Monte Carlo on a single sample

TL;DR

This work addresses parameter-dependent Monte Carlo pricing by learning the conditional mean $h_0(\Theta)=\mathbb E[X|\Theta]$ across a model class $\Gamma$ through a minimization of $V(h)=\mathbb E[(h(\Theta)-X)^2]$ in a reproducing kernel Hilbert space. By introducing empirical, ridge-regularized objectives $V_{N,\lambda}$ and proving convergence of the minimizers $h_{N,\lambda_N}$ to $h_0$ as $N\to\infty$ and $\lambda_N\to 0$ (with $\lambda_NN^{1/4}\to\infty$), the paper establishes a nonparametric-rate result $\mathbb E[\|h_{N,\lambda_N}-h_0\|_{L^2(\mu)}] \le C\max\{\lambda_N^{-2}N^{-1/2}, \sqrt{\lambda_N}\}$ and identifies an optimal tuning $\lambda_N=N^{-1/5}$ giving rate $N^{-1/10}$. The method leverages kernel ridge regression and SGD-trained neural networks to approximate $h_N$, with $M=1$ samples per model sufficing in theory, and demonstrates substantial empirical gains over standard MC across Black-Scholes and Heston-type cases under the same total sampling budget. The results suggest MinMC as a robust benchmark for parameter-dependent MC tasks in finance and broader scientific computing, enabling efficient, simultaneous evaluation of option prices and risk measures across model spaces.

Abstract

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.

Figures (8)

• Figure 1: Model approximations for a European call option in the Black-Scholes model for different values of $\lambda$, but based on the same samples, compared to the true price (Black–Scholes formula).
• Figure 2: Mean squared error in the Black-Scholes model based on $500$ simulation runs for $N=100\, 000$ compared to Monte Carlo simulations (each with $1\, 000$ samples) for $100$$\theta$ values chosen equidistant in $[0,1]$.
• Figure 3: Mean squared error in the Black-Scholes model based on $500$ simulation runs for $\lambda=0.01$ and $\lambda=0.001$ with fixed number of random variables $M\cdot N=120\, 000$. For fixed $M$ the simulations are based on the same samples.
• Figure 4: Mean squared error in the Black-Scholes model for the feature map setting based on $500$ simulation runs for $N=100\, 000$ compared to Monte Carlo simulations (each with $1\, 000$ samples) for $100$$\theta$ values chosen equidistant in $[0.1,0.2]$.
• Figure 5: Mean squared error in the Black-Scholes model for the feature map setting based on $500$ simulation runs for $\lambda=0.01$ and $\lambda=0.001$ with fixed number of random variables $M\cdot N=120\, 000$. For fixed $M$ the simulations are based on the same samples.

Theorems & Definitions (45)

• Theorem 2.2
• Remark 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• Example 2.6
• Example 2.7
• Example 2.8
• Example 2.9
• Proposition 3.1