PDE and Optimal Stopping Views of Perpetual American Put Pricing
Summary
The document contrasts two mathematical characterizations of the perpetual American put discussed in a stochastic calculus text. The analytical treatment describes the option value through partial differential inequalities and linear complementary conditions. This frames pricing as a boundary-value problem related to the Black–Scholes–Merton equation, with the exercise boundary separating continuation from immediate exercise.
The probabilistic treatment instead casts the value as an optimal stopping problem. Under the risk-neutral measure, the discounted option value is bounded below by the discounted exercise payoff and behaves as a supermartingale; an optimal stopping time specifies when exercise is appropriate. The answer presents these approaches as different languages for the same valuation problem, with analytical and numerical methods applicable to the PDE formulation and stochastic-process tools applicable to the stopping formulation. The example is mathematically instructive, but perpetual American puts are described as uncommon in actual markets, so the discussion is chiefly a framework for understanding American exercise valuation rather than a trading strategy.
Key ideas
- The analytical characterization expresses American exercise pricing as partial differential inequalities with complementary conditions.
- The probabilistic characterization formulates the option value as an optimal stopping problem.
- A discounted American option value is a supermartingale and remains a martingale until the optimal exercise time.
- Both formulations describe the same value through different mathematical tools and conditions.
Tags
Full text
# Put price characterisation
# Put price characterisation
I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models.
I am trying to understand the below two concepts:
- Topic 8.3.3 Analytical Characterization of the Put price on Page 351
- Topic 8.3.4 Probabilistic Characterization of the Put price on Page 353
Can someone please help me understand the difference between the above two approaches? I have notes but I am not able to summarize the difference between the approaches, how are those two approaches different.
Thank you
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/71625
All of 8.3 is concerned with the value of the Perpetual American Put. Although there are few or no PAP's actually traded in the real world, it is an interesting topic because it is an example of an American exercise security that is well understood mathematically. So in the book it serves as an example of the valuation of such securities, which are considerably more complicated than European exercise ones.
In section 8.3.3 we look at partial derivatives of the value function V, leading to a "system of partial differential inequalities" 8.3.18 8.3.19 8.3.20 called the Linear Complementary Conditions. In principle all the tools (including numerical tools) for addressing partial differential inequalities and Linear Complementarity Problems can be used to find the solution V (example). In the European exercise case we found the Black Scholes Merton PDE, here we uncover a generalization of BSM-PDE for American exercise.
In section 8.3.4 we look instead at the problem as an optimal stopping problem in the context of stochastic processes. V is the solution of such a problem, so again the tools of the trade for such problems can be applied (examples). Key insights in this section include "Discounted European option prices are martingales under the risk-neutral probability measure. Discounted American option prices are martingales up to the time they should be exercised. If they are not exercised when they should be, they tend downward." So the discounted price is a supermartingale. In conclusion the stochastic process for V satisfies 3 conditions, listed at the end pf this section: V is greater than or equal the immediate exercise value, $e^{-r T}V(t)$ is a supermartingale, and there exists an optimal stopping time $\tau_*$. As in the previous section 3 conditions were found but they are expressed in very different language.
HTHShown 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.