Path Dependence in Derivatives and Valuation
Summary
The document offers two perspectives on classifying path dependence in derivatives. One practitioner-oriented criterion treats a payoff as non-path-dependent when its present value can be written as a discounted expectation at a single future tenor. Under this view, American options differ because their value involves optimizing over possible exercise times, while barriers and interim fixings also introduce dependence on events before maturity.
Some path-dependent products can still be modeled with limited added state information, such as a running average. A second answer associates path dependence with non-ergodic stochastic processes and discusses how a Markov chain's long-run behavior may depend on its starting distribution or periodicity. These answers use different conceptual frames and do not establish one universal formal definition. The discussion is therefore useful as an introduction, while more precise classification depends on the model and purpose.
Key ideas
- A single-tenor discounted expectation is offered as a practical criterion for non-path-dependent valuation.
- American options require optimization over exercise times, unlike ordinary fixed-maturity valuation.
- Barriers and interim fixings can make value depend on events before maturity.
- Some path dependence can be represented by adding state variables such as a running average.
- A separate answer links path dependence to non-ergodic stochastic processes, a distinct framing.
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Full text
# When can a derivative be considered to be path dependant?
# When can a derivative be considered to be path dependant?
The typical example of path dependant derivatives are knock-ins and knock-outs. At the same time vanilla American options can also be considered to be highly path dependant.
Does a more or less formal defintion/classification of "path dependency" exist ?
## Answer by Brian B (score 4)
https://quant.stackexchange.com/a/11022
For practitioners, a derivative is not path-dependent if its value can be expressed as an expectation of discounted future values at some specific tenor $T$
$$ V(0) = E\left[ \left. V(T) \exp{\left(-\int_0^T r(s)ds\right)} \right| {\cal{I}_0} \right] $$
Obviously this is convenient when it happens because one only needs to worry about probability densities inside the expectation for the single tenor $T$.
American-exercise options fail to meet this criterion since their value depends on an exercise strategy, written here as a stopping time $\tau$
$$ A(0) = \sup_{\tau \leq T} E\left[ \left. V(\tau) \exp{\left(-\int_0^\tau r(s)ds\right)} \right| {\cal{I}_0} \right] $$
Other options fail to meet the criterion because their value depends on fixings $t<T$, or barrier conditions, etc. etc. In many such cases, the path dependence is weak, in the sense that one can introduce a single extra dimension to the SDE/PDE, for example a current tabulation of running average price, and solve accordingly.
## Answer by Jacob M. Morley (score 0)
https://quant.stackexchange.com/a/11017
A stochastic process that is non-ergodic is inherently "path dependent." You can think of this a number of ways, but for me, most intuitively in the MC context.
A Markov chain is non-ergodic if it is not positive recurrent or if it is periodic. Alternatively, as the transitions in the chain increase, a non-ergodic chain's probability measure will not be greater than zero AND independent of its initial probability distribution (more on this in the famed Feller II). That is, an ergodic MC can eventually reach any other state with some positive probability.
Another useful source might be Omri Sarig's notes on ergodicity, depending on level of formality you are looking for.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.