Pricing American Options with LSM Without Deriving a PDE
Summary
The document explains that Longstaff–Schwartz least squares Monte Carlo can value an American option without first deriving its pricing PDE. The method simulates paths from the underlying asset’s stochastic process, then works backward across exercise dates. At each date, it compares immediate exercise value with the discounted expected value of continuing.
LSM estimates continuation value by regression on selected projection functions, avoiding a separate Monte Carlo simulation from every simulated state. The stated requirements are a model for generating asset paths and suitable regression functions. The discussion gives the dynamic-programming recursion as its rationale, but no numerical example or validation results. It does not explain how to choose regression functions, discretize a jump-diffusion with time-varying jump frequency, or assess approximation error; those choices still matter for implementation quality.
Key ideas
- LSM can estimate American option value without deriving a pricing PDE.
- The underlying process is needed to generate simulated paths.
- Backward recursion compares immediate exercise value with discounted continuation value.
- Regression estimates continuation value across simulated states.
- Projection functions and time discretization affect the implementation.
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Full text
# Do we need to derive the PDE for the option price when applying Least Squares Monte Carlo?
# Do we need to derive the PDE for the option price when applying Least Squares Monte Carlo?
I want to price an American call option based on an underlying that follows a jump-diffusion process with an inhomogeneous jump frequency function.
My mathematical skills are not sufficient to derive the respective PDE for the option price (due to complexity, there we cannot find a closed-form solution anyway).
I want to apply the LSM by Longstaff and Schwartz (2001) to find the optimal exercise strategy and calculate the value of an american call option.
To do this, it still necessary to derive the respective PDE? If yes, why?
## Answer by Antoine Conze (score 5)
https://quant.stackexchange.com/a/37427
You do not need the PDE to implement the LSM algorithm.
The $T$ maturity American call price on time $t$ is $$v_t = \max_{\tau} E_t\left[e^{-\int_t^\tau r(u) du} (S_\tau - K)^+\right]$$ where the max is over all the stopping times $t \leq \tau \leq T$.
After discretizing time along a discrete time line $t_k$, this leads to the recursion $$ v_{t_k} = \max\left\{(S_{t_k} - K)^+, E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right] \right\} $$ where $E_{t_k}\left[v_{t_{k+1}} e^{-\int_{t_k}^{t_{k+1}} r(u) du}\right]$ is the option continuation value (the value if you choose not to exercise on $t_k$).
The Longstaff and Schwartz LSM algorithm is a "trick" to compute the continuation value at any point in the Monte Carlo simulation without having to resort to a new Monte Carlo simulation that would originate at that point.
As you can see you do not need a PDE for $v$ to implement the algorithm. The only thing you require is the SDE for $S_t$ for generating the Monte Carlo paths, as well as appropriate projection functions for the LS algorithm itself (there are many good references on the latter, for instance Monte Carlo Methods in Finance - Peter Jäckel).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.