Black Scholes Merton and treatment of Dividend

[anand99]

Hello,

When a Dividend is given as $ amount how does that impact the D1 calculation?

Do you use the ln(S0*De^-rt/k) amount instead of ln(S0/k)?

When a dividend is give as % amount how does that impact the D1 calcuation?

Do you subtract dividend from r as (r-q)+sigma^2/2)

Thanks.

 

[ShaktiRathore]

Yes Dividends represents a net outflow from the stock price so just replace S0 with So-PVD or S0 e^-qT where q is the continuous dividend yield. we assume continous dividend yield since bsm itself is based on continous returns. In case continous yoield is not given convert the yiled to continous using the formula ln(1+R)
d1=ln(S0/X)+T*(r+.5*stdDev^2)/(stdDev*sqrt(T)
replace S0 with S0 e^-qT in case of dividends,
so that d1=ln(S0 e^-qT/X)+T*(r+.5*stdDev^2)/(stdDev*sqrt(T)
d1=ln(S0/X)+ln(e^-qT)+T*(r+.5*stdDev^2)/(stdDev*sqrt(T)
d1=ln(S0/X)-qT+T*(r+.5*stdDev^2)/(stdDev*sqrt(T)
d1=ln(S0/X)+T*(r-q+.5*stdDev^2)/(stdDev*sqrt(T)

thanks

 

[anand99]

Thank you!