Finite difference methods for (continuously) strike-resettable American options

Question

For simplicity, let us consider an American call/put with a continuously resettable strike price. Current time is $t=0$, maturity is at $t=T$, and the initial strike is $K_0$. We consider a "resettable period" $t\in [T_0, T]\subset [0, T]$. In this resettable period, if the stock price ever falls below $80\%$ (say) of the current strike, then the strike is reset so that the option becomes ATM. Formally, define a ratio $a>0$ as the reset floor ($a=80\%$ in the above example), a sequence of stopping times $\tau_{1,2,\cdots}$ denoting possible resets and a sequence of strikes $K_{0,1,2,...}$ denoting the initial and reset strikes: $$ \begin{align} \tau_1 &:=\inf_{t>T_0}\{t\le T\mid S_t\le aK_0\},\quad K_1:=S_{\tau_1}I(\tau_1<\infty) + K_0I(\tau_1=\infty)\\ \tau_2 &:=\inf_{t>\tau_1}\{t\le T\mid S_t\le aK_1\},\quad K_2:=S_{\tau_2}I(\tau_2<\infty)+K_1I(\tau_2=\infty)\\ \end{align} $$ and so on. (However, if an unlimited number of reset opportunities are too complicated to analyse, we may simplify the situation to only one reset opportunity, i.e. we only consider the first reset time $\tau_1$ and the first reset strike $K_1$.)

I want to use FD with the Crank-Nicolson scheme to solve the problem. Let's suppose I already know all the top/bottom/terminal boundary conditions, and the only thing left to do is roll back in time. For a regular American option, Crank-Nicolson can be easily implemented using a modified SOR (Successive Over-Relaxation), see Paul Wilmott On Quantitative Finance section 78.9.2. The basic idea is simply to compare, at each node being computed, the value of continuation against the value of immediate exercise, and take whichever is greater to be the value at this node.

But now, with continuous resettability of the strike, comparing values at each node is no longer as straightforward (at least in my opinion). What are the possible approaches to circumvent this difficulty? Thanks!

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EDIT: just to clarify, in reality the resettability is usually a right of the issuer or the holder, rather than an obligation. However, in certain cases, it can be determined that if both parties are rational then when the stock price triggers certain conditions it will be optimal for one party (or both) to exercise the right. Therefore, for the purpose of modelling, it is sometimes good enough to assume that when certain conditions are met, the reset isn't just allowed to happend but indeed must happen.

Answer 1

You need to add an auxiliary state variable that represents the current strike $K_t$, with dynamics $K_{t} = K_{t^-}$ if $S_t > 0.8 K_{t^-}$, $K_{t} = S_t$ if $S_t \leq 0.8 K_{t^-}$. You will get a jump/PDE with 2 state variables which you can then solve. Some people call that "1.5" PDE because the second state variable updates only depend on the first state variable.