonormal predictors; the source of sparsity. §11.4
Logit link
\[\log\frac{p}{1-p}=x^\top\beta,\quad p=\sigma(x^\top\beta)\]
Linear in log-odds; sigmoid maps back to a probability. §11.5
LDA discriminant
\[\delta_k(x)=x^\top\Sigma^{-1}\mu_k-\tfrac12\mu_k^\top\Sigma^{-1}\mu_k+\log\pi_k\]
Linear score; assign to the largest. §11.5
Bagged variance
\[\Var\approx\rho\sigma^2+\tfrac{1-\rho}{B}\sigma^2\]
Averaging cuts the \(1/B\) term; decorrelation (forests) cuts \(\rho\sigma^2\). §11.6
SVM margin
\[\min\tfrac12\|w\|^2\ \text{s.t.}\ y_i(w^\top x_i+b)\ge1\]
Margin width \(2/\|w\|\); soft version = hinge loss + \(\ell_2\). §11.6
PCA objective
\[w_1=\argmax_{\|w\|=1} w^\top\hat\Sigma w\]
Top eigenvector of the covariance; variance captured = eigenvalue. §11.7
Overfit Sharpe bound
\[\E[\max_i \widehat{SR}_i]\approx\sqrt{2\log N}/\sqrt{T}\]
Apparent best-of-\(N\) Sharpe from pure luck; deflate by \(N\). §11.7
Phase 12 - Time-Series Analysis
ACF
\[\rho(h)=\gamma(h)/\gamma(0),\quad \rho(0)=1\]
Normalized autocovariance; \(\pm1.96/\sqrt n\) white-noise bands. §12.1
ARMA lag form
\[\phi(L)X_t=\theta(L)\varepsilon_t\]
Compact model; roots of \(\phi\)/\(\theta\) give stationarity/invertibility. §12.2
Yule-Walker
\[\rho(k)=\textstyle\sum_{i}\phi_i\,\rho(k-i)\]
Solve \(R\phi=r\) for AR coefficients from the ACF. §12.2
ARIMA
\[\phi(L)(1-L)^d X_t=\theta(L)\varepsilon_t\]
ARMA on the \(d\)-th difference; \(d\) kills unit roots. §12.3
Forecast-error variance
\[\sigma_h^2=\sigma^2\textstyle\sum_{j=0}^{h-1}\psi_j^2\]
Widens with \(h\); the basis of prediction intervals. §12.3
GARCH(1,1)
\[\sigma_t^2=\omega+\alpha r_{t-1}^2+\beta\sigma_{t-1}^2\]
Persistent conditional variance; the volatility workhorse. §12.4
Unconditional variance
\[\bar\sigma^2=\omega/(1-\alpha-\beta)\]
Finite iff \(\alpha+\beta\lt 1\); persistence \(\alpha+\beta\) sets the half-life. §12.4
Kalman gain
\[K_t=P_{t\mid t-1}H^\top(HP_{t\mid t-1}H^\top+R)^{-1}\]
Precision weight on the innovation; balances model vs data. §12.5
Covariance recursion
\[P_{t\mid t-1}=FP_{t-1\mid t-1}F^\top+Q,\ \ P_{t\mid t}=(I-K_tH)P_{t\mid t-1}\]
Predict inflates, update deflates uncertainty. §12.5
Error-correction model
\[\Delta y_t=\alpha(y_{t-1}-\beta x_{t-1})+\dots\]
Changes respond to the lagged equilibrium error; \(\alpha\lt 0\) pulls back. §12.6
Overfit Sharpe
\[\E[\max_i \widehat{SR}_i]\approx\sqrt{2\log N}/\sqrt T\]
Best of \(N\) trials from luck alone; deflate by \(N\). §12.6
Phase 15 - Derivatives & Volatility
Black–Scholes PDE
\[\frac{\partial V}{\partial t}+\tfrac12\sigma^2S^2\frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0\]
Equivalently the gamma–theta identity that a delta-hedged book earns. §15.1
Hedged P&L
\[\tfrac12\,\Gamma\,S^2\big(\sigma_{\text{real}}^2-\sigma_{\text{imp}}^2\big)\,dt\]
A delta-hedged long option is a bet on realized minus implied variance. §15.1
Implied vol
\[C_{\text{BS}}(S,K,r,\sigma_{\text{imp}},T)=C_{\text{mkt}}(K,T)\]
Unique because vega \(\gt 0\); solved by Newton using the vega slope. §15.2
Breeden–Litzenberger
\[q(K)=e^{rT}\,\partial^2 C/\partial K^2\]
Smile curvature is the risk-neutral density; must stay non-negative (no butterfly arbitrage). §15.2
Dupire local volatility
\[\sigma_{\text{loc}}^2(K,T)=\frac{\partial_T C+rK\,\partial_K C}{\tfrac12 K^2\,\partial_{KK}C}\]
Reads local vol straight off the observed call surface; positivity encodes no calendar/butterfly arbitrage. §15.3
Heston dynamics
\[dS=rS\,dt+\sqrt v\,S\,dW^S,\quad dv=\kappa(\theta-v)dt+\xi\sqrt v\,dW^v,\quad d\langle W^S,W^v\rangle=\rho\,dt\]
Mean-reverting CIR variance correlated with the spot; \(\rho\) sets skew, \(\xi\) convexity. §15.4
Total variance & calendar rule
\[w(k,T)=\sigma_{\text{imp}}^2 T,\qquad \partial_T w(k,T)\ge 0\]
No calendar arbitrage: variance grows with maturity. §15.5
Butterfly (discrete)
\[C(K-\Delta)-2C(K)+C(K+\Delta)\ge 0\]
Non-negative risk-neutral density: no butterfly arbitrage. §15.5
Phase 16 - Algorithmic & High-Frequency Trading
Book observables
\[\text{mid}=\tfrac12(P_a+P_b),\ \ \text{micro}=\frac{P_aQ_b+P_bQ_a}{Q_a+Q_b}\]
The micro-price leans toward the heavier-depth side and better predicts the next mid. §16.1
Average execution price
\[\bar P=\tfrac1Q\sum_i P_i\min(q_i,\,Q-\textstyle\sum_{j\lt i}q_j)\]
Walking the book; slippage \(=\bar P-\text{mid}\). §16.1
Flow price impact
\[m_t=m_{t-1}+\lambda\,x_t+\varepsilon_t\]
Kyle’s lambda: permanent mid impact per unit of signed order flow. §16.2
Roll’s spread
\[s=2\sqrt{-\operatorname{Cov}(\Delta p_t,\Delta p_{t-1})}\]
Recovers the spread from bid–ask bounce (valid only when the covariance is negative). §16.2
Linear impact model
\[\tilde P_t=P_0+\gamma X_t+\eta v_t\]
Permanent \(\gamma X_t\) (persists, size-driven) plus temporary \(\eta v_t\) (decays, rate-driven). §16.3
Square-root impact law
\[\text{impact}\approx Y\,\sigma\sqrt{Q/V}\]
Empirical, sublinear in size; used to calibrate expected cost. §16.3
Mean–variance execution cost
\[\E[\mathcal{C}]+\lambda\Var[\mathcal{C}]=\big(\tfrac12\gamma X^2+\sum_k\tfrac{\eta}{\tau}n_k^2\big)+\lambda\sigma^2\sum_k\tau x_k^2\]
Impact (trade fast is costly) versus timing risk (trade slow is risky). §16.4
Optimal exponential trajectory
\[x_k=X\,\frac{\sinh(\kappa(T-t_k))}{\sinh(\kappa T)},\ \ \kappa=\sqrt{\lambda\sigma^2/\eta}\]
Urgency rises with \(\lambda,\sigma\), falls with \(\eta\); \(\lambda\to0\) gives TWAP. §16.4
Reservation price
\[r(s,q,t)=s-q\,\gamma\,\sigma^2\,(T-t)\]
Inventory-adjusted fair value; long inventory pulls quotes down to sell it off. §16.5
Optimal spread
\[\delta^a+\delta^b=\gamma\sigma^2(T-t)+\tfrac{2}{\gamma}\ln\!\big(1+\tfrac{\gamma}{k}\big)\]
Inventory/vol risk premium plus an order-flow-shape term. §16.5
Net of costs
\[r_{\text{net}}\approx r_{\text{gross}}-\text{turnover}\times\text{cost per unit}\]
High-turnover strategies are fragile; frictions routinely flip a backtest negative. §16.6
Data-snooping benchmark
\[\E\big[\max_i \widehat{SR}_i\big]\approx\sqrt{2\ln n/T}\]
The in-sample Sharpe a real strategy must beat after \(n\) trials over \(T\) observations. §16.6