Digital Call Option value = exp(-rt)N(d2)

Black Scholes says C= SN(d1) - Xexp(-rt)N(d2) = exp(-rt){SN(d1)exp(rt)-XN(d2)},
So for risk-neutral, N(d2)= Pr(Price end up above Strike).

Digital Call = Pay M if expired above X. So Digital Call Value =exp(-rt)MN(d2).

In general with Yield,

C=exp(Y-r)tMN(d2)
P=exp(Y-r)tMN(-d2)

Forward Start Call Option = Straight Call Option

  This only be true when all of the following are true:
   (1) Forward Start is designed to be at-money-call.
   (2) Interest Rate and Volatility are constant.

Using Black-Scholes Model, give Forward Start time periods [t1,T1] and calls
are at the money at t=0 and t=t1, we would have call value c, c1 both are proportional to stock price S, S1. Therefore c1= c(S1/S).
Under risk-nerual, S1= Random variable but E[S1]*exp(-rt1]=S or E[S1]=exp(rt1]S 
==> value of forward start call at t=0 = exp(-rt1)E[c1]=exp(-rt1)cE[S1]/S=c

That is V-ForwardStart Call(t=0) = C expiring in T1-t1 days.

To really make interest rate and Volality constant, we will need to enter IR swap and Volatility Swap (Pay floating, receiv fixed)