Sequence to remember Black-Scholes-Merton model 
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For a call option:

  Step 1: draw a time line

              |---------------------|
              0                     T

  Step 2: on maturity, the payoff of call = Max(sT-X,0)

  Step 3: take out the parentheses
          since we would consider      sT-X
          distribution later 
                                      sT
       s0 <-------------<--------------|
                                       X
       exp(-rT)X<-------<--------------|

  Step 4: consider present values

          S0    -   exp(-rT)X

  Step 5: add distribution in front of those two values

        call = S0 * N()  - exp(-r*T)*X* N()

  N() is the cumulative standard normal distribution. 
      It will be defined soon. 

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Nobel prize winning formula
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Black-Scholes-Merton call option (European)
   input value: S, X, T, r, sigma
         S  : today's stock price
         X  : exercise price (strike price)
         T  : maturity day in years
         r  : continuously compounded risk-free rate
       sigma: volatility (standard deviation of stock returns)

         ln(S/X) + (r+ 0.5*sigma^2)T
   d1= ------------------------------
             sigma* sqrt(T)

   d2 = d1 - sigma * sqrt(T)

   call = S*N(d1) - exp(-r*T)*N(d2)

          where N(x) =normdist(x,0,1,TRUE)

   put = exp(-r*T)*N(-d2) - S*N(-d1)

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Put-call parity       
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First, let's look at the logic.
 Portfolio 1: cash plus a call option 
              assume the exercise price is X
              payoff at the maturity
                 if sT>x   we exercise our call get stock 
                 if sT<=x  we let the option expires
                               i.e., keep our money X
             the value of our portfolio is
                 max(sT,X)

 Portfolio 2: one share plus a put option 
              assume the exercise price is X
              payoff at the maturity
                 if sT<=x  we exercise our put to have X
                 if sT>x  we let the option expires
                               i.e., keep the stock
             the value of our portfolio is
                 max(sT,X)

  Since the final values of those two portfolios are the same, 
    their present values should be the same as well. 

         C + exp(r*T)*X = p + S
     
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