Modeling Continuously Resettable American Options with a Strike State
Summary
The document describes an American call or put whose strike resets to the stock price when the stock falls to a specified fraction of the current strike during a defined period. Reset events can occur repeatedly, although the problem statement allows restricting the model to a single reset. It asks how to apply a Crank-Nicolson finite-difference method when a reset changes the option's strike state as well as its value.
The proposed approach adds the current strike as an auxiliary state variable. The option value then depends on both stock price and strike, with strike remaining unchanged above the reset threshold and jumping to the stock price when the threshold is reached. This produces a two-state jump/PDE problem for numerical solution. The account is a modeling suggestion rather than a numerical recipe: it provides no grid design, boundary treatment, convergence evidence, or implementation results. It also notes that real reset rights may be optional, while the simplified model assumes a trigger makes resetting mandatory.
Key ideas
- A continuous reset feature changes the strike when the stock crosses a threshold tied to the current strike.
- Repeated resets may be modeled, or the problem may be simplified to one reset event.
- The current strike should be represented as an auxiliary state variable alongside the stock price.
- The resulting valuation problem has two state variables and includes strike jumps at the trigger.
- The suggested method does not specify discretization details or provide numerical validation.
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# Finite difference methods for (continuously) strike-resettable American options
# Finite difference methods for (continuously) strike-resettable American options
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!
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 by Antoine Conze (score 1, accepted)
https://quant.stackexchange.com/a/44817
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.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.