Black–Scholes Put Signs and Dynamic Replication
Summary
The document clarifies why the Black–Scholes put price has a positive discounted strike term and a negative stock term. The signs follow the put payoff: the holder benefits from the strike and is exposed negatively to the underlying price. Setting the underlying price to zero provides a simple check that the put value remains positive and equals the discounted strike in that limiting case.
It also corrects the idea that a put can generally be replicated by a fixed short-stock and lending position. Replication requires dynamically rebalancing the stock and cash holdings, and the direction of those trades can change over time. The explanation is conceptual rather than a worked hedge simulation, and it assumes the standard Black–Scholes framework with the stated rate and dividend yield inputs.
Key ideas
- The put price contains a positive discounted-strike contribution and a negative underlying contribution.
- The signs correspond to the put payoff, which increases with the strike and decreases with the terminal stock price.
- A zero underlying price is a useful limiting case for checking the put formula’s signs.
- Option replication generally requires dynamic rebalancing of stock and cash positions.
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# Understanding the relationship between the Black-Scholes formula and a replicating portfolio
# 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.
## Answer by Quantuple (score 3, accepted)
https://quant.stackexchange.com/a/29469
> It is my understanding that a replicating portfolio for a put involves short selling stock and lending money.
You cannot statically replicate an option. So this is not true in general, you'll need to re-balance your replicating portfolio (underlying + cash) dynamically if you want to replicate the option. This will imply sometimes buying stock and borrowing money.
> Basically I'm looking for clarification on the signs of the terms in the formula
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: $\max(K-S_T,0)$.
Extending this idea:
- 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): $$\max({\color{blue}{+K}}{\color{blue}{-S}}_T,0) \to P(K,T) = {\color{blue}{+K}}e^{-rT}N(-d_2) {\color{blue}{-S}} e^{-\delta T} N(-d_1)$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.