# Understanding the relationship between the Black-Scholes formula and a replicating portfolio

I'm self-studying and I'm considering the below example. The specific example is not especially relevant, but I included it for reference.

I'm trying to understand the relationship between a replicating portfolio and the Black-Scholes equation.

It is my understanding that a replicating portfolio for a put involves short selling stock and lending money. There would be a positive cash flow of $Se^{-\delta T} N(-d_1)$ and a negative cashflow of $Ke^{-rT} N(-d_2).$

However, the Black-Scholes formula for a put is: $P = Ke^{-rT}N(-d_2) - Se^{-\delta T} N(-d_1)$.

This formula suggests that a long position in a put is a positive cashflow of $Ke^{-rT} N(-d_2)$ and a negative cashflow of $Se^{-\delta T} N(-d_1)$.

So wouldn't the replicating portfolio create a short position put, while the Black Scholes formula provides a long position put (since the signs are swapped comparing the replicating portfolio to the Black Scholes formula for a put)?

Should examples like the one below clarify what position the put is in?

Basically I'm looking for clarification on the signs of the terms in the formula.

• I think you should look for a better book. amazon.com/… – MJ73550 Aug 3 '16 at 16:22

Take the case $S=0$ and you'll see that the signs $$P(K,T) = Ke^{-rT}N(-d_2) - Se^{-\delta T} N(-d_1)$$ are indeed correct since in that case you need to find $P(K,T) = K e^{-rT} > 0$, because the payout of a put option is: $\max(K-S_T,0)$.
• Call (Payoff $\to$ Price): $$\max({\color{blue}{+S}}_T{\color{blue}{-K}},0) \to C(K,T) = {\color{blue}{+S}}e^{-\delta T}\phi(d_1) {\color{blue}{-K}} e^{-rT}\phi(d_2)$$
• Put (Payoff $\to$ Price): $$\max({\color{blue}{+K}}{\color{blue}{-S}}_T,0) \to P(K,T) = {\color{blue}{+K}}e^{-rT}N(-d_2) {\color{blue}{-S}} e^{-\delta T} N(-d_1)$$