State Variables and the Black-Scholes PDE for Options
Summary
The document explores which derivatives can be described by the standard Black-Scholes PDE. Its main explanation is that the PDE applies when the derivative’s value can be represented using time and the current underlying price as the changing state variables, with other inputs held fixed. It uses European options and barrier options as examples, arguing that a fixed barrier level does not itself evolve over time.
By contrast, an Asian option depends on a running average that changes as the underlying moves, so that average must be added as a state variable and the pricing equation expanded. The discussion also raises early exercise for American options but does not resolve the American put case. A second answer challenges the barrier example, highlighting that the barrier feature changes option values and must be reflected in the contract’s boundary conditions. The exchange is illustrative rather than a complete derivation; path dependence may require additional state variables or boundary conditions, not simply exclusion from PDE methods.
Key ideas
- A pricing PDE tracks the state variables that evolve over time in the model.
- A fixed contract parameter, such as a barrier level, does not necessarily need to be a dynamic state variable.
- An Asian option’s running average evolves and must be included in its pricing state.
- Barrier options require their feature to be represented in suitable boundary conditions.
- The document raises but does not settle how early exercise affects the standard equation for American puts.
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Full text
# What class of derivatives satisfy the Black-Scholes PDE? # 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? ## Answer by Dhruv Gupta (score 2) https://quant.stackexchange.com/a/49035 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. ## Answer by Chris (score 0) https://quant.stackexchange.com/a/47399 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.
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