Monte Carlo on a single sample

In this paper, we consider a Monte Carlo simulation method (MinMC) that approximates prices and risk measures for a range $Γ$ of model parameters at once. The simulation method that we study has recently gained popularity [HS20, FPP22, BDG24], and we provide a theoretical framework and convergence rates for it. In particular, we show that sample-based approximations to $\mathbb{E}_θ[X]$, where $θ$ denotes the model and $\mathbb{E}_θ$ the expectation with respect to the distribution $P_θ$ of the model $θ$, can be obtained across all $θ\in Γ$ by minimizing a map $V:H\rightarrow \mathbb{R}$ with $H$ a suitable function space. The minimization can be achieved easily by fitting a standard feedforward neural network with stochastic gradient descent. We show that MinMC, which uses only one sample for each model, significantly outperforms a traditional Monte Carlo method performed for multiple values of $θ$, which are subsequently interpolated. Our case study suggests that MinMC might serve as a new benchmark for parameter-dependent Monte Carlo simulations, which appear not only in quantitative finance but also in many other areas of scientific computing.