State Variables for Barrier and Lookback Option PDEs
Summary
The document explains why barrier and lookback options can require different state variables in a partial differential equation. For a knock-out barrier option that is still alive, the current spot price is sufficient to describe the future payoff distribution: whether the barrier was crossed earlier is captured by the option’s survival status. Earlier highs do not otherwise enter the payoff.
A lookback payoff directly depends on the maximum price reached over the option’s life, so the running maximum must be tracked alongside spot, giving a two-variable pricing problem. The answer notes that homogeneity in spot and the running maximum can reduce the lookback problem to a single transformed variable. This distinction relies on the specific payoff structures discussed; other barrier features or path-dependent payoffs may require additional state information.
Key ideas
- A surviving knock-out barrier option can be valued using current spot as its state variable.
- Past barrier events matter through whether the option remains alive.
- A lookback payoff depends directly on the running maximum, which must be tracked as a second state variable.
- Homogeneity can reduce the dimensionality of the lookback pricing problem.
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# PDE of barrier and lookback options
# PDE of barrier and lookback options
In Shreve's book, he obtain the PDE of barrier option by
Payment function $$V(T) = (S(T) - K)^+\mathbb{II}_{\{S_{\textrm{max}}(T) > B\}}$$ Then use the risk neutral pricing formula and Markov property of $S(t),$ we have its value $$V(t) = v(t,S(t))$$ for some function $v(t,x),$ then we can have the PDE of barrier option respect to $v(t,x).$
But in lookback option, he regards $S_{\textrm{max}}(t)$ as a new variable $y$ and value function becomes $v(t,x,y),$ although finally $v(t,x,y)$ can be changed into one variable $u(t,z).$
So my question is, why there is one variable in barrier but two variables in lookback, or they are just equivalent?
## Answer by LocalVolatility (score 5, accepted)
https://quant.stackexchange.com/a/31968
The difference is that the barrier option is weakly path dependent while the lookback option is strongly path dependent.
In case of a knock-out barrier option, conditional on the option being alive at the pricing time you don't need to carry any additional state variables except for the current asset price. The payoff doesn't directly depend on the level of $S_{\text{max}}(0, T)$ except through the survival indicator. Given the current spot $S(t)$ and conditional on no prior knock-out you can compute the joint density of $\left( S(T), S_{\max}(t, T) \right)$. The information on $S_{\max}(0, t)$ is irrelevant for the payoff in that case.
In case of a lookback option, the payoff directly depends on $S_{\max}(0, T)$. You thus need to introduce a second state variable and obtain a two-dimensional PDE. It turns out that the dimension of the latter can be reduced in case of a lookback option since the price is homogeneous of degree one in the two state variables $S(t)$ and $S_{\max}(0, t)$.
See also Chapter 22 "An Introduction to Exotic and Path-Dependent Derivatives" in Wilmott (2006) "Paul Wilmott on Quantitative Finance", Wiley.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.