Formula sheet · Markets: A Rigorous Course

Microstructure & execution

Call-auction executable volume M01· M01 (1.2)

$V(p) = \min\{D(p), S(p)\}$
Round-trip transaction cost (Indian charges) M01· M01 (1.3)

$C = \underbrace{2F(1+g)}_{K} + \underbrace{\big(\sigma_b + \delta + (e+\phi)(1+g)\big)}_{\alpha}\,B + \underbrace{\big(\sigma_s + (e+\phi)(1+g)\big)}_{\beta}\,S$
Break-even price move after costs M01· M01 (1.4)

$\Delta^* = \frac{K/Q + (\alpha + \beta) P}{1 - \beta}, \qquad \frac{\Delta^*}{P} = \frac{\alpha+\beta}{1-\beta} + \frac{K}{(1-\beta)\,PQ}$
Effective spread = realised spread + price impact M05· M05 (5.1)

$\underbrace{q_t(p_t - m_t)}_{\text{effective half-spread}} = \underbrace{q_t(p_t - m_{t+\Delta})}_{\text{realised half-spread}} + \underbrace{q_t(m_{t+\Delta} - m_t)}_{\text{price impact}}$
Roll spread estimator M05· M05 (5.4)

$\hat s = 2\sqrt{-\hat\gamma_1}, \qquad \hat\gamma_1 = \text{sample first-order autocovariance of } \Delta p$
Glosten–Milgrom quotes and spread M05· M05 (5.6)

$a = \bar V + \tfrac{\mu}{2}\Delta V, \qquad b = \bar V - \tfrac{\mu}{2}\Delta V, \qquad a - b = \mu\,\Delta V$
Kyle equilibrium: price impact λ and insider intensity β M05· M05 (5.7)

$\lambda = \frac{\sigma_v}{2\sigma_u}, \qquad \beta = \frac{\sigma_u}{\sigma_v}, \qquad \mu = p_0, \qquad \alpha = -\beta p_0$
Square-root market impact law M05· M05 (5.13)

$\frac{I(Q)}{P} \approx Y\,\sigma_D\,\Big(\frac{Q}{V_D}\Big)^{\delta}, \qquad \delta \approx \tfrac12, \quad Y = O(1)$
Implementation shortfall decomposition M05· M05 (5.15)

$\text{IS} = \underbrace{Q_e(p_0 - p_d)}_{\text{delay}} + \underbrace{Q_e(\bar p - p_0)}_{\text{execution}} + \underbrace{(Q - Q_e)(p_T - p_d)}_{\text{opportunity}} + \text{fees}$
Almgren–Chriss expected implementation shortfall M05· M05 (5.20)

$E[\text{IS}] = \tfrac12\gamma X^2 + \varepsilon\sum_k|n_k| + \frac{\tilde\eta}{\tau}\sum_k n_k^2, \qquad \tilde\eta = \eta - \tfrac12\gamma\tau$
Almgren–Chriss optimal holdings trajectory M05· M05 (5.23)

$x(t) = X\,\frac{\sinh\big(\kappa(T - t)\big)}{\sinh(\kappa T)}, \qquad \kappa = \sqrt{\lambda\sigma^2/\eta}$
Annual Sharpe ratio after per-trade costs M05· M05 (5.25)

$\mu_A = N(e - c), \qquad \sigma_A = \sqrt N\,\sigma_t, \qquad \text{SR}_A = \sqrt N\,\frac{e - c}{\sigma_t} = \text{SR}_A^{\text{gross}} - \sqrt N\,\frac{c}{\sigma_t}$
Net asset value per unit, with daily fee accrual M06B· M06B (6B.1)–(6B.2)

$\text{NAV}_t = \frac{A_t - L_t}{N_t}, \qquad E_t = \frac{c}{365}\,NA_t^{\text{pre}}$
Wealth transfer from dealing at a stale NAV M06B· M06B (6B.3)–(6B.4)

$T_{\text{buy}} = \frac{S\,\varepsilon}{1 + f}, \quad f = \frac{S}{Nv_s}; \qquad T_{\text{red}} = -R\,\varepsilon$
Swing price (s = trading cost leaves stayers whole) M06B· M06B (6B.5)

$p^{\text{sw}} = \text{NAV}\,(1 \mp s)$
ETF no-arbitrage premium band M06B· M06B (6B.26)

$I\,(1 - C_r) \le P \le I\,(1 + C_c), \quad C_c = \kappa_b + \varphi_c + h + s/2, \; C_r = \kappa_s + \varphi_r + h + s/2$
ETF versus index fund crossover horizon M06B· M06B (6B.27)

$T^* = \frac{\ln[(1 - k_I)/(1 - k_E)]}{\ln[(1 + r - c_E)/(1 + r - c_I)]} \approx \frac{k_E - k_I}{c_I - c_E}$

Returns & statistics

Free-float index return as weighted constituent return M01· M01 (1.7)

$\frac{I_t}{I_{t-1}} - 1 = \sum_i w_{i,t-1}\, R_{i,t}, \qquad w_{i,t-1} = \frac{P_{i,t-1}N_i f_i}{M_{t-1}}, \quad R_{i,t} = \frac{P_{i,t}}{P_{i,t-1}} - 1$
Index divisor adjustment M01· M01 (1.8)

$D_t' = D_t \cdot \frac{M_t'}{M_t}$
Total return index M01· M01 (1.9)

$\text{TRI}_t = \text{TRI}_{t-1}\cdot \frac{I_t + \text{ID}_t}{I_{t-1}}$
Dividend price-adjustment factor M01· M01 (1.12)

$F_{\text{div}} = \frac{P_c - d}{P_c} = 1 - \frac{d}{P_c}$
Simple and log returns M02· M02 (2.1)

$R_t = \frac{P_t + D_t}{P_{t-1}} - 1 \quad\text{(simple, or arithmetic, return)}, \qquad r_t = \ln(1 + R_t) \quad\text{(log, or continuously compounded, return).}$
Volatility drag M02· M02 (2.3)

$E[\ln(1+R)] = \ln(1+\mu) - \frac{\sigma^2}{2(1+\mu)^2} + O\!\left(E|R-\mu|^3\right) \approx \mu - \frac{\sigma^2}{2}$
GBM: lognormal terminal price, mean and median M02· M02 (2.4)

$\ln\frac{S_T}{S_0} \sim N\!\left((\mu - \tfrac12\sigma^2)T,\ \sigma^2 T\right), \quad E[S_T] = S_0 e^{\mu T}, \quad \operatorname{median}(S_T) = S_0 e^{(\mu - \sigma^2/2)T}$
Variance ratio M02· M02 (2.7)

$VR(q) = \frac{\operatorname{Var}(r_t(q))}{q \operatorname{Var}(r_t)} = 1 + 2\sum_{j=1}^{q-1}\Big(1 - \frac{j}{q}\Big)\rho_j$
Estimating the mean: standard error and t-stat M02· M02 (2.13)

$\hat\mu = \frac{\ln P_T - \ln P_0}{T}, \qquad \operatorname{sd}(\hat\mu) = \frac{\sigma}{\sqrt{T}}, \qquad t\text{-stat} = \frac{\hat\mu}{\sigma/\sqrt{T}} \approx S\sqrt{T}$
Jarque–Bera normality test M02· M02 · 2.4

$JB = n\left(\frac{\hat S^2}{6} + \frac{\hat K^2}{24}\right) \Rightarrow \chi^2_2$
Hill tail-index estimator M02· M02 (2.16)

$\hat\alpha_k = \Big[\frac1k\sum_{i=1}^{k}\ln X_{(i)} - \ln X_{(k+1)}\Big]^{-1}$
Gap-fill probability M04A· M04A (4A.4)

$P(\text{fill})=2\Phi\!\left(-\frac{|g|}{\sigma_{in}}\right)$
SIP average cost is the harmonic mean of prices M06B· M06B (6B.14)–(6B.15)

$\bar c = \frac{n}{\sum_i 1/p_i} \le \frac1n\sum_i p_i, \qquad H \approx A\,(1 - \sigma_\delta^2)$

Portfolio & risk

Normal VaR and expected shortfall M02· M02 · 2.8

$\text{VaR}_c = -\mu + \sigma\Phi^{-1}(c), \qquad \text{ES}_c = -\mu + \sigma\,\phi(\Phi^{-1}(c))/(1-c)$
Equal-weight portfolio variance and the diversification floor M02· M02 (2.17)

$\sigma_p^2 = \frac{\sigma^2}{n}\big(1 + (n-1)\rho\big) = \sigma^2\Big(\rho + \frac{1-\rho}{n}\Big) \xrightarrow[n\to\infty]{} \rho\,\sigma^2$
Betting-against-beta return M06A· M06A (6A.25)

$r^{BAB} = \frac{r_L - r_f}{\beta_L} - \frac{r_H - r_f}{\beta_H}$
Fee drag: terminal wealth ratio and bounds M06B· M06B (6B.7)–(6B.8)

$\rho_T = \left(\frac{1 + r - c_1}{1 + r - c_2}\right)^{T}, \quad e^{-T\Delta c/(1 + r - c_1)} \le \rho_T \le e^{-T\Delta c/(1 + r - c_2)}$
Active share M06B· M06B (6B.20)

$AS = \tfrac12\sum_i |w_{p,i} - w_{b,i}|$
Berk–Green equilibrium fund size and flow M06B· M06B (6B.22)–(6B.23)

$q_t^* = \frac{\phi_t - f}{b}, \qquad q^*_{t+1} - q^*_t = \frac{\kappa_t}{b}\,\eta_{t+1}$
Decomposition of an index fund's tracking difference M06B· M06B (6B.25)

$TD \approx c + w_c\,(R_I - r_c) + \kappa - \ell + \delta - \textstyle\sum_i d_iR_i$
Dilution of stayers in a fund run, and the run threshold M06B· M06B (6B.30)–(6B.31)

$D(\rho) = \frac{\rho\,k}{(1 - k)(1 - \rho)}, \qquad \rho^* = \frac{b\,(1 - k)}{k + b\,(1 - k)}$
Portfolio overlap M06B· M06B (6B.32)

$O(A, B) = \sum_i \min(w^A_i, w^B_i) = 1 - \tfrac12\sum_i |w^A_i - w^B_i|$
Active expense ratio M06B· M06B (6B.33)

$c_A = \frac{c - (1 - AS)\,c_I}{AS}$
Global minimum-variance portfolio M14· M14 (14.1)

$w_{GMV} = \frac{\Sigma^{-1}\mathbf 1}{A}, \qquad \sigma^2_{GMV} = \frac1A, \qquad \mu_{GMV} = \frac BA$
Mean–variance frontier portfolio M14· M14 (14.2)

$w(m) = \frac{(C - Bm)\,\Sigma^{-1}\mathbf 1 + (Am - B)\,\Sigma^{-1}\mu}{D}, \qquad \sigma^2(m) = \frac{Am^2 - 2Bm + C}{D}$
Tangency (maximum-Sharpe) portfolio M14· M14 (14.3)

$w_T = \frac{\Sigma^{-1}(\mu - r_f\mathbf 1)}{\mathbf 1^\top\Sigma^{-1}(\mu - r_f\mathbf 1)} = \frac{\Sigma^{-1}(\mu - r_f\mathbf 1)}{B - Ar_f}$
Maximum Sharpe ratio M14· M14 · 14.1

$S_T^2 = e^\top\Sigma^{-1}e = C - 2Br_f + Ar_f^2$
Optimal covariance shrinkage intensity M14· M14 (14.4)

$\delta^* = \frac{\mathbb E\|S - \Sigma\|_F^2}{\mathbb E\|S - \Sigma\|_F^2 + \|\Sigma - F\|_F^2}$
Euler risk contributions M14· M14 (14.6)

$\sigma(w) = \sum_{i=1}^n w_i\frac{\partial\sigma}{\partial w_i} = \sum_{i=1}^{n} RC_i, \qquad RC_i := \frac{w_i(\Sigma w)_i}{\sigma(w)}$
Sharpe ratio of a volatility-targeted strategy M14· M14 · 14.5

$S_{VT} = \frac{\mu\,\mathbb E[1/\sigma_t]}{\sqrt{1 + \mu^2\operatorname{Var}(1/\sigma_t)}}$
Maximum Sharpe of two combined strategies M14· M14 (14.7)

$S^* = \sqrt{\frac{S_1^2 + S_2^2 - 2\rho S_1S_2}{1 - \rho^2}}$
Sharpe of N equally combined strategies M14· M14 · 14.7

$S\sqrt{N/(1 + (N-1)\bar\rho)}$
Prospect-theory value function M14· M14 · 14.9.1

$v(x) = \begin{cases} x^\alpha, & x \ge 0,\\ -\lambda(-x)^\beta, & x < 0,\end{cases}$
Disposition effect: PGR and PLR M14· M14 · 14.9.2

$\text{PGR} = \frac{\text{realised gains}}{\text{realised gains} + \text{paper gains}}, \qquad \text{PLR} = \frac{\text{realised losses}}{\text{realised losses} + \text{paper losses}}$
VaR and expected shortfall (quantile definition) M14· M14 (14.10)

$\operatorname{VaR}_\alpha(L) = q_\alpha, \qquad \operatorname{ES}_\alpha(L) = \frac{1}{1-\alpha}\int_\alpha^1 q_u\,du$
Delta–gamma–vega scenario P&L (return form) M14· M14 (14.14)

$\Delta V \approx \Theta\,\Delta t + \sum_i D_i\,r_i + \tfrac12\sum_{i} G_i\,r_i^2 + \sum_k \mathcal V_k\,\Delta\sigma_k, \qquad D_i = S_i\Delta_i,\quad G_i = S_i^2\Gamma_{ii}$
Implied stock borrow fee from futures M14· M14 (14.17)

$b_{\text{imp}} = r - q - \frac{1}{T}\ln\frac FS$
Constant (rebalanced) leverage: log-equity and volatility decay M14· M14 (14.19)

$\ln\frac{E_T}{E_0} = \lambda\ln\frac{P_T}{P_0} - \tfrac12\lambda(\lambda - 1)\sigma^2T - (\lambda - 1)cT$
Price fall that triggers a margin call M14· M14 (14.21)

$d^* = 1 - \frac{(1 - e_0)(1 + ct)}{1 - m}, \qquad d^* = \frac{e_0 - m}{1 - m}\ \text{ when } c = 0$
Probability of a margin call before T (first passage) M14· M14 (14.23)

$P(\tau_b \le T) = \Phi\Big(\frac{b - \nu T}{\sigma\sqrt T}\Big) + e^{2\nu b/\sigma^2}\,\Phi\Big(\frac{b + \nu T}{\sigma\sqrt T}\Big)$

Betting & sizing

Expectancy per trade (in R) M03· M03 (3.1)

$E = \frac1N\sum_{i=1}^{N} R_i = p\,\bar W - (1-p)\,\bar L$
Kelly fraction, binary bet M03· M03 (3.5)

$f^* = p - \frac{q}{b} = \frac{pb - q}{b}$
Fractional Kelly: share of full-Kelly growth M03· M03 (3.7)

$\frac{g(cf^*)}{g(f^*)} \approx 2c - c^2$
Continuous-time Kelly fraction and growth M03· M03 (3.10)

$f^* = \frac{\mu - r}{\sigma^2}, \qquad g(f^*) = r + \tfrac12 S^2$
Probability of ever falling to x·W0 at c× Kelly M03· M03 (3.12)

$P\big(W_t \le xW_0 \text{ for some } t\big) = x^{2/c - 1}$
Optimal Kelly shrinkage under estimation error M03· M03 (3.14)

$k^* = \frac{\mu^2}{\mu^2 + \text{se}^2} = \frac{t^2}{1 + t^2}$
Multi-asset Kelly M03· M03 (3.15)

$g(f) = r + f^\top(\mu - r\mathbf 1) - \tfrac12 f^\top\Sigma f, \qquad f^* = \Sigma^{-1}(\mu - r\mathbf 1), \qquad g(f^*) = r + \tfrac12(\mu - r\mathbf 1)^\top\Sigma^{-1}(\mu - r\mathbf 1)$
Gambler's ruin probability M03· M03 (3.16)

$P_a = \begin{cases}\dfrac{\rho^a - \rho^N}{1 - \rho^N}, & p \ne q,\\[2mm] 1 - \dfrac aN, & p = q.\end{cases}$
Fixed-fractional position size (lot-rounded) M03· M03 (3.23)

$Q = \ell\left\lfloor\frac{1}{\ell}\left\lfloor\frac{fE_q}{|P_e - P_s|}\right\rfloor\right\rfloor$
Break-even win rate with costs M04· M04 (4.5)

$p>p^*=\frac{1+c}{1+W}$
Bracket order: probability target hit first (with drift) M04A· M04A (4A.6)

$p=\frac{1-e^{\gamma b}}{e^{-\gamma a}-e^{\gamma b}}$
Bracket order: expected holding time M04A· M04A (4A.7)

$\mathbb E\tau=\frac{ap-b(1-p)}{\mu}$
Exits cannot create expectancy (optional stopping) M04A· M04A (4A.8)

$\mathbb E[X_\tau]=\mu\,\mathbb E[\tau]$
Probability a stop is hit within horizon T M04A· M04A (4A.9)

$P\Big(\min_{t\le T}X_t\le-b\Big)=\Phi\!\Big(\frac{-b-\mu T}{\sigma\sqrt T}\Big)+e^{-2\mu b/\sigma^2}\,\Phi\!\Big(\frac{-b+\mu T}{\sigma\sqrt T}\Big)$
Discrete-monitoring barrier shift M04A· M04A (4A.10)

$\beta\,\sigma\sqrt{\Delta t},\qquad \beta=-\frac{\zeta(1/2)}{\sqrt{2\pi}}\approx0.5826$
Trailing stop: mean peak gain before give-back d M04A· M04A (4A.14)

$m(d)=\frac{e^{\gamma d}-1}{\gamma},\quad\gamma=\frac{2\mu}{\sigma^2}\qquad(m=d\text{ if }\mu=0)$

Backtesting statistics

System quality number (SQN) M03· M03 (3.2)

$\text{SQN} = \sqrt{\min(N, 100)}\;\frac{\bar R}{s_R}$
Variance of the mean of overlapping h-period returns M04· M04 (4.4)

$\operatorname{Var}(\bar Y)\approx\frac{h^2\sigma^2}{N},\qquad\text{while the naive formula gives }\frac{\operatorname{Var}(Y_t)}{N}=\frac{h\sigma^2}{N}$
Survivorship bias in mean return M04· M04 (4.9)

$\mathbb E[R\mid\text{survive}]=\mu+\sigma\,\frac{\phi(a)}{1-\Phi(a)},\qquad a=\frac{-d-\mu}{\sigma}$
Break-even cost per side M04· M04 (4.10)

$c^*=\mu_g/\mathrm{TO}$
Standard error of the Sharpe ratio (normal returns) M04· M04 (4.13)

$\operatorname{SE}(\hat S)\approx\sqrt{\frac{1+S^2/2}{T}}$
Sharpe ratio aggregation with autocorrelation M04· M04 (4.14)

$S(q)=\eta(q)\,S,\qquad\eta(q)=\frac{q}{\sqrt{q+2\sum_{k=1}^{q-1}(q-k)\rho_k}}$
Probabilistic Sharpe ratio (PSR) M04· M04 (4.15)

$\mathrm{PSR}(S^*)=\Phi\!\left(\frac{(\hat S-S^*)\sqrt{T-1}}{\sqrt{1-\hat\gamma_3\hat S+\frac{\hat\gamma_4-1}{4}\hat S^2}}\right)$
Minimum track record length M04· M04 (4.16)

$\mathrm{MinTRL}=1+\Big(1-\hat\gamma_3\hat S+\frac{\hat\gamma_4-1}{4}\hat S^2\Big)\left(\frac{z_{1-\alpha}}{\hat S-S^*}\right)^2$
Deflated Sharpe ratio (DSR) M04· M04 (4.18)

$\mathrm{DSR}=\mathrm{PSR}(S_0),\qquad S_0=\sqrt{V}\left[(1-\gamma)\Phi^{-1}\!\Big(1-\frac1N\Big)+\gamma\Phi^{-1}\!\Big(1-\frac1{Ne}\Big)\right]$
Trades needed to detect an edge with power 1−β M14· M14 (14.9)

$N \ge \left(\frac{(2 + z_{1-\beta})\,s}{E}\right)^2$

Volatility

ATR and the implied volatility estimate M04A· M04A (4A.12)

$\mathrm{ATR}\approx1.596\,\sigma S,\qquad \hat\sigma_{\mathrm{ATR}}=\frac{\mathrm{ATR}}{1.596\,S}$
Implied volatility: Newton starting value M08· M08 (8.30)

$\sigma^*=\sqrt{2|x|/\tau}=\sqrt{2|\ln(F/K)|/\tau}$
Close-to-close variance estimator M09· M09 (9.1)

$\hat\sigma^2_{CC}=\frac{1}{n-1}\sum_{t=1}^{n}(r_t-\bar r)^2$
Parkinson range estimator M09· M09 (9.6)

$\hat\sigma^2_{P,t}=\frac{(\ln H_t/L_t)^2}{4\ln 2}$
Garman–Klass estimator M09· M09 (9.7)

$\hat\sigma^2_{GK,t}=\tfrac12(u_t-d_t)^2-(2\ln2-1)\,c_t^2$
Yang–Zhang estimator M09· M09 (9.9)

$\hat\sigma^2_{YZ}=V_O+k\,V_C+(1-k)\,V_{RS},\qquad k=\frac{0.34}{1.34+\frac{n+1}{n-1}}$
Daily P&L of a delta-hedged option (hedged at implied vol) M09· M09 (9.12)

$\text{P\&L}_{\text{day}}\approx\tfrac12\Gamma^iS^2\left[\left(\frac{\Delta S}{S}\right)^2-\sigma_i^2\Delta t\right]$
Total P&L when hedging at realised volatility M09· M09 (9.15)

$e^{-rT}X_T=BS(\sigma_r)-BS(\sigma_i)$
Variance risk premium M09· M09 (9.16)

$VRP_t=E^{\mathbb Q}_t[RV_{t,T}]-E^{\mathbb P}_t[RV_{t,T}]$
Breeden–Litzenberger risk-neutral density M09· M09 (9.17)

$q(K)=e^{rT}\frac{\partial^2C}{\partial K^2}$
25-delta risk reversal and butterfly M09· M09 (9.18)

$RR_{25}=\sigma_{25C}-\sigma_{25P},\qquad BF_{25}=\tfrac12(\sigma_{25C}+\sigma_{25P})-\sigma_{ATM}$
Dupire local volatility M09· M09 (9.20)

$\sigma_{loc}^2(K,T)=\frac{\partial_TC+(r-q)K\,\partial_KC+qC}{\tfrac12K^2\,\partial^2_KC}$
Heston stochastic volatility model M09· M09 (9.21)

$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^1_t,\qquad dv_t=\kappa(\theta-v_t)dt+\xi\sqrt{v_t}\,dW^2_t,\qquad d\langle W^1,W^2\rangle_t=\rho\,dt$
Variance swap fair strike (option replication) M09· M09 (9.22)

$K_{var}=\frac1TE^{\mathbb Q}\left[\int_0^T\sigma_t^2dt\right]=\frac{2e^{rT}}{T}\left[\int_0^F\frac{P(K)}{K^2}dK+\int_F^\infty\frac{C(K)}{K^2}dK\right]$
VIX / India VIX discretised variance M09· M09 (9.25)

$\sigma^2_{VIX}=\frac2T\sum_i\frac{\Delta K_i}{K_i^2}e^{rT}Q(K_i)-\frac1T\left(\frac F{K_0}-1\right)^2$
Dealer-gamma hedging feedback on volatility M09A· M09A (9A.6)

$dS=\frac{dS^0}{1+\lambda G},\qquad \sigma_{\text{eff}}=\frac{\sigma}{1+\lambda G}\quad(1+\lambda G>0)$
ATM straddle price (straddle-implied move) M09A· M09A (9A.8)

$\Pi_{ATM}=e^{-r\tau}F\sigma\sqrt\tau\sqrt{\tfrac2\pi}\left(1-\frac{\sigma^2\tau}{24}+O(\sigma^4\tau^2)\right)\le e^{-r\tau}F\sigma\sqrt\tau\sqrt{\tfrac2\pi}\approx0.8\,e^{-r\tau}F\sigma\sqrt\tau$
Expected absolute move and probability it is not exceeded M09A· M09A (9A.9)

$\mathbb E|S_T-S|=S\sigma\sqrt\tau\sqrt{\tfrac2\pi},\qquad \mathbb P\big(|S_T-S|<\mathbb E|S_T-S|\big)=2N\!\left(\sqrt{2/\pi}\right)-1=0.5751$
IV rank M09A· M09A (9A.10)

$\mathrm{IVR}_t=\frac{\sigma_t-\min_{t-n<s\le t}\sigma_s}{\max_{t-n<s\le t}\sigma_s-\min_{t-n<s\le t}\sigma_s}$
IV percentile M09A· M09A (9A.11)

$\mathrm{IVP}_t=\frac{1}{n-1}\#\{s:\ t-n<s<t,\ \sigma_s<\sigma_t\}$
Forward variance between two expiries M09A· M09A (9A.12)

$\sigma_{1,2}^2=\frac{w_2-w_1}{\tau_2-\tau_1}$
Event variance from two expiries after the event M09A· M09A (9A.13)

$e=\frac{\tau_1\tau_2\,(\sigma_1^2-\sigma_2^2)}{\tau_2-\tau_1},\qquad \sigma_d^2=\frac{\sigma_2^2\tau_2-\sigma_1^2\tau_1}{\tau_2-\tau_1}$
GARCH(1,1) M13· M13 (13.13)

$\varepsilon_t = \sigma_t z_t, \quad z_t \overset{iid}{\sim} (0,1), \qquad \sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2$
GARCH unconditional variance M13· M13 (13.14)

$\bar\sigma^2 \equiv \mathbb E[\varepsilon_t^2] = \mathbb E[\sigma_t^2] = \frac{\omega}{1-\alpha-\beta}$
GARCH k-step variance forecast M13· M13 (13.17)

$\mathbb E_T[\sigma_{T+k}^2] = \mathbb E_T[\varepsilon_{T+k}^2] = \bar\sigma^2 + (\alpha+\beta)^{k-1}\left(\sigma_{T+1}^2 - \bar\sigma^2\right)$
GARCH volatility half-life M13· M13 (13.18)

$h_{1/2} = \frac{\ln 0.5}{\ln(\alpha+\beta)}$
GARCH(1,1) kurtosis with normal innovations M13· M13 (13.22)

$\kappa = \frac{\mathbb E\varepsilon_t^4}{(\mathbb E\varepsilon_t^2)^2} = \frac{3\,[1-(\alpha+\beta)^2]}{1-(\alpha+\beta)^2-2\alpha^2} > 3$

Valuation

NOPAT and free cash flow to the firm M06· M06 (6.5)

$\text{NOPAT}_t = EBIT_t (1-\tau), \qquad \text{FCFF}_t = \text{NOPAT}_t + \text{D\&A}_t - \text{Capex}_t - \Delta NWC_t$
Growing perpetuity (Gordon) M06· M06 (6.8)

$\sum_{t=1}^{\infty} \frac{c_1(1+g)^{t-1}}{(1+r)^t} = \frac{c_1}{r-g}$
Two-stage DCF with terminal value M06· M06 (6.9)

$V_0 = \sum_{t=1}^{N}\frac{F_0(1+g)^t}{(1+W)^t} + \frac{F_0(1+g)^N(1+g_T)}{(W-g_T)(1+W)^N}$
Value-driver formula M06· M06 (6.11)

$V_0 = \frac{\text{NOPAT}_1\left(1 - \dfrac{g}{\text{RONIC}}\right)}{W - g}$
CAPM M06· M06 (6.14)

$\mu_i - r_f = \beta_i(\mu_M - r_f), \qquad \beta_i = \frac{\operatorname{Cov}(R_i, R_M)}{\sigma_M^2}$
Levered beta (Hamada) M06· M06 (6.15)

$\beta_L = \beta_U\left[1 + (1-\tau)\frac{D}{E}\right]$
Weighted average cost of capital M06· M06 (6.16)

$W = \frac{E}{D+E}k_e + \frac{D}{D+E}k_d(1-\tau)$
Justified P/E M06· M06 (6.17)

$\frac{P_0}{E_1} = \frac{p}{k-g} = \frac{1 - g/\text{ROE}}{k-g}, \qquad \frac{P_0}{E_0} = \frac{p(1+g)}{k-g}$
Beneish M-score M06· M06 (6.19)

$M = -4.84 + 0.920\,\text{DSRI} + 0.528\,\text{GMI} + 0.404\,\text{AQI} + 0.892\,\text{SGI} + 0.115\,\text{DEPI} - 0.172\,\text{SGAI} + 4.679\,\text{TATA} - 0.327\,\text{LVGI}$
EPS after a debt-funded buyback M06A· M06A (6A.5)

$\text{EPS}' = \frac{E - r(1-\tau)C}{N - C/P_b}$
Promoter pledge invocation price M06A· M06A (6A.15)

$P^* = \frac{cL}{n}, \qquad 1 - \frac{P^*}{P_0} = 1 - \frac{c}{c_0}, \quad c_0 = \frac{nP_0}{L}$
Residual income valuation M06A· M06A (6A.16)

$V_0 = B_0 + \sum_{t=1}^{\infty} \frac{(\text{ROE}_t - k)\,B_{t-1}}{(1+k)^t}$
Justified price-to-book M06A· M06A (6A.17)

$\frac{V_0}{B_0} = 1 + \frac{\text{ROE} - k}{k - g} = \frac{\text{ROE} - g}{k - g}$
Bank ROA and ROE M06A· M06A (6A.19)

$\text{ROA} = (\text{NIM} + \text{fees} - \text{opex} - \text{credit cost})(1 - \tau), \qquad \text{ROE} = \text{ROA} \times \frac{A}{B}$
Net NPA from gross NPA and provision coverage M06A· M06A (6A.20)

$\text{NNPA} = \frac{\text{GNPA}(1 - \text{PCR})}{1 - \text{GNPA}\cdot\text{PCR}}$
Beta of AMC profits to the market (operating leverage) M06B· M06B (6B.12)

$\beta_\Pi = \beta_A\,\frac{(1 - v)\,y_eA_e}{\Pi}; \quad \text{all equity: } \beta_A\Big(1 + \frac{F}{\Pi}\Big)$
Present value of fees on a SIP customer M06B· M06B (6B.13)

$PV = y_mP\,\frac{q}{1 - q}\cdot\frac{1 - a^{n}}{1 - a}, \quad a = \frac{1 - \lambda}{1 + k}, \; q = (1 + g)\,a$

Event studies

Standardised unexpected earnings (SUE) M06A· M06A (6A.1)

$\text{SUE}_q = \frac{d_q - \hat\delta_q}{\hat s_q}$
CAR t-test under the market model (results-day event study) M06A· M06A (6A.4)

$t_i = \frac{CAR_i}{\hat\sigma_i\sqrt{v_i}} \sim t_{L_1-2}, \qquad v_i = L_2 + \gamma' X^*(X'X)^{-1}X^{*\prime}\gamma$
Merger arbitrage: implied completion probability M06A· M06A (6A.12)

$P_T(1 + rT) = p\,D + (1 - p)F + c \quad\Longrightarrow\quad p = \frac{P_T(1 + rT) - c - F}{D - F}$
Market model for normal returns M12· M12 (12.1)

$R_{i\tau}=\alpha_i+\beta_iR_{m\tau}+\varepsilon_{i\tau},\qquad \mathbb E[\varepsilon_{i\tau}]=0,\ \operatorname{Var}(\varepsilon_{i\tau})=\sigma_i^2$
Abnormal return M12· M12 (12.4)

$AR_{i\tau}=R_{i\tau}-\hat\alpha_i-\hat\beta_iR_{m\tau}, \qquad \tau\in[T_1+1,T_2]$
Cumulative abnormal return and its variance M12· M12 (12.8)

$CAR_i=\gamma'AR_i^*,\qquad \operatorname{Var}(CAR_i)=\gamma'V_i\gamma=\sigma_i^2\left[L_2+\gamma'X^*(X'X)^{-1}X^{*\prime}\gamma\right]$
Standardised CAR M12· M12 (12.9)

$SCAR_i=\frac{CAR_i}{\sqrt{\hat\sigma_i^2\,\gamma'[I+X^*(X'X)^{-1}X^{*\prime}]\gamma}} \sim t_{L_1-2}$
Cross-sectional t-test of mean CAR M12· M12 (12.11)

$t_{CS}=\frac{\overline{CAR}}{s_{CAR}/\sqrt N},\qquad s_{CAR}^2=\frac1{N-1}\sum_i(CAR_i-\overline{CAR})^2$
Boehmer–Musumeci–Poulsen (BMP) test M12· M12 (12.13)

$t_{\text{BMP}}=\frac{\overline{SCAR}}{s_{SCAR}/\sqrt N},\qquad s^2_{SCAR}=\frac1{N-1}\sum_i(SCAR_i-\overline{SCAR})^2$
Sequential Bayesian updating in log-odds M12· M12 (12.18)

$\ln\frac{P(A\mid n_{1:k})}{P(B\mid n_{1:k})}=\ln\frac{P(A)}{P(B)}+\sum_{j=1}^k\ln\frac{P(n_j\mid A)}{P(n_j\mid B)}$
Expected value of perfect information M12· M12 (12.20)

$EVPI=\sum_sP(s)\max_ax(s,a)-\max_a\sum_sP(s)x(s,a)\ \ge 0$

Rates & FX

Debt fund NAV change for a parallel yield shift M06B· M06B (6B.28)

$\frac{\Delta \text{NAV}}{\text{NAV}} \approx -D_p\,\Delta y + \tfrac12 C_p\,(\Delta y)^2$
Bond price from yield (m compounding periods a year) M11· M11 (11.2)

$P(y)=\sum_{k=1}^{N} CF_k\,(1+y/m)^{-m t_k}$
Macaulay duration M11· M11 (11.4)

$D_{\text{mac}} = \frac{1}{P}\sum_{k=1}^{N} t_k\,CF_k\,(1+y/m)^{-mt_k}$
Modified duration M11· M11 (11.5)

$D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}$
Convexity M11· M11 (11.7)

$C = \frac{1}{P}\frac{d^2P}{dy^2}$
Duration–convexity price approximation M11· M11 (11.9)

$\frac{\Delta P}{P} = -D_{\text{mod}}\,\Delta y + \tfrac12 C\,(\Delta y)^2 + O\big((\Delta y)^3\big)$
Equity duration (Gordon model) M11· M11 (11.11)

$D_{\text{mod}}^{eq} \equiv -\frac{1}{P}\frac{\partial P}{\partial r} = \frac{1}{r-g} = \frac{P}{D_1}, \qquad D_{\text{mac}}^{eq} = \frac{1+r}{r-g}$
Long yield = expected short rates + term premium M11· M11 (11.14a)

$y_n = \frac1n\sum_{k=0}^{n-1}\mathbb E_t[r_{t+k}] + TP_n$
Taylor rule M11· M11 (11.15)

$i = r^* + \pi + a_\pi(\pi-\pi^*) + a_y\,x$
Covered interest parity (FX forward) M11· M11 (11.17)

$F = S\,\frac{1+r_dT}{1+r_fT}$
Uncovered interest parity M11· M11 (11.19)

$\mathbb E_t[s_{t+1}] - s_t = i_t - i_t^*$
Currency carry excess return M11· M11 (11.22)

$xr_{t+1} = (i_t - i_t^*) - \Delta s_{t+1}$
Dollar return on a rupee asset M11· M11 (11.24)

$1+R_{\text{USD}} = (1+R_{\text{INR}})\frac{S_0}{S_1}$
USDINR futures fair value and annualised premium M11· M11 (11.26)

$F = S\,\frac{1 + r_d\tau}{1 + r_f\tau}, \qquad \frac{F/S - 1}{\tau} = \frac{r_d - r_f}{1 + r_f\tau} \approx r_d - r_f$
Effective rate of a currency futures hedge M11· M11 (11.27)

$\text{effective rate} = F_1 + (S_2 - F_2) = F_1 - b_2$
Garman–Kohlhagen currency option prices M11· M11 · 11.17.5

$c = Se^{-r_f\tau}N(d_1) - Ke^{-r_d\tau}N(d_2),\qquad p = Ke^{-r_d\tau}N(-d_2) - Se^{-r_f\tau}N(-d_1)$

Futures

Cost-of-carry forward price M07· M07 (7.1)

$F_0 = S_0\,e^{(r-q)T}$
Forward price with known income M07· M07 (7.3)

$F_0 = (S_0 - I)\,e^{rT}$
Value of a forward contract M07· M07 (7.4)

$f_t = (F_t - K)\,e^{-r\tau} = S_te^{-q\tau} - Ke^{-r\tau}, \qquad \tau = T - t$
No-arbitrage band with costs and borrow/lend rates M07· M07 (7.5)

$S_0(1-c)\,e^{(r_\ell - q)T} \;\le\; F_0 \;\le\; S_0(1+c)\,e^{(r_b - q)T}$
Basis (futures minus spot) M07· M07 (7.9)

$b_t = S_t\left(e^{(r-q)\tau} - 1\right) = S_t(r-q)\tau + O(\tau^2)$
Minimum-variance hedge ratio M07· M07 (7.15)

$h^* = \frac{\operatorname{Cov}(\Delta S, \Delta F)}{\operatorname{Var}(\Delta F)} = \rho\,\frac{\sigma_S}{\sigma_F}$
Optimal number of futures contracts M07· M07 (7.17)

$N^* = \frac{h^*Q_A}{Q_F}$
Contracts to change portfolio beta M07· M07 (7.18)

$N = (\beta^\star - \beta)\,\frac{V_P}{V_F}$
Tailing the hedge M07· M07 (7.19)

$N_{\text{tail}} = N^*\,e^{-r\tau}$

Options

Put–call parity (European, dividend yield q) M08· M08 (8.4)

$c - p = Se^{-qT} - Ke^{-rT}$
One-period binomial price and risk-neutral probability M08· M08 (8.9)

$V_0=\Delta S_0+B = e^{-r\Delta t}\big[pV_u+(1-p)V_d\big],\qquad p=\frac{e^{r\Delta t}-d}{u-d}\in(0,1)$
Cox–Ross–Rubinstein up and down factors M08· M08 (8.11)

$u=e^{\sigma\sqrt{\Delta t}},\qquad d=1/u$
Geometric Brownian motion solution M08· M08 (8.16)

$S_t=S_0\exp\!\Big(\big(\mu-\tfrac12\sigma^2\big)t+\sigma W_t\Big)$
Black–Scholes–Merton PDE M08· M08 (8.18)

$\frac{\partial V}{\partial t}+\tfrac12\sigma^2S^2\frac{\partial^2V}{\partial S^2}+(r-q)S\frac{\partial V}{\partial S}-rV=0$
Risk-neutral valuation M08· M08 (8.19)

$V(t,S)=e^{-r(T-t)}\,\mathbb E\big[h(S_T)\,\big|\,S_t=S\big]$
Black–Scholes d1 and d2 M08· M08 (8.22)

$d_1=\frac{\ln(S/K)+(r-q+\tfrac12\sigma^2)\tau}{\sigma\sqrt\tau},\qquad d_2=d_1-\sigma\sqrt\tau=\frac{\ln(S/K)+(r-q-\tfrac12\sigma^2)\tau}{\sigma\sqrt\tau}$
Black–Scholes call M08· M08 (8.23)

$c=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2)$
Black–Scholes put M08· M08 (8.24)

$p=Ke^{-r\tau}N(-d_2)-Se^{-q\tau}N(-d_1)$
Black-76 (options on futures) M08· M08 (8.25)

$c=e^{-r\tau}\big[FN(d_1)-KN(d_2)\big],\quad p=e^{-r\tau}\big[KN(-d_2)-FN(-d_1)\big],\quad d_{1,2}=\frac{\ln(F/K)\pm\frac12\sigma^2\tau}{\sigma\sqrt\tau}$
Call delta M08· M08 · 8.9

$\Delta_c=e^{-q\tau}N(d_1)$
Put delta M08· M08 · 8.9

$\Delta_p=\Delta_c-e^{-q\tau}=-e^{-q\tau}N(-d_1)$
Gamma M08· M08 · 8.9

$\Gamma=\dfrac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt\tau}$
Vega M08· M08 · 8.9

$\mathcal V=Se^{-q\tau}\varphi(d_1)\sqrt\tau$
Call theta M08· M08 · 8.9

$\Theta_c=-\dfrac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt\tau}-rKe^{-r\tau}N(d_2)+qSe^{-q\tau}N(d_1)$
Theta–gamma relation M08· M08 (8.27)

$\Theta+\tfrac12\sigma^2S^2\Gamma+(r-q)S\Delta-rV=0$
Delta-hedged P&L (gamma vs variance) M08· M08 (8.28)

$d\Pi\approx\tfrac12\Gamma S^2\big[(dS/S)^2-\sigma^2dt\big]$
Parity-implied forward from the option chain M09A· M09A (9A.1)

$F = K + e^{r\tau}\,(C-P)$
Put–call ratios (OI and volume) M09A· M09A (9A.3)

$\mathrm{PCR}_{OI}=\frac{\sum_K p_K}{\sum_K c_K},\qquad \mathrm{PCR}_{vol}=\frac{\sum_K v^p_K}{\sum_K v^c_K}$
Option holders' payout at expiry (max pain = its argmin) M09A· M09A (9A.4)

$W(x)=\sum_{j=1}^{n}\Big[c_{K_j}(x-K_j)^+ + p_{K_j}(K_j-x)^+\Big]$

Commodities

Commodity futures with storage cost and convenience yield M10· M10 (10.1)

$F_t(T) = S_t\, e^{c\,(T-t)} = S_t\, e^{(r+u-y)(T-t)}$
Implied convenience yield M10· M10 (10.4)

$y \equiv r + u - \frac{1}{T}\ln\frac{F_0(T)}{S_0}$
Futures return = spot return + change in log basis M10· M10 (10.8)

$\ln\frac{F_{t'}(T)}{F_t(T)} = \underbrace{\ln\frac{S_{t'}}{S_t}}_{\text{spot return}} + \underbrace{\left[\ln\frac{F_{t'}(T)}{S_{t'}} - \ln\frac{F_t(T)}{S_t}\right]}_{\text{change in log basis}}$
Rolling long futures: return minus cumulative carry M10· M10 (10.10)

$\sum_{k=0}^{N-1}\ln\frac{S_{t_{k+1}}}{F_{t_k}(t_{k+1})} = \ln\frac{S_{t_N}}{S_{t_0}} - \Delta\sum_{k=0}^{N-1} c_k$
Hotelling rule for exhaustible resources M10· M10 (10.13)

$p_t - c = (p_0 - c)\, e^{rt}$
Domestic gold price (INR per 10 g) M10· M10 (10.15)

$P^{\text{INR}}_{10g} = \frac{P^{\text{USD}}_{oz} + q}{G}\times 10 \times X \times (1 + d)$
Mean-reversion half-life from AR(1) coefficient M10· M10 (10.19)

$h_{1/2} = -\frac{\ln 2}{\ln \varphi}$
3-2-1 crack spread M10· M10 (10.20)

$\text{3-2-1 crack (\$/bbl)} = \frac{2P_{\text{gasoline}} + P_{\text{distillate}} - 3P_{\text{crude}}}{3}$

Forecasting

Yule–Walker equations for AR(p) M13· M13 (13.5)

$\rho_k = \sum_{j=1}^p \phi_j \rho_{k-j}, \quad k \ge 1, \qquad \sigma^2 = \gamma_0\Big(1 - \sum_{j=1}^p \phi_j\rho_j\Big)$
AR(1) h-step forecast and its variance M13· M13 (13.6)

$\mathbb E_t[X_{t+h}] = \mu + \phi^h (X_t - \mu), \qquad \operatorname{Var}_t(X_{t+h}) = \sigma^2\,\frac{1-\phi^{2h}}{1-\phi^2}$
Predictive regression with persistent predictor M13· M13 (13.7)

$r_{t+1} = \alpha + \beta x_t + u_{t+1}, \qquad x_{t+1} = \theta + \rho x_t + v_{t+1}$
Out-of-sample R² M13· M13 (13.8)

$R^2_{OS} = 1 - \frac{\sum_{t} (r_{t+1} - \hat r_{t+1})^2}{\sum_t (r_{t+1} - \bar r_t)^2}$
Sharpe ratio of a timing strategy from predictive R² M13· M13 (13.9)

$S^{*2} \equiv \mathbb E\!\left[\frac{(\mu+x_t)^2}{\sigma_\varepsilon^2}\right] = \frac{S^2 + R^2}{1 - R^2}$
Stambaugh bias in predictive regressions M13· M13 (13.12)

$\mathbb E[\hat\beta - \beta] = \gamma\,\mathbb E[\hat\rho - \rho] \approx -\gamma\,\frac{1+3\rho}{T}$
Hamilton filter (regime probability update) M13· M13 (13.23)

$\xi_{t\mid t}(j) = \frac{\xi_{t\mid t-1}(j)\,f_j(y_t)}{\sum_k \xi_{t\mid t-1}(k)\,f_k(y_t)}$
Triple-barrier label M13· M13 (13.29)

$\text{label} = \begin{cases}+1 & R_{e,t}\ge \text{pt first},\\ -1 & R_{e,t}\le -\text{sl first},\\ 0 & \text{vertical barrier } t = e+v \text{ reached first.}\end{cases}$
Brier score M13· M13 (13.32)

$\mathrm{BS} = \frac1N\sum_{i=1}^N(p_i - y_i)^2$
Murphy decomposition of the Brier score M13· M13 (13.36)

$\mathrm{BS} = \underbrace{\frac1N\sum_k n_k(f_k - \bar y_k)^2}_{\text{reliability (REL)}} - \underbrace{\frac1N\sum_k n_k(\bar y_k - \bar y)^2}_{\text{resolution (RES)}} + \underbrace{\bar y(1-\bar y)}_{\text{uncertainty (UNC)}}$

Stat arb

Dickey–Fuller regression M13A· M13A (13A.2)

$\Delta y_t = \gamma\, y_{t-1} + e_t, \qquad \gamma = \rho - 1$
Dickey–Fuller limiting distributions M13A· M13A (13A.4)

$T(\hat\rho - 1) \Rightarrow \frac{\tfrac12\left(W(1)^2 - 1\right)}{\int_0^1 W(r)^2\,dr}, \qquad \hat\tau \equiv \frac{\hat\rho-1}{\operatorname{se}(\hat\rho)} \Rightarrow \frac{\tfrac12\left(W(1)^2 - 1\right)}{\left(\int_0^1 W(r)^2\,dr\right)^{1/2}}$
Augmented Dickey–Fuller regression M13A· M13A (13A.5)

$\Delta y_t = \gamma\,y_{t-1} + \sum_{j=1}^{p-1}\delta_j\,\Delta y_{t-j} + e_t$
Vector error-correction model (VECM) M13A· M13A (13A.9)

$\Delta y_t = \nu + \Pi\,y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i\,\Delta y_{t-i} + \varepsilon_t,\qquad \Pi = -\Big(I - \sum_{i=1}^p A_i\Big),\quad \Gamma_i = -\sum_{j=i+1}^p A_j$
Reduced-rank factorisation (adjustment × cointegrating vectors) M13A· M13A (13A.10)

$\Pi = \alpha\beta', \qquad \alpha,\beta \in\mathbb R^{n\times r}\ \text{of full column rank}$
Engle–Granger cointegrating regression M13A· M13A (13A.14)

$p^1_t = \mu + b\,p^2_t + u_t$
Johansen trace statistic M13A· M13A (13A.20)

$\lambda_{\text{trace}}(r) = -T\sum_{i=r+1}^n\ln(1-\hat\lambda_i)$
Ornstein–Uhlenbeck process M13A· M13A (13A.22)

$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t, \qquad \kappa>0$
OU exact discretisation as AR(1) M13A· M13A (13A.23)

$X_{t+\Delta} = a + b\,X_t + e_{t+\Delta},\qquad b = e^{-\kappa\Delta},\quad a = \theta(1-b),\quad \sigma_e^2 = \frac{\sigma^2(1-b^2)}{2\kappa}$
OU parameters from AR(1) estimates M13A· M13A (13A.24)

$\hat\kappa = -\frac{\ln\hat b}{\Delta},\qquad \hat\theta = \frac{\hat a}{1-\hat b},\qquad \hat\sigma = \hat\sigma_e\sqrt{\frac{2\hat\kappa}{1-\hat b^2}}$
Profit rate of an OU entry/exit cycle M13A· M13A (13A.26)

$\mu(a,m) = \frac{m - a - c}{\mathbb E[\mathcal T]},\qquad \mathbb E[\mathcal T] = \mathbb E_a[\tau_m] + \mathbb E_m[\tau_a]$
Total least squares hedge ratio M13A· M13A (13A.30)

$\hat\beta_{\text{TLS}} = \frac{(S_{yy}-S_{xx}) + \sqrt{(S_{yy}-S_{xx})^2 + 4S_{xy}^2}}{2S_{xy}}$
Time-varying hedge ratio state-space model M13A· M13A (13A.31)

$y_t = x_t'\beta_t + \varepsilon_t,\ \ \varepsilon_t\sim N(0,R);\qquad \beta_t = \beta_{t-1} + \eta_t,\ \ \eta_t\sim N(0,Q)$
Kalman filter innovation and its variance M13A· M13A (13A.33)

$e_t = y_t - x_t'b_{t|t-1},\qquad F_t = x_t'P_{t|t-1}x_t + R$
Kalman gain and update M13A· M13A (13A.34)

$K_t = P_{t|t-1}x_t/F_t,\qquad b_{t|t} = b_{t|t-1} + K_te_t,\qquad P_{t|t} = P_{t|t-1} - K_tx_t'P_{t|t-1}$
MCX–COMEX gold log spread M13A· M13A (13A.40)

$s_t = \ln F^{\text{MCX}}_t - \Big[\ln F^{\text{COMEX}}_t + \ln X^{\text{fwd}}_t + \ln(1+d_t) + \ln\frac{10\times0.995}{31.1034768}\Big]$