Separating Black–Scholes PDE Terms in a Call Price Decomposition
Summary
The document examines a proposed derivation of the Black–Scholes call formula. It changes variables to log spot, writes the call price as a combination of an asset-or-nothing term and a discounted cash-or-nothing term, and substitutes that form into the pricing PDE. The resulting equation is a weighted sum of two differential expressions. The question is whether this single equation implies that each expression must vanish on its own.
An answer initially argues that varying the strike forces the two parts to zero separately, linking them to binary option prices. However, an edit explicitly retracts that reasoning and notes that alternative choices of the component functions need not satisfy separate PDEs. Thus the exchange is useful as a caution about assumptions in decompositions: one equation involving multiple unknown functions does not by itself establish independent equations. It does not provide the promised corrected derivation, so it should not be treated as a complete proof of the Black–Scholes formula.
Key ideas
- A proposed call-price decomposition leads to a weighted sum of two PDE expressions.
- The original answer claims strike variation forces each expression to vanish separately.
- An edit retracts that claim and acknowledges that alternative component functions can satisfy the combined equation.
- The exchange highlights the need for additional assumptions before splitting a PDE into separate equations.
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Full text
# Alternative derivation of the Black Scholes formula
# Alternative derivation of the Black Scholes formula
I encountered the following derivation of the Black Scholes formula for call price. It may very well be an established method but I had never seen it before so I called it an alternative derivation.
I immediately start with the slightly transformed version of the Black Scholes PDE. $z$ denoting the log spot price and $C$ denoting the call price we have
$$\frac{\partial C}{\partial t} + \left(r-\frac{1}{2}\sigma^2\right)\frac{\partial C}{\partial z} + \frac{1}{2}\sigma^2\frac{\partial^2 C}{\partial z^2} = rC$$ with the boundary condition $C(T,z) = \max(e^z-K,0)$ where $K$ is strike and $T$ is expiry.
The derivation then assumes the following form for $C$
$$C(t,z) = e^zP(t,z) - Ke^{-r(T-t)}Q(t,z)$$
and substitutes the partial derivatives into the original PDE to obtain
$$e^z\left(\frac{\partial P}{\partial t} + \left(r+\frac{1}{2}\sigma^2\right)\frac{\partial P}{\partial z} + \frac{1}{2}\sigma^2\frac{\partial^2 P}{\partial z^2}\right) - Ke^{-r(T-t)}\left(\frac{\partial Q}{\partial t} + \left(r-\frac{1}{2}\sigma^2\right)\frac{\partial Q}{\partial z} + \frac{1}{2}\sigma^2\frac{\partial^2 Q}{\partial z^2}\right)= 0$$
The bit that I don't get is that according to the derivation this somehow implies the following.
$$\frac{\partial P}{\partial t} + \left(r+\frac{1}{2}\sigma^2\right)\frac{\partial P}{\partial z} + \frac{1}{2}\sigma^2\frac{\partial^2 P}{\partial z^2} = 0$$ $$\frac{\partial Q}{\partial t} + \left(r-\frac{1}{2}\sigma^2\right)\frac{\partial Q}{\partial z} + \frac{1}{2}\sigma^2\frac{\partial^2 Q}{\partial z^2} = 0$$
Why would the individual terms be zero?
## Answer by RLK (score 4)
https://quant.stackexchange.com/a/42720
The option pricing formula must satisfy the PDE you have derived for all values of $K$. The only way this can be the case is if the two parts that you separate are both equal to zero. Suppose the joint PDE (before you separate it into two) is satisfied for some value of $K$. But suppose that the $Q$-part in parentheses on the second line is not zero. Change the value of $K$, and your PDE will no longer be satisfied. So the second line of your PDE must be zero. Since the second line must be zero, the first line must also be zero.
The solution to the $P$ differential equation is related to the price of an asset-or-nothing call, and the solution of the $Q$ differential equation is related to the price of a cash-or-nothing option.
Edited to add - this is actually not right, there are alternate $P$ and $Q$ such that the PDEs are not satisfied separately. Working on a fix . . .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.