Summary of key results · M08 Option pricing

Result	Formula
Payoffs	$C_T=(S_T-K)^+$ , $P_T=(K-S_T)^+$
Forward value	$Se^{-qT}-Ke^{-rT}=e^{-rT}(F-K)$ , $F=Se^{(r-q)T}$
Put–call parity	$c-p=Se^{-qT}-Ke^{-rT}=e^{-rT}(F-K)$
Bounds	$(Se^{-qT}-Ke^{-rT})^+\le c\le Se^{-qT}$ ; $(Ke^{-rT}-Se^{-qT})^+\le p\le Ke^{-rT}$
American call, $q=0$	$C=c$ (never exercise early)
One-period	$\Delta=\frac{V_u-V_d}{S_0(u-d)}$ , $V_0=e^{-r\Delta t}[pV_u+(1-p)V_d]$ , $p=\frac{e^{(r-q)\Delta t}-d}{u-d}$ , no arbitrage iff $d<e^{(r-q)\Delta t}<u$
CRR	$u=e^{\sigma\sqrt{\Delta t}}$ , $d=1/u$ ; error $O(1/n)$ , oscillating
Itô	$df=(f_t+af_x+\frac12b^2f_{xx})dt+bf_xdW$
GBM	$S_T=S_0e^{(\mu-\sigma^2/2)T+\sigma W_T}$ ; $\mathbb E S_T=S_0e^{\mu T}$ ; median $S_0e^{(\mu-\sigma^2/2)T}$
BSM PDE	$V_t+\frac12\sigma^2S^2V_{SS}+(r-q)SV_S-rV=0$
Risk-neutral pricing	$V=e^{-r\tau}\mathbb E^{\mathbb Q}[h(S_T)]$ , drift $r-q$ under $\mathbb Q$
$d_1,d_2$	$d_{1,2}=\frac{\ln(S/K)+(r-q\pm\frac12\sigma^2)\tau}{\sigma\sqrt\tau}$
BSM	$c=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2)$ ; $p=Ke^{-r\tau}N(-d_2)-Se^{-q\tau}N(-d_1)$
Black-76	$c=e^{-r\tau}[FN(d_1)-KN(d_2)]$ , $d_{1,2}=\frac{\ln(F/K)\pm\frac12\sigma^2\tau}{\sigma\sqrt\tau}$
Identity	$Se^{-q\tau}\varphi(d_1)=Ke^{-r\tau}\varphi(d_2)$
Delta	call $e^{-q\tau}N(d_1)$ ; put $-e^{-q\tau}N(-d_1)$ ; futures delta (Black-76) $e^{-r\tau}N(d_1)$
Gamma, vega	$\Gamma=\frac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt\tau}$ ; $\mathcal V=Se^{-q\tau}\varphi(d_1)\sqrt\tau$
Theta (call)	$-\frac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt\tau}-rKe^{-r\tau}N(d_2)+qSe^{-q\tau}N(d_1)$
Rho	call $K\tau e^{-r\tau}N(d_2)$ ; put $-K\tau e^{-r\tau}N(-d_2)$
Theta–gamma	$\Theta+\frac12\sigma^2S^2\Gamma+(r-q)S\Delta-rV=0$
Hedged P&L	$d\Pi\approx\frac12\Gamma S^2[(dS/S)^2-\sigma^2dt]$
Vomma	$\partial^2c/\partial\sigma^2=\mathcal V d_1d_2/\sigma$
IV start	$\sigma^*=\sqrt{2
Reference	$S=K=100,\tau=1,r=5\%,\sigma=20\%$ : $c=10.4505835722$ , $p=5.5735260223$