Reference · Look-ups · Chapter R2

The maths on one page

Every formula, with where it is derived

This page collects the formulas the book relies on, grouped by part. Each one links to the numbered display where the book derives it or first works it with numbers, so you can go back to the reasoning when the formula alone is not enough.

Use it as a reminder, not a first introduction. A formula on its own hides its assumptions: the square-root-of-time rule needs independent returns, and Black-Scholes needs a volatility that does not change. The chapter behind each link states what the formula assumes and where it breaks.

1 · Symbols

The book uses the same symbol for the same thing everywhere, with a few exceptions. A handful of letters, $G$ , $L$ and $D$ among them, mean different things in different chapters, so each table below says what they mean there.

Symbol	Meaning
$S$ , $S_t$	the price of the underlying, at time $t$
$K$	an option's strike price
$T$	time to expiry in years; 30 days is $30/365$
$R_t$ , $r_t$	the simple return and the log return over period $t$
$r$ , $r_f$	the risk-free interest rate
$q$	the dividend yield
$\mu$ , $\sigma$	the mean return and the volatility, annualized unless a subscript says otherwise
$\sigma_I$ , $\sigma_R$	implied and realized volatility
$p$	a probability, often a win rate
$\mathbb{E}[X]$ , $\operatorname{Var}(X)$	expected value and variance
$N(x)$ , $\varphi(x)$	the standard normal distribution function and its density
$\Delta$ , $\Gamma$ , $\Theta$ , $\nu$	delta, gamma, theta and vega
$\rho$	correlation; the Greek rho is written as a word
$\text{SR}$	the Sharpe ratio

2 · Returns

From Part I. $S_t$ is the price at the end of period $t$ , and $G$ is the compound growth rate.

Quantity	Formula	Where
Simple return	$R_t = S_t/S_{t-1} - 1$	(1A.1)
Compounding	$1 + R_{\text{total}} = \prod_{t=1}^{n} (1 + R_t)$	(1A.2)
Log return	$r_t = \ln(S_t/S_{t-1})$	(1A.10)
From one to the other	$r_t = \ln(1 + R_t)$ and $R_t = e^{r_t} - 1$	(1A.11)
Log returns add	$r_{\text{total}} = \ln(S_n/S_0) = \sum_{t=1}^{n} r_t$	(1A.14)
Square-root-of-time rule	$\sigma_{n\text{-day}} = \sigma_{\text{daily}}\sqrt{n}$	(1A.16)
Annualized volatility	$\sigma_{\text{annual}} = \sigma_{\text{daily}}\sqrt{252}$	(1A.17)
Growth rate after variance drag	$G = \mu - \sigma^2/2$	(1A.19)

3 · Distributions and edge

From Chapters 1B and 1C. A bet wins $G$ with probability $p$ , loses $L$ otherwise, and costs $c$ .

Quantity	Formula	Where
Expected value	$\mathbb{E}[X] = \sum_i p_i\,x_i$	(1B.1)
Skewness	$\gamma_3 = \mathbb{E}\big[((R - \mu)/\sigma)^3\big]$	(1B.2)
Kurtosis	$\gamma_4 = \mathbb{E}\big[((R - \mu)/\sigma)^4\big]$	(1B.3)
Skewness of a two-outcome bet	$\gamma_3 = (1 - 2p)/\sqrt{p(1-p)}$	(1B.6)
Expected shortfall	$\text{ES}_{95} = \mathbb{E}[R \mid R \le \text{VaR}_{95}]$	(1B.8)
A bet's expected value	$\mathbb{E}[X] = pG - (1-p)L - c$	(1C.2)
Breakeven win rate	$p^{*} = (L + c)/(G + L)$	(1C.3)
Log utility	$U(W) = \ln W$	(1C.7)

4 · Risk-adjusted return

From Chapter 1D. $\widehat{\text{SR}}$ is a Sharpe ratio measured on $n$ returns, and $K$ is the number of strategies tried.

Quantity	Formula	Where
Sharpe ratio	$\text{SR} = (\mathbb{E}[R] - r_f)/\sigma_R$	(1D.1)
Annualized Sharpe ratio	$\text{SR}_{\text{annual}} = \text{SR}_{\text{daily}}\sqrt{252}$	(1D.3)
Sharpe ratio over $N$ periods	$\text{SR}_N \approx \text{SR}_1\sqrt{N}$	(1D.4)
Chance of ever falling to a fraction $f$ of wealth	$\Pr = f^{\,2\mu/\sigma^2 - 1}$	(1D.5)
Sortino ratio	$(\mathbb{E}[R] - r_f)/\sigma_{\text{down}}$	(1D.7)
Information ratio	$\mathbb{E}[R_s - R_b]/\sigma(R_s - R_b)$	(1D.9)
Treynor ratio	$(\mathbb{E}[R] - r_f)/\beta$	(1D.10)
Calmar ratio	$\mathbb{E}[R_{\text{annual}}]/\lvert\text{MaxDD}\rvert$	(1D.11)
Tail beta	$\operatorname{Cov}(R_s, R_m \mid \text{tail})/\operatorname{Var}(R_m \mid \text{tail})$	(1D.12)
Standard error of a Sharpe ratio	$s = \sqrt{\big(1 - \gamma_3\widehat{\text{SR}} + \tfrac{\gamma_4 - 1}{4}\widehat{\text{SR}}^2\big)/(n - 1)}$	(1D.13)
Probabilistic Sharpe ratio	$\text{PSR} = N\big((\widehat{\text{SR}} - \text{SR}^*)/s\big)$	(1D.14)
The best of $K$ tries by luck alone	$\text{SR}_0 = \sqrt{V}\,\big[(1 - \gamma_E)N^{-1}(1 - \tfrac1K) + \gamma_E N^{-1}(1 - \tfrac{1}{Ke})\big]$	(1D.15)

5 · Drawdown, leverage and portfolios

From Chapters 1E to 1G. $W_t$ is wealth, $P_t$ its running peak, $D$ a drawdown, and $k$ the leverage.

Quantity	Formula	Where
Drawdown	$D_t = 1 - W_t/P_t$	(1E.1)
Gain needed to recover	$G = D/(1 - D)$	(1E.2)
Years to recover at growth $g$	$t = -\ln(1 - D)/\ln(1 + g)$	(1E.3)
Leverage	$k = \text{exposure}/\text{capital}$	(1E.4)
Levered return, borrowing at $r_b$	$R_k = kR - (k - 1)\,r_b$	(1E.5)
Correlation	$\rho = \operatorname{Cov}(X, Y)/(\sigma_X\sigma_Y)$	(1F.1)
Variance of two assets	$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2$	(1F.2)
Variance of many assets	$\sigma_p^2 = \mathbf{w}^\top\boldsymbol{\Sigma}\,\mathbf{w}$	(1F.6)
Two equal strategies combined	$\text{SR}_p = \text{SR}\sqrt{2/(1 + \rho)}$	(1F.7)
A risk premium	$\mathbb{E}[R_i] - r = \beta_i\,\pi$	(1G.2)

6 · Markets, costs and taxes

From Part II. $d$ is +1 for a buy and −1 for a sell, and $N$ is one trade's notional.

Quantity	Formula	Where
The index level	$\text{SPX}_t = \sum_i P_{i,t}F_{i,t}/D_t$	(2A.1)
The index return	$R_{\text{index}} = \sum_i w_i R_i$	(2A.2)
Divisor after a change	$D_{\text{new}} = D_{\text{old}}\,V_{\text{after}}/V_{\text{before}}$	(2A.5)
Notional of a contract	$\text{notional} = m \times L$	(2A.6)
Fair value of an index future	$F = S\,e^{(r - q)T}$	(2A.7)
Effective spread	$2d\,(P - M_0)$	(2B.2)
Realized spread	$2d\,(P - M_1)$	(2B.3)
Spread and midpoint	$s = a - b$ , $m = (a + b)/2$	(2C.1)
Break-even half-spread	$h = \pi\,\delta$	(2C.3)
Trade only if	$\mathbb{E}[\text{gain}] - c_{\text{RT}} > 0$	(2D.1)
Exit-cost ratio	$(s/2)\,nM/\mathbb{E}[\text{profit}]$	(2D.3)
A year's friction drag	$D = n\,(s + 2i + 2k/N)$	(2D.4)
Square-root impact	$\text{slippage} \approx k\sqrt{Q/\text{ADV}}$	(2D.5)
Fall that triggers a Reg-T call	$V/V_0 = (1 - i)/(1 - m)$	(2E.1)
Section 1256 blended rate	$t_{1256} = 0.6\,t_{\text{LT}} + 0.4\,t_{\text{ST}}$	(2E.2)
Growth after tax	$W_n = W_0\,\big(1 + R(1 - t)\big)^n$	(2E.4)

7 · Models and sizing

From Part III. In the Kelly formulas a bet wins $G$ or loses $L$ per dollar staked, and $c$ is the multiple of full Kelly.

Quantity	Formula	Where
Error decomposition	$\text{MSE} = \text{bias}^2 + \text{variance}$	(3A.2)
AR(1) process	$r_t = w\,r_{t-1} + b + \varepsilon_t$	(3B.2)
Its long-run variance	$V = \sigma^2/(1 - w^2)$	(3B.3)
Variance ratio	$\text{VR}(k) = \operatorname{Var}(r_t + \cdots + r_{t+k-1})/(k\operatorname{Var}(r_t))$	(3B.9)
Least squares	$\hat\beta = (X^\top X)^{-1}X^\top y$	(3C.4)
Gradient descent step	$w_{i+1} = w_i - \eta\,\partial L/\partial w$	(3C.8)
Ridge and lasso	$L_{\text{data}} + \lambda\sum_i w_i^2$ and $L_{\text{data}} + \lambda\sum_i \lvert w_i\rvert$	(3C.11)
Growth of a bet sized $f$	$g(f) = p\ln(1 + fG) + (1 - p)\ln(1 - fL)$	(3D.1)
Kelly fraction	$f^* = p/L - (1 - p)/G$	(3D.2)
Kelly for a continuous return	$f^* = \mu/\sigma^2$	(3D.5)
Chance of ever halving at $c$ times Kelly	$0.5^{\,2/c - 1}$	(3D.6)
Sharpe ratio of $N$ independent bets	$\text{SR}_{\text{total}} \approx \text{SR}_{\text{per bet}}\sqrt{N}$	(3D.8)

8 · Futures and options

From Parts IV and V. $m$ is the contract multiplier, 100 for SPX options.

Quantity	Formula	Where
Long future at expiry	$S_T - K$	(4B.1)
Cost of carry	$F_0 = S_0\,e^{(r - q)T}$	(4B.2)
Call and put payoffs	$\max(S_T - K, 0)$ and $\max(K - S_T, 0)$	(4B.4)
Jensen's inequality, $f$ convex	$\mathbb{E}[f(S_T)] \ge f(\mathbb{E}[S_T])$	(4B.5)
Dollars paid for $n$ contracts	$\text{quote} \times m \times n$	(5A.3)
An option's premium	$\text{intrinsic} + \text{extrinsic}$	(5B.1)
Put-call parity	$C - P = S e^{-qT} - K e^{-rT}$	(5F.2)

9 · Black-Scholes and the Greeks

From Chapters 5C and 5D. With no dividends, set $q = 0$ .

Quantity	Formula	Where
$d_1$	$\big(\ln(S/K) + (r - q + \tfrac12\sigma^2)T\big)/(\sigma\sqrt{T})$	(5C.3)
$d_2$	$d_1 - \sigma\sqrt{T}$	(5C.4)
Call price	$S e^{-qT}N(d_1) - K e^{-rT}N(d_2)$	(5C.5)
Put price	$K e^{-rT}N(-d_2) - S e^{-qT}N(-d_1)$	(5C.6)
A small move, to second order	$\Delta V \approx \Delta\cdot\Delta S + \tfrac12\Gamma(\Delta S)^2$	(5C.2)
Delta	$e^{-qT}N(d_1)$ for a call, $-e^{-qT}(1 - N(d_1))$ for a put	(5C.8)
Gamma	$e^{-qT}\varphi(d_1)/(S\sigma\sqrt{T})$	(5C.10)
Theta of a call, $q = 0$	$-S\varphi(d_1)\sigma/(2\sqrt{T}) - rKe^{-rT}N(d_2)$	(5C.12)
Theta of a call against its gamma, $q = 0$	$\Theta = -\tfrac12\sigma^2S^2\Gamma - rKe^{-rT}N(d_2)$	(5C.13)
Vega	$S e^{-qT}\varphi(d_1)\sqrt{T}$	(5C.15)
Rho of a call	$KTe^{-rT}N(d_2)$	(5C.17)
Vanna	$-e^{-qT}\varphi(d_1)\,d_2/\sigma$	(5D.4)
Charm, $q = 0$	$-\varphi(d_1)\,(2rT - d_2\sigma\sqrt{T})/(2T\sigma\sqrt{T})$	(5D.9)
Volga	$\nu\,d_1d_2/\sigma$	(5D.14)
Speed	$-(\Gamma/S)\big(d_1/(\sigma\sqrt{T}) + 1\big)$	(5D.17)
Color, with $k = (2rT - d_2\sigma\sqrt{T})/(\sigma\sqrt{T})$	$(\Gamma/2T)(1 + d_1k)$	(5D.19)

10 · Volatility

From Chapters 5E, 5G, 9A and 9B. $O_i$ , $H_i$ , $L_i$ and $C_i$ are day $i$ 's open, high, low and close.

Quantity	Formula	Where
Rule of 16	$\text{daily move} \approx \text{annual IV}/16$	(5E.3)
Close-to-close variance	$\tfrac1N\sum_i [\ln(C_i/C_{i-1})]^2$	(5E.4)
Parkinson variance	$\tfrac{1}{4N\ln 2}\sum_i [\ln(H_i/L_i)]^2$	(5E.5)
Garman-Klass variance	$\tfrac1N\sum_i \big[\tfrac12(\ln\tfrac{H_i}{L_i})^2 - (2\ln 2 - 1)(\ln\tfrac{C_i}{O_i})^2\big]$	(5E.6)
An at-the-money call	$C \approx 0.4\,S\sigma\sqrt{T}$	(5E.7)
IV rank	$(\text{IV}_{\text{now}} - \text{IV}_{\text{low}})/(\text{IV}_{\text{high}} - \text{IV}_{\text{low}})$	(5G.1)
Forward volatility	$\sigma_f = \sqrt{(\sigma_2^2T_2 - \sigma_1^2T_1)/(T_2 - T_1)}$	(5G.3)
Skew	$\text{IV}_{25\Delta\,\text{put}} - \text{IV}_{25\Delta\,\text{call}}$	(5G.4)
The VIX, as a variance	$\sigma^2 = \tfrac2T\sum_i \tfrac{\Delta K_i}{K_i^2}e^{rT}Q(K_i) - \tfrac1T\big(\tfrac{F}{K_0} - 1\big)^2$	(9A.1)
Gamma at two expiries	$\Gamma(T_1)/\Gamma(T_2) \approx \sqrt{T_2/T_1}$	(9B.1)

11 · Hedging

From Parts VI and VII. $n$ is the number of contracts, $\$\Gamma_{1\%}$ is dollar gamma per 1% move, and $\Pi$ is the value of the hedged book.

Quantity	Formula	Where
Shares to hedge	$n \times m \times \Delta$	(6C.1)
Variance of one step's hedging error	$\tfrac12\Gamma^2S^4\sigma^4(\Delta t)^2$	(6C.2)
A short-gamma dealer's P&L	$d\Pi \approx \tfrac12\Gamma S^2\big[\sigma^2dt - (dS/S)^2\big]$	(6D.2)
Dollar gamma per 1% move	$\$\Gamma_{1\%} = \Gamma \times S^2 \times 0.01$	(6D.3)
Gain or loss on a $p$ % move	$\tfrac12 \times \$\Gamma_{1\%} \times p^2/100$	(6D.4)
Spread cost against gamma gain	$2s/h$	(6E.2)
Geometric Brownian motion	$dS = \mu S\,dt + \sigma S\,dW$	(7A.2)
Itô's lemma in Greeks	$dV = \Theta\,dt + \Delta\,dS + \tfrac12\Gamma\sigma^2S^2\,dt$	(7A.6)
The log price	$d\ln S = (\mu - \tfrac12\sigma^2)\,dt + \sigma\,dW$	(7A.7)
The Black-Scholes-Merton equation	$\tfrac{\partial V}{\partial t} + rS\tfrac{\partial V}{\partial S} + \tfrac12\sigma^2S^2\tfrac{\partial^2 V}{\partial S^2} = rV$	(7A.13)
A delta-hedged book's P&L	$d\Pi - r\Pi\,dt \approx \tfrac12\Gamma S^2\big[(dS/S)^2 - \sigma_I^2\,dt\big]$	(7A.15)
How delta changes	$d\Delta \approx \Gamma\,dS + \text{vanna}\,d\sigma + \text{charm}\,dt$	(7B.1)

12 · Gamma exposure

From Part VIII. $s_i$ is +1 where dealers are long a line and −1 where they are short, which is an assumption (Chapter 8B). $G$ is the negative of published GEX, and $\kappa$ measures how far prices move per unit of hedging flow (Chapter 8D). $\lambda$ is the index points that $1 billion of hedging moves the index, with $G$ then in billions of dollars per 1% move. At one strike, $w$ is the damping of the dealers' gamma and $c$ the drift of their charm (Chapter 8E). $\mathbb{E}\lvert r\rvert$ is the average size of a day's return (Chapter 8G).

Quantity	Formula	Where
One line's dollar gamma	$\Gamma_{\$,i} = s_i\Gamma_iS^2MN_i \times 0.01$	(8A.1)
Gamma exposure	$\text{GEX} = \sum_i \Gamma_{\$,i}$	(8A.2)
Dealers' hedge flow	$\Delta H \approx -\text{GEX} \times (\Delta S/S)/0.01$	(8A.3)
The flow in ES contracts	$n_{\text{ES}} = \Delta H/(50\,S)$	(8A.4)
Volatility with feedback	$\sigma_{\text{eff}}^2 = \sigma^2/(1 - \kappa G)^2$	(8D.1)
The loop gain in dollars	$\kappa G = \lambda G/(0.01\,S)$	(8D.5)
A move at a pinning strike	$dS = (\text{news} - c)/(1 + w)$	(8E.3)
Vanna exposure	$\text{VEX} = \sum_i s_i\,\text{vanna}_i \times 0.01\,SMN_i$	(8F.1)
Charm exposure, per day	$\text{CHEX} = \sum_i s_i\,(\text{charm}_i/365)\,SMN_i$	(8F.2)
Charm from vanna	$\text{charm} = -\text{vanna} \times \sigma/(2T)$	(8F.3)
Daily moves to annual volatility	$\sigma = \sqrt{252}\sqrt{\pi/2}\;\mathbb{E}\lvert r\rvert \approx 19.9\,\mathbb{E}\lvert r\rvert$	(8G.1)
Directional persistence	$P = \text{MA}_{21}(r)/\text{MA}_{21}(\lvert r\rvert)$	(8I.1)

13 · Edges

From Part X. $\rho$ is the correlation between strategies, and $Y$ is a number of years.

Quantity	Formula	Where
Extra return from a higher Sharpe ratio	$\Delta\mu = \Delta\text{SR} \times \sigma_p$	(10A.1)
$N$ strategies with correlation $\rho$	$\text{SR}_N = \text{SR}\sqrt{N/(1 + (N-1)\rho)}$	(10A.3)
The equity risk premium	$\mathbb{E}[R_{\text{eq}}] - r = \beta_{\text{eq}}\,\pi_{\text{cons}} + \text{behavioral premia}$	(10B.1)
Expected return implied by dividends	$k = D_1/P + g$ and $\text{ERP} = k - r$	(10B.4)
The variance risk premium	$\text{VRP}_t = \sigma_{I,t}^2 - \mathbb{E}_t[\sigma_{R,t,T}^2]$	(10C.1)
Implied minus realized volatility	$\sigma_{I,t} - \sigma_{R,\,t-21,\,t}$	(10H.3)
Time-series momentum weight	$w_{i,t} = \operatorname{sign}(R_{i,t-12,t})/\sigma_{i,t}$	(10D.1)
When adding $B$ to $P$ raises the Sharpe ratio	$\text{SR}_B > \rho\,\text{SR}_P$	(10D.2)
Carry	$C_{i,t} = (S_{i,t} - F_{i,t})/F_{i,t}$	(10E.1)
A long-short factor	$F_t = R^{\text{top}}_t - R^{\text{bottom}}_t$	(10F.1)
t-statistic of a Sharpe ratio	$t = \text{SR}\sqrt{Y}$	(10F.2)
The capital asset pricing model	$\mathbb{E}[R_i] - r = \beta_i\,(\mathbb{E}[R_m] - r)$	(10F.6)
Betting against beta	$\text{BAB}_t = \tfrac{1}{\beta_L}(R_{L,t} - r) - \tfrac{1}{\beta_H}(R_{H,t} - r)$	(10F.7)
What an overlay earns	$\mathbb{E}[k_tR_t] = \mathbb{E}[k_t]\,\mathbb{E}[R_t] + \operatorname{Cov}(k_t, R_t)$	(10G.1)
A rebalancing imbalance	$\text{Imbalance}_t = w_t - w^*$	(10G.3)
A regime filter scales a position	$Q_t = m(x_t)\,Q$	(10H.1)
The VIX term-structure ratio	$F_t = \text{VIX}_t/\text{VIX3M}_t$	(10H.2)
Compound annual growth rate	$c = (V_T/V_0)^{1/T} - 1$	(10I.2)

The Edge Book's own summary card gives the term-structure filter with other thresholds than its chapter on filters, which Chapter 10H follows. The card halts new short-volatility entries when the ratio stays above 1.05 for 3 days, cuts short volatility by 50% above 1.10 for 3 days, and re-enters in phases once the ratio has been below 1.00 for 5 days. Chapter 10H halts new entries above 1.00, halves the sleeve above 1.05 and keeps only deep tails above 1.15, each after 3 closes. The source does not reconcile the two, and 10H's is the fuller rule.

14 · Swing trading

From Part XI. $C_t$ , $H_t$ and $L_t$ are bar $t$ 's close, high and low, $n$ is a look-back in bars, and $\bar U_t$ and $\bar D_t$ are the smoothed average up and down moves.

Quantity	Formula	Where
Simple moving average	$M_t = \tfrac1n\sum_{i=0}^{n-1} C_{t-i}$	(11A.1)
Exponential moving average	$E_t = \alpha C_t + (1 - \alpha)\,E_{t-1}$	(11A.2)
An at-the-money straddle	$\text{straddle}/S_0 \approx 0.8\,\sigma\sqrt{T}$	(11A.3)
Signal against noise over $h$ days	$\mu h/(\sigma\sqrt{h}) = (\mu/\sigma)\sqrt{h}$	(11B.2)
Wilder smoothing	$W_t = W_{t-1} + (x_t - W_{t-1})/n$	(11C.1)
Relative strength index	$\text{RSI}_t = 100\,\bar U_t/(\bar U_t + \bar D_t)$	(11C.4)
MACD	$\text{MACD}_t = E^{(12)}_t - E^{(26)}_t$	(11C.5)
True range	$\max\{H_t - L_t,\ \lvert H_t - C_{t-1}\rvert,\ \lvert L_t - C_{t-1}\rvert\}$	(11C.6)
Average true range	$\text{ATR}_t = \big((n-1)\,\text{ATR}_{t-1} + \text{TR}_t\big)/n$	(11C.7)
Directional index	$\text{DX}_t = 100\,\lvert\text{DI}^+_t - \text{DI}^-_t\rvert/(\text{DI}^+_t + \text{DI}^-_t)$	(11C.9)
Where a close sits in its Bollinger bands	$\%B_t = \tfrac12 + z_t/4$	(11C.11)
Independent readings among $k$ indicators	$n_{\text{eff}} = k/(1 + (k-1)\rho)$	(11C.17)
Mansfield relative strength	$\text{MRS}_t = R_t/\bar R_t - 1$	(11D.1)
How often a pattern that fires is real	$\text{PPV} = s\pi/\big(s\pi + f(1 - \pi)\big)$	(11D.4)
The composite score	$\text{CS} = 100\sum_{k=1}^{8} w_k s_k$	(11G.2)
Units for a risk budget $B$ and stop distance $d$	$n = \lfloor B/(d\,m) \rfloor$	(11H.1)
Internal bar strength	$\text{IBS} = (C - L)/(H - L)$	(11H.2)
Implied volatility with an event of size $J$	$\sigma_I = \sqrt{\sigma_a^2 + J^2/T}$	(11I.2)
The implied move	$(C + P)/S$	(11I.3)
Volume-weighted average price	$\sum_i p_iv_i/\sum_i v_i$	(11J.1)
Chance a release reaches a stop $d$ away	$\Pr(\text{stop}) = N(-0.8\,d/m)$ , with $m$ the expected move	(11K.3)
One-factor model	$R_i = \beta_iR_m + \varepsilon_i$	(11L.1)
Correlation from one factor	$\rho_{ij} = \beta_i\beta_j\sigma_m^2/(\sigma_i\sigma_j)$	(11L.4)
Independent bets among $N$ positions	$N_{\text{eff}} = N/(1 + (N-1)\rho)$	(11L.5)
An at-the-money option's daily decay, $D$ days out	$(\lvert\Theta\rvert/365)/C \approx 1/(2D)$	(11M.1)
Share of value lost holding $h$ days	$L(D, h) = 1 - \sqrt{(D - h)/D}$	(11M.2)

15 · Strategies and margin

From Part XII.

Quantity	Formula	Where
A structure's slope at expiry	$c - p + h$ : net calls struck below, less net puts struck above, plus shares	(12A.1)
The rate a box spread implies	$r = \big((W - C)/C\big) \times 365/n$ , for width $W$ , credit $C$ , $n$ days	(12E.2)
Reg-T margin on a short uncovered option, per point	$P + \max(0.20\,S - O,\ 0.10\,S)$	(12G.1)
When exercising a put early can pay	$K\,r\,T > \text{time value}$	(12H.1)

16 · Discipline

From Part XIII and Chapter 1F. The Edge Book puts the typical product of the haircut's factors at 0.30 to 0.60, and calls a conviction score of 0.5 or more high conviction.

Quantity	Formula	Where
The retail haircut	$\text{SR}_{\text{user}} = \text{SR}_{\text{paper}}\,h_1h_2h_3h_4$	(13A.1)
The conviction score	$C_i(t) = E_i\,M_i\,R_i(t)\,X_i(t)\,S_i$	(13B.2)
The average factor behind a score	$g = C^{1/5}$	(13B.3)
The Real Value score, for single stocks	$\text{RV} = F\,T\,O\,G\,V\,X\,S$	(13B.4)
Trade only if	$\hat\mu_{\text{net}} - \lambda\,\text{ES}_{95} - \kappa\,C_{\text{exec}} > 0$	(1F.12)
Positions from a loss budget	$n \le \lfloor B_{\exp}/L_{\max} \rfloor$	(13C.1)
The portfolio's tail budget	$\sum_i \text{ES}_{95,i} \le B_{\text{portfolio}}$	(13C.2)
Readiness to trade	$\prod_{k=1}^{7} s_k$	(13C.3)
Risk contribution of sleeve $i$	$\text{RC}_i = w_i\,(\boldsymbol{\Sigma}\mathbf{w})_i/\sigma_p$	(13G.2)
Risk contributions add up to the whole	$\sum_i \text{RC}_i = \sigma_p$	(13G.5)
One position's vega, for $n_i$ contracts held long ( $s_i = +1$ ) or short ( $-1$ )	$V_i = s_i\,n_i\,m\,\nu_i$	(13H.1)
The book's vega, weighted by each instrument's VIX beta	$V = \sum_i \beta_i V_i$	(13H.2)