Put price characterisation

Question

I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models.

I am trying to understand the below two concepts:

1. Topic 8.3.3 Analytical Characterization of the Put price on Page 351

2. 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 1

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.

HTH