# 02 Black-Scholes Formula

#### 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

#### Exam FM/2

Financial MathematicsDerivative Markets

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