A Derivative Shortcut for Checking Black–Scholes PDE Compliance
Summary
The question asks how to quickly determine whether a candidate claim price satisfies the Black–Scholes partial differential equation, without fully differentiating a complicated expression involving the normal cumulative distribution function. The answer uses the structure of the derivatives instead of calculating every term. For the candidate expression, its time derivative contains a term proportional to the claim price itself, while its first derivative with respect to the underlying does not; consequently, its second underlying-price derivative has no claim-price term either.
The PDE requires certain price-proportional terms to balance. Since the relevant term from the time derivative cannot be canceled by the second-derivative contribution, the candidate fails the equation. This is a targeted algebraic check that can reduce work when an expression has recognizable product structure. It applies to the particular candidate and stated parameters in the exercise; it is not a general test for arbitrary claims, and correct conclusions depend on using the PDE assumptions and dividend treatment appropriate to the claim.
Key ideas
- A candidate pricing function can sometimes be checked without fully expanding every derivative.
- Inspecting whether derivatives contain a term proportional to the original price can reveal an imbalance in the PDE.
- In the example, the time derivative has a price-proportional component, while the second underlying-price derivative does not.
- The shortcut rules out the candidate for the stated setup but does not replace full verification in general.
- The relevant Black–Scholes equation depends on assumptions such as the underlying’s dividend yield.
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Full text
# Is there a quick way to see why this claim $C(S, t)$ on $S$ does not satisfy the Black-Scholes PDE?
# Is there a quick way to see why this claim $C(S, t)$ on $S$ does not satisfy the Black-Scholes PDE?
I'm self-studying for an actuarial exam on financial economics and encountered the below practice exam problem.
An exam problem should typically take 5-6 minutes to complete, so I'm wondering if there is a "quick" way to confirm that answer choice (D) does not satisfy the Black-Scholes PDE.
Assuming for the moment that $C(S, t)$ does not pay dividends (which in my opinion cannot be assumed just from the information provided), the PDE implies that $r = 0.04$, $\delta = 0.02$ and $\sigma = 0.3$.
So I would think that any asset that has these parameters will satisfy the PDE. Let's check:
(A) is the price of a risk-free bond with maturity value 1.
(B) is the price of a cash-or-nothing call that pays 1 when the stock price is above 100.
(C) is the price of a cash-or-nothing put that pays 1 when the stock is below 100.
(E) is the price of an asset-or-nothing put that pays the stock when the stock price is below 100.
By elimination, that leaves (D) as the claim that does not satisfy the PDE.
But what if I wanted to show that (D) cannot satisfy the PDE? I can only think to find $C_s$, $C_{ss}$ and $C_t$. However, this would be messy as $C(S, t)$ would require differentiating $N(d_1)$. Is there an quicker or better way of convincing myself that (D) cannot satisfy the equation?
## Answer by LocalVolatility (score 2, accepted)
https://quant.stackexchange.com/a/32394
You already associated the valuation function ins A, B, C and E with the corresponding products. In order to explicitly exclude D, you don't have to compute all the derivatives but just note that
\begin{eqnarray} C_S & = & e^{-0.02 (T - t)} \mathcal{N}' \left( d_1 \right) \frac{\partial d_1}{\partial S}\\ C_t & = & 0.02 \underbrace{e^{-0.02 (T - t)} \mathcal{N} \left( d_1 \right)}_{=C(S, t)} + e^{-0.02 (T - t)} \mathcal{N}' \left( d_1 \right) \frac{\partial d_1}{\partial t}. \end{eqnarray}
From the expression for $C_S$ you can infer that the expression for $C_{SS}$ does not contain a $C(S, t)$-term. Thus, you can conclude that the $C(S, t)$-terms in the PDE coming from $C_t$ and the r.h.s. don't cancel each other out and you are done.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.