Longstaff–Schwartz Exercise Decisions and American Option Boundaries
Summary
The document explains how least-squares Monte Carlo (LSM) values an American option and identifies when exercising is preferable to continuing to hold it. Working backward from terminal payoff, the method estimates each time step’s discounted continuation value by regressing on basis functions of the underlying price. At each step, the option value is the greater of immediate exercise value and estimated continuation value; the regression functions are specific to their respective time steps. The stopping rule exercises when immediate payoff reaches or exceeds the estimated continuation value.
It distinguishes the pathwise stopping decisions produced by Monte Carlo from a deterministic exercise boundary often defined in a PDE framework. It also describes computing continuation value by integrating against a transition density and splitting the integral at a point where exercise and continuation values meet. The answer focuses on American puts, despite the question mentioning calls, and does not provide numerical results or guidance on selecting basis functions, so regression accuracy and boundary quality remain implementation concerns.
Key ideas
- LSM estimates continuation value by regressing discounted future option values on basis functions of the current state.
- The regression is performed separately at each time step.
- Exercise when immediate payoff is at least as large as estimated continuation value.
- Monte Carlo stopping decisions are path dependent, while a PDE exercise boundary is expressed as a state-price curve.
- Transition-density integration offers an alternative to regression for estimating continuation value.
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# Estimating optimal exercise boundary for an American call by LSM method
# Estimating optimal exercise boundary for an American call by LSM method
I'm trying to derive optimal exercise boundary using LSM method and got some weird outcome. So, I evaluated an American call option by LSM method and now need to find the optimal exercise curve.
Do I understand it correct that it is necessary to do the following: on each time step I have to solve non-linear equation (Continuation value which is approximated by basis polynomials of some unknown price S_opt = S_opt - Strike)? And the coefficients of these basis polynomials are the same that I got previously (while I was regressing the discounted payoffs when I was evaluating the option)?
Could you please give me the cue?
Thank you
## Answer by ir7 (score 5)
https://quant.stackexchange.com/a/54846
With $V$ American option value, $H$ holding (aka continuation) value, and $B$ bank account value, we have:
$$V_N(S_N) = (K-S_N)^+$$
and for $i$ backwards from $N-1$ down to 0, we have:
$$ H_i(S_i) = \mathbf{E}\left[B_{i}B_{i+1}^{-1}V_{i+1}(S_{i+1})|S_i\right]$$
$$ V_i(S_i) = \max (K-S_i, H_i(S_i)) $$
The algorithm result is $V_0(S_0)$.
The different regression functions $r_i$ (linear combinations of basis functions) give the conditional expectations needed at every step $i$:
$$\mathbf{E}\left[B_{i}B_{i+1}^{-1}V_{i+1}(S_{i+1})|S_i\right] = r_i(S_i)$$
Optimal stopping time index $\eta$ taking values in $\{1,..., N \}$ is defined as: $$ \eta = \min \{k\geq 1 \mid K-S_k \geq r_k(S_k) \} \wedge N$$
Edit: What is the exercise boundary in the probabilistic (Monte Carlo) framework?
When solving for the optimal stopping time index $\eta$ above, one needs to introduce the probability state set $\Omega =\{\omega^1,...,\omega^J\}$, which in Monte Carlo context represents the labels of the primitive market variables fully (all the way to the end of the financial contract) simulated paths, as $\eta$, like all variables here, are functions on it:
$$ \eta (\omega^j)= \min \: \{k\geq 1 \mid K-S_k(\omega^j) \geq r_k(S_k(\omega^j)) \} \wedge N$$
The underlying price on path $\omega_j$ where one stops is $$ S_{\eta{(\omega_j)}} (\omega_j).$$
If we introduce a new variable $\Gamma: \{1,...,n\}\times \Omega \rightarrow \{0,1\}$ defined as:
$$ \Gamma (i, \omega) = 1 {\rm \: if \:} i > \eta(\omega), $$
and $0$ otherwise, then the (topological) boundary of the set $$\Gamma^{-1}(1) = \{(i,\omega) | i > \eta(\omega) \} $$ is called the exercise boundary.
Edit2: If you are referring to the PDE framework, then indeed the exercise boundary is defined as the deterministic curve $S^{\rm opt}_k$
$$ S^{\rm opt}_k = \inf \: \{x | (K - x)^+ = V_k(x) \} $$
Edit3: If one uses the conditional density $\phi_i(y\mid x)$ (transition from $S_i$ to $S_{i+1}$) to compute the conditional expectation for holding value by integration (rather than using regression function $r_i$), we have $$ H_i(x) = \mathbf{E}\left[B_{i}B_{i+1}^{-1}V_{i+1}(S_{i+1})|S_i =x\right]$$ $$ = B_{i}B_{i+1}^{-1}\int_0^\infty \max (K-x, H_{i+1}(x))\phi_i(y\mid x)dx $$ The integral calculation can be improved if we split its integration domain at $x^*$, the root of the equation: $$ K-x = H_{i+1}(x)$$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.