By Andreas J. Grau
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Additional resources for Applications of Least-Squares Regressions to Pricing and Hedging of Financial Derivatives
37), we get Ai 19 For a European call option S m = e10σ i Vj+1 − Vji ∂Vτ ds ≈ Ai . 21]. 4 Numerical Methods for Option Valuation 31 All the above equations finally lead to Ai i − Vji Vj+1 ∆τ = σ2 i 2 (S ) 2 V i+1 − V i V i−1 − V i + i i+1 i S −S S − S i−1 + rS i V i+1 − V i−1 − rV i Ai 2 where we have to define the time level of the right-hand side. Choosing a time level j for the right hand side, we get a so called explicit discretization; choosing a level of j + 1, the discretization is called implicit.
05. Different numbers of basis functions are used within the 50 time steps Least-Squares Monte Carlo. 66761. 3 Direct PDE Methods There are several direct PDE solver methods. The common ones include finite differences [103, 117], finite elements  and finite volume methods . Other methods are Laplace and Fourier transform methods  as well as meshless methods . In the following, we present a simple finite element method, which is sufficient for our purposes as a pricing engine. In some cases, this thesis will refer to more advanced techniques.
If we now focus on column D, where both functions p˜StT (s) and f˜(s) are estimated from the sample, we see that the Std is very close to the value of the Reference MC. Again, this is expected because besides smoothing, no additional information was added to the valuation process. The question arises, how the Std values of the different methods are connected. In the following we give some empirical intuition for the interdependency of the different Monte Carlo errors using the obtained numerical values.