\begin{algorithm}[H]
\caption{Optimal Regress-later with Neural Networks (OPT-RLNN)}\label{alg-opt-rlnn}
\begin{algorithmic}[1]
\State Setup the target portfolio information (with strike $K$ and exercise points $\{t_0, t_1, t_2, \ldots, t_M\}$) and time-zero market data ($S_0$, $r$, $\sigma$)
\State Generate $S_{t_m}\left(\omega_j\right)$  for paths $j=1,\ldots,N, \, m = 0,\ldots,M$ 
\State $V_{t_M} \leftarrow h \left(S_{t_M}\right)$  evaluate option value for each path at $t_M$
\For{$m=M\ldots,1$}
	\State Initialize $\beta_{t_m}$ as per the proposed parameter initialisation (i.e. static hedge portfolio weights and constituent option strikes) in Section \ref{Parameter Initialisation}
       \State  $\beta_{t_m} \leftarrow \underset{\beta} {\mathrm{argmin}} \left(\frac{1}{N} \sum_{j = 1}^{N} \frac{1}{2} \left(V_{t_m}(\mathbf{S}_{t_m}(\omega_j))-G^{\beta}\left(\mathbf{S}_{t_m}(\omega_j)\right)\right)^2\right) = \underset{W, \boldsymbol{b}} {\mathrm{argmin}} \ L(t_m; W, \boldsymbol{b})$; Fitting the network with the proposed optimisation technique in Section \ref{Optimisation Methodology}
\For{$j=1,\ldots,N$}
\State $Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right) \leftarrow \mathbb{E}\left[\tilde{G}^{\beta_{t_m}} (S_{t_m}) \mid S_{t_{m-1}} (\omega_j)\right]$, which is the continuation value evaluated using Black-Scholes pricing model      
       \If{$h(S_{t_{m-1}}(\omega_j)) > Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right)$}
    			\State$V_{t_{m-1}}(S_{t_{m-1}}(\omega_j)) \leftarrow h(S_{t_{m-1}}(\omega_j))$
  	  \Else
    			\State $V_{t_{m-1}}(S_{t_{m-1}}(\omega_j)) \leftarrow Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right)$
      \EndIf
      \EndFor
    \EndFor
 
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[H]
\caption{Optimal Regress-later with Neural Networks (OPT-RLNN)}\begin{algorithmic}
[1]
\State Setup the target portfolio information (with strike $K$ and exercise points $\{t_0, t_1, t_2, \ldots, t_M\}$) and time-zero market data ($S_0$, $r$, $\sigma$)
\State Generate $S_{t_m}\left(\omega_j\right)$ for paths $j=1,\ldots,N, \, m = 0,\ldots,M$
\State $V_{t_M} \leftarrow h \left(S_{t_M}\right)$ evaluate option value for each path at $t_M$
\For{$m=M\ldots,1$}
\State Initialize $\beta_{t_m}$ as per the proposed parameter initialisation (i.e. static hedge portfolio weights and constituent option strikes) in Section \ref{Parameter Initialisation}
\State $\beta_{t_m} \leftarrow \underset{\beta} {\mathrm{argmin}} \left(\frac{1}{N} \sum_{j = 1}^{N} \frac{1}{2} \left(V_{t_m}(\mathbf{S}_{t_m}(\omega_j))-G^{\beta}\left(\mathbf{S}_{t_m}(\omega_j)\right)\right)^2\right) = \underset{W, \boldsymbol{b}} {\mathrm{argmin}} \ L(t_m; W, \boldsymbol{b})$; Fitting the network with the proposed optimisation technique in Section \ref{Optimisation Methodology}
\For{$j=1,\ldots,N$}
\State $Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right) \leftarrow \mathbb{E}\left[\tilde{G}^{\beta_{t_m}} (S_{t_m}) \mid S_{t_{m-1}} (\omega_j)\right]$, which is the continuation value evaluated using Black-Scholes pricing model
\If{$h(S_{t_{m-1}}(\omega_j)) > Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right)$}
\State$V_{t_{m-1}}(S_{t_{m-1}}(\omega_j)) \leftarrow h(S_{t_{m-1}}(\omega_j))$
\Else
\State $V_{t_{m-1}}(S_{t_{m-1}}(\omega_j)) \leftarrow Q_{t_{m-1}}\left(S_{t_{m-1}}(\omega_j)\right)$
\EndIf
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
{
    "title": "optimizing neural networks for bermudan option pricing: convergence   acceleration, future exposure evaluation and interpolation in counterparty   credit risk",
    "id": "2402.15936",
    "abstract": "this paper presents a monte-carlo-based artificial neural network framework for pricing bermudan options, offering several notable advantages. these advantages encompass the efficient static hedging of the target bermudan option and the effective generation of exposure profiles for risk management. we also introduce a novel optimisation algorithm designed to expedite the convergence of the neural network framework proposed by lokeshwar et al. (2022) supported by a comprehensive error convergence analysis. we conduct an extensive comparative analysis of the present value (pv) distribution under markovian and no-arbitrage assumptions. we compare the proposed neural network model in conjunction with the approach initially introduced by longstaff and schwartz (2001) and benchmark it against the cos model, the pricing model pioneered by fang and oosterlee (2009), across all bermudan exercise time points. additionally, we evaluate exposure profiles, including expected exposure and potential future exposure, generated by our proposed model and the longstaff-schwartz model, comparing them against the cos model. we also derive exposure profiles at finer non-standard grid points or risk horizons using the proposed approach, juxtaposed with the longstaff schwartz method with linear interpolation and benchmark against the cos method. in addition, we explore the effectiveness of various interpolation schemes within the context of the longstaff-schwartz method for generating exposures at finer grid horizons.",
    "categories": "q-fin.cp q-fin.pr q-fin.rm",
    "doi": "",
    "created": "2024-02-24",
    "updated": "",
    "authors": [
        "vikranth lokeshwar dhandapani",
        "shashi jain"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2402.15936"
}
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