When Option Values Reduce from a PDE to an ODE
Summary
The document explains why option values generally depend on both the underlying price and time, so they are described by a partial differential equation rather than an ordinary differential equation. It then gives perpetual American options as an important exception: with no expiry date, the value can be time-independent, reducing the Black–Scholes equation to an ODE in the underlying price.
For a perpetual American put, solving the ODE yields an exercise threshold and a value function, illustrating how an optimal stopping boundary enters the solution. The note also mentions bonds and Poisson-model credit default swaps as simpler cases governed by time-based ODEs with suitable boundary conditions. These examples clarify that an ODE is possible when the problem's structure removes a state variable; the result does not extend to ordinary finite-maturity options, whose value usually varies with both price and time.
Key ideas
- Most finite-maturity option values depend on both the underlying price and time, leading to a PDE.
- A perpetual American option can be time-independent and described by an ODE in the underlying price.
- The perpetual American put solution includes a price threshold for exercise.
- Bond values and Poisson-model CDS values can also satisfy ODEs with appropriate boundary conditions.
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Full text
# Can option prices be characterised by an ODE?
# Can option prices be characterised by an ODE?
If a stock price, $S(t)$, is governed by a geometric brownian motion. Is it possible to characterise the value of an option $V(S,t)$ as an ODE rather than a PDE (given $S$ is itself a function of $t$)? Hence is it possible to write a closed form solution as just a function of $t$?
Apologies if this is a stupid question (I'm new to QF).
## Answer by Brian B (score 8, accepted)
https://quant.stackexchange.com/a/9066
It is not possible for what most people think of as options, but there are classes of options for which an ODE is used.
For a nontrivial example, think of perpetual American-exercise options. Because of perpetual exercise, the option value is independent of time. In place of the Black-Scholes PDE
$$ \frac{\partial f}{\partial t} = \frac12 \sigma^2 x^2 f^{\prime \prime} + r x f^\prime-rf $$
we obtain the time-homogeneous ODE
$$ 0= \frac12 \sigma^2 x^2 f^{\prime \prime} + r x f^\prime-rf $$
Solving this ODE, one finds there is a barrier, $x^\star$, beyond which a perpetual American put should be executed. The solution is a relatively simple function
$$ K\left( \frac{K}{S} \left( 1-\frac{2r}{2r+\sigma^2} \right) \right)^{2r/\sigma^2} $$
This solution was, as far as I can tell, first derived by McKean in 1965. As you can tell, it mainly works because we were able to remove one of the (underlying price, time) variable from the ODE. Most options quite clearly have value that depends on both.
More trivial examples include bonds, which don't have any optionality to speak of but do follow the ODE
$$ \frac{dB}{dt}= -(r+h) B $$
and CDS which in the Poisson model follow the same ODE with different boundary conditions.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.