Financial Meaning of Stock and Cash Solutions to Black–Scholes
Summary
The document explains why the stock price and the value of interest-bearing cash satisfy the Black–Scholes pricing PDE. The stock itself is a traded asset whose value is the underlying price, while a bank balance grows deterministically at the risk-free rate. Neither example requires an option payoff or other optionality.
The central idea is that, under the standard Black–Scholes assumptions, different contracts can satisfy the same pricing equation while their boundary or terminal conditions determine their distinct prices. For a deterministic value depending only on time, the PDE reduces to the risk-free growth equation; any other deterministic growth rate would conflict with the no-arbitrage premise. The explanation is qualitative and relies on the standard model assumptions. It does not derive the PDE or discuss how the interpretation changes with market frictions, dividends, or alternative models.
Key ideas
- The stock price is a solution because the underlying asset itself is a traded claim.
- Interest-bearing cash satisfies the PDE by growing at the risk-free rate.
- A shared pricing PDE can produce different contract values through different boundary and terminal conditions.
- The deterministic-value argument depends on the standard no-arbitrage Black–Scholes framework.
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Full text
# Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?
# Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?
It's readily verified mathematically that $V=S$ and $V=e^{rt}$ are solutions to the Black-Scholes PDE $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2} \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0$. How can we motivate/explain this from a financial perspective?
## Answer by Magic is in the chain (score 4, accepted)
https://quant.stackexchange.com/a/42257
Under the standard assumptions, generally speaking, any contract that depends on the current values of t and S, and which are paid for at the start satisfy this PDE. In the financial context, the boundary conditions would be different for the different contracts, so the solutions of the PDE would be different for the different contracts. Thus different contracts would be identified by different boundary conditions, and these conditions would determine the solution.
In summary, different payoffs, same PDE, different prices.
Re-comment, your commentary is correct; however for deterministic function of t only, the PDE reduces to $\frac{\partial V}{\partial t}=rV$ and this has the solution that you mentioned in your question. Anything deterministic must have this form, otherwise there is an arbitrage. In other words, risk free or deterministic must grow at this rate.
## Answer by vonjd (score 4)
https://quant.stackexchange.com/a/42258
These are well known trivial solutions to the Black-Scholes PDE. The first one is just the price of the underlying stock and the second is interest bearing money in a bank. These are trivially true because there is no optionality involved (which is expressed in the boundary and terminal condition of the respective contract to price).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.