Generic B-S Formula
$C_t = [+F^P(S)]N(+d_1) – [F^P(K)]N(+d_2)$
$P_t = [-F^P(S)]N(-d_1) + [F^P(K)]N(-d_2)$
$v = \sigma \sqrt{T-t}$
$d_1 = \frac{ln[\frac{F^P(S)}{F^P(K)}]+0.5v^2}{v}$ $d_2 = \frac{ln[\frac{F^P(S)}{F^P(K)}] – 0.5v^2}{v}$ $d_2 = d_1 – v$
B-S for Stocks
$C_t = [+F^P(S)]N(+d_1) – [F^P(K)]N(+d_2)$
$P_t = [-F^P(S)]N(-d_1) + [F^P(K)]N(-d_2)$
Stock with :
- No dividends $F^P(S) = S_t $
- Discrete dividends $F^P(S) = S_t – \sum PV(dividends)$
- Continuous dividends $F^P(S) = S_te^{-\delta(T-t)} $
$F^P(K) = Ke^{-r(T-t)}$ | $v^2 = \sigma^2(T-t)$
$d_1 = \frac{ln[\frac{S_T}{K}]+(r- \delta + 0.5 \sigma^2)(T-t)}{v}$
$d_2 = \frac{ln[\frac{S_T}{K}] +(r – \delta – 0.5 \sigma^2)(T-t)}{v}$
$d_2 = d_1 – v$
To convert above equations from risk-neutral pricing to realistic pricing
- $r \rightarrow \alpha$
- $d_1,d_2 \rightarrow \hat{d_1}, \hat{d_2}$
B-S for Currency
To determine black-scholes formula for currency exchange
$x_t = \frac{domestic}{foreign}$
- $S_t \rightarrow x_t$
- $\delta \rightarrow r_f$
- $r \rightarrow r_d$
B-S for Futures
To determine black-scholes formula for Futures
$Stocks \rightarrow futures$
- Calculate the future price $F_{0,T_F} = S_0e^{(r-\delta)T_F}$ [btw]*note that $T_F$ is used here [/btw]
- Replace $\delta \rightarrow r$
- $F_{0,T_F} ^P(S) = F_{0,T_F}e^{-\delta T} \rightarrow F_{0,T_F}e^{-rT} = S_0e^{(r-\delta)T_F}e^{-rT}$ [btw] *note that $T$ is used here [/btw]
- $F_{0,T_F} ^P(K) = Ke^{-rT}$
- $d_1 = \frac{ln[\frac{F^P(S)}{F^P(K)}] + 0.5v^2}{v}$ = $\frac{ln[\frac{F_{0,T_F}}{K}] + 0.5v^2}{v}$
- $d_2 = \frac{ln[\frac{F^P(S)}{F^P(K)}] – 0.5v^2}{v}$ = $\frac{ln[\frac{F_{0,T_F}}{K}] – 0.5v^2}{v}$ $d_2 = d_1 – v$ [btw]*No change here- Just the regular formulae for $d_1$ and $d_2$ [/btw]
- Substitute in the regular Black-scholes formula for stocks