Summary of key results · M07 Forwards and futures

Result	Formula	Where
Forward price, continuous yield	$F_0 = S_0e^{(r-q)T}$	Prop. 7.1, (7.1)
Cash-and-carry profit	$\Pi_T = F_0 - S_0e^{(r-q)T}$	(7.2)
Forward price, discrete dividends	$F_0 = (S_0 - I)e^{rT}$ , $I = \sum D_ie^{-rt_i}$	Prop. 7.2, (7.3)
Value of a long forward	$f_t = (F_t - K)e^{-r\tau} = S_te^{-q\tau} - Ke^{-r\tau}$	Prop. 7.3, (7.4)
No-arbitrage band	$S_0(1-c)e^{(r_\ell-q)T} \le F_0 \le S_0(1+c)e^{(r_b-q)T}$	Prop. 7.4, (7.5)
Daily MTM	$m_t = Q(F_t - F_{t-1})$ , $\sum m_t = Q(F_n - F_{\text{entry}})$	(7.6)
Futures = forward (deterministic $r$ )	$F_0 = G_0$ ; tail $\delta_i = 1/A(t_{i+1},T)$	Prop. 7.5
Futures price, stochastic $r$	$F_t = \mathbb{E}^{\mathbb{Q}}_t[S_T]$	(7.7)
Futures–forward gap	$F_0 - G_0 = -\operatorname{Cov}^{\mathbb{Q}}(D_T,S_T)/P(0,T)$	Prop. 7.6, (7.8)
Basis and convergence	$b = F - S$ , $b_T = 0$ , $b_t \approx S_t(r-q)\tau$	Prop. 7.7, (7.9)
Hedger's effective price	$F_1 - b_2$	(7.10)
Futures + cash = total-return index	$S_T - F_0 + S_0e^{rT} = S_T + S_0e^{rT}(1-e^{-qT})$	(7.11)
Return on margin	$R_M = (\Delta F/F)/m$	(7.12)
Roll cost, calendar carry	$Q(F_{\text{far}} - F_{\text{near}})$ ; $\ln(F_{\text{far}}/F_{\text{near}})/\Delta T$	(7.13)
Hedged variance	$\sigma_S^2 - 2h\rho\sigma_S\sigma_F + h^2\sigma_F^2$	(7.14)
Minimum-variance ratio	$h^* = \rho\sigma_S/\sigma_F = \operatorname{Cov}/\operatorname{Var}$	Prop. 7.8, (7.15)
Hedge effectiveness	$\rho^2$ ; penalty $\sigma_F^2(h - h^*)^2$	(7.16)
Contracts	$N^* = h^*Q_A/Q_F$	(7.17)
Beta hedge	$N = (\beta^\star - \beta)V_P/V_F$	Prop. 7.9, (7.18)
Interest tail	$N_{\text{tail}} = N^*e^{-r\tau}$	(7.19)
Storage (investment asset)	$F_0 = S_0e^{(r+u)T} = (S_0+U)e^{rT}$	Prop. 7.10, (7.20)
Convenience yield (consumption asset)	$F_0 = S_0e^{(r+u-y)T}$ , only $F_0 \le S_0e^{(r+u)T}$ is forced	Prop. 7.11, (7.21)
OI accounting	$\Delta\text{OI} = N_{oo} - N_{cc}$ , volume $\ge	\Delta \text{OI}