# What class of derivatives satisfy the Black-Scholes PDE?

The title pretty much sums up the question, but I will provide some context.

There is a large class of derivatives—such as those the payoffs from which depend only on the share price at maturity—which do satisfy the standard Black-Scholes PDE. At the same time, there are several path-dependent derivatives, such as Asian and Lookback options, which do not satisfy the standard Black-Scholes PDE.

However, there is some grey area: Barrier options, which are clearly path-dependent, do in fact satisfy the standard Black-Scholes PDE.

An additional layer of complexity creeps in from American-style options where the contract can be exercised before maturity. I am actually not sure if American puts satisfy the standard Black-Scholes PDE; I know that American calls do, because it's never optimal to exercise an American call early—assuming no dividends on the share—which renders it equivalent to a European call.

All of this begs the original question: How do we know if a particular derivative satisfies the standard Black-Scholes PDE?

## 2 Answers

I think I have found the answer to my question: While deriving the Black-Scholes PDE, we write out the derivative price $$f$$ as a function of 2 things—the current time $$t$$ and the price of the underlying $$S_t$$. This is not to say that $$f$$ doesn't depend on other factors; it clearly depends on 5 other inputs $$(r, \sigma, q, K, T)$$. But the reason for writing out the derivative price as $$f(t, S_t)$$ is that $$t$$ and $$S_t$$ are the inputs which change with time—the other 5 inputs being constant.

Choosing to explicitly show the dependence on $$t$$ and $$S_t$$, while side-tracking the dependence on the other 5 inputs plays a crucial role: it reminds us that the change in the derivative price $$df$$ arises only out of the changes in time $$dt$$ and the changes in the price of the underlying $$dS_t$$.

So to answer the question: the price of a derivative will depend on multiple inputs. But if only two of those inputs—$$t$$ and $$S_t$$—change as we move ahead in time will the derivative price satisfy the Black-Scholes PDE. Let's look at some standard examples to clarify this thought.

1) In our model, the price of standard European options only change due to changes in $$t$$ and $$S_t$$, so they must satisfy the Black-Scholes PDE.

2) Barrier options depend on an additional input: the barrier level. Even though there is an extra input involved, we ask ourselves 'Does this input change as we move ahead in time?'. The answer is a resounding No, which makes us conclude—just like point (1)—that only two of the inputs, namely $$t$$ and $$S_t$$ are changing, making the price of a barrier option satisfy the Black-Scholes PDE.

3) Asian options also depend on an additional input: the average share price $$A_t$$ upto the current time $$t$$. Now this is an input—unlike the constant barrier level in point (2)—that does change as we move ahead in time. So the price of an Asian option no longer satisfies the Black-Scholes PDE. In order to find a PDE for Asian options, we will have to write out $$f$$ as a function of 3 inputs: $$(t, S_t, A_t)$$. Consequently, while finding the differential $$df$$, we will also have to consider $$dA_t$$—this will change the form of the PDE.

Do you have proof of your proposed grey area? While a standard european option clearly falls under the purview of BS, it's not clear to me that barrier options do as well.

For example, consider a vanilla call written on some underlying, S, with strike, K, and time to expiry, t, ...and an up-and-in call with the same terms outside of the barrier feature, set at price B. Even assuming the two options are worth the same for S < K and S > B, they obviously aren't worth the same for K < S < B, despite having the same BS inputs. Thus their expected payoff can't both be represented accurately using BS.

• A "PDE problem" consists of a PDE and also a domain, a boundary condition, and an initial condition. Barrier options satisfy the Black Scholes PDE, and the barrier features are accomodated by changing the other 3 things. Google "pde methods barrier options". Aug 30, 2019 at 1:38
• I think you misunderstood my question, Chris. The price of a barrier option will surely be different from the price of the corresponding vanilla option—of course, assuming that the barrier hasn't been triggered yet. But both the prices still satisfy the same PDE, with the difference coming in through the solution domain and the boundary conditions. Thank you for sharing your thoughts anyway! Aug 30, 2019 at 4:35