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 $F_{0,T_F} = S_0e^{(r-\delta)T_F}$ + *note that $T_F$ is used here
  • Replace $\delta \rightarrow r$
    • $F_{0,T_F} ^P(S) = F_{0,T_F}e^{-\delta T} \rightarrow F_{0,T_F}e^{-rT} = S_0e^{(r-\delta)T_F}e^{-rT}$ +  *note that $T$ is used here
    • $F_{0,T_F} ^P(K) = Ke^{-rT}$
    • $d_1 = \frac{ln[\frac{F^P(S)}{F^P(K)}] + 0.5v^2}{v}$ = $\frac{ln[\frac{F_{0,T_F}}{K}] + 0.5v^2}{v}$  
    • $d_2 = \frac{ln[\frac{F^P(S)}{F^P(K)}] – 0.5v^2}{v}$ = $\frac{ln[\frac{F_{0,T_F}}{K}] – 0.5v^2}{v}$  $d_2 = d_1 – v$ + *No change here- Just the regular formulae for $d_1$ and $d_2$
  • Substitute in the regular Black-scholes formula for stocks

 

 


Leave a comment

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.