[Paper Review] A Monte Carlo method for exponential hedging of contingent claims

Martino Grasselli, T. R. Hurd|ArXiv.org|Nov 25, 2002

Stochastic processes and financial applications25 references3 citations

TL;DR

This paper introduces a novel Monte Carlo algorithm for exponential hedging in incomplete markets using a least-squares regression approach inspired by Longstaff-Schwartz, enabling efficient computation of indifference prices and certainty equivalent values. The method achieves high accuracy in pricing despite limitations in approximating optimal hedging strategies due to subspace and sample size constraints.

ABSTRACT

Utility based methods provide a very general theoretically consistent approach to pricing and hedging of securities in incomplete financial markets. Solving problems in the utility based framework typically involves dynamic programming, which in practise can be difficult to implement. This article presents a Monte Carlo approach to optimal portfolio problems for which the dynamic programming is based on the exponential utility function U(x)=-exp(-x). The algorithm, inspired by the Longstaff-Schwartz approach to pricing American options by Monte Carlo simulation, involves learning the optimal portfolio selection strategy on simulated Monte Carlo data. It shares with the LS framework intuitivity, simplicity and flexibility.

Motivation & Objective

• To bridge the gap between theoretical exponential hedging in incomplete markets and practical numerical implementation.
• To develop a flexible, intuitive, and computationally efficient Monte Carlo algorithm for optimal portfolio selection under exponential utility.
• To enable numerical exploration of key concepts like indifference prices and certainty equivalents in complex market models.
• To assess the performance of the method in approximating both pricing and hedging strategies, particularly in a geometric Brownian motion setting.
• To identify limitations such as subspace approximation errors and finite sample size effects in the algorithm’s implementation.

Proposed method

• The algorithm employs backward induction via dynamic programming to learn optimal hedging strategies over discrete time steps.
• It uses a low-dimensional subspace of strategies, parameterized by basis functions, to approximate the optimal control at each time step.
• At each time step, the method applies least-squares regression to fit the optimal hedge to simulated paths, minimizing the expected exponential utility loss.
• The method leverages the time-consistent structure of dynamic programming, reducing computational cost from O(NR²K²) to O(NR²K) by solving R-dimensional optimizations sequentially.
• The algorithm is validated in a geometric Brownian motion model with known analytical solutions for comparison.
• It computes indifference prices and certainty equivalent values by simulating profit/loss distributions across Monte Carlo paths.

Experimental results

Research questions

• RQ1 Can a Monte Carlo method effectively approximate optimal hedging strategies under exponential utility in incomplete markets?
• RQ2 How accurately can the proposed algorithm estimate indifference prices compared to theoretical values?
• RQ3 What are the primary sources of error in the algorithm’s performance, particularly regarding strategy approximation and sample size?
• RQ4 Why does the algorithm perform better in pricing than in hedging strategy estimation?
• RQ5 To what extent can the method be extended to more complex derivatives, such as American-style options?

Key findings

• The algorithm achieves high accuracy in estimating indifference prices, even with relatively small sample sizes, suggesting strong reliability for pricing applications.
• The computed indifference prices were found to be more accurate than the two certainty equivalent values that define them, indicating robustness in the core pricing metric.
• Hedging strategies learned by the algorithm showed significant deviation from the theoretical Merton strategy along individual sample paths, indicating poor strategy approximation.
• With N = 100,000 Monte Carlo simulations, the method produced reliable price estimates but still yielded unstable and fluctuating hedging strategies over time.
• The main limitations were identified as subspace approximation (restricting strategy form) and finite sample size effects, both of which degrade strategy accuracy.
• Dynamic programming reduced computational cost from O(NR²K²) to O(NR²K), preserving accuracy due to the block-diagonal structure of the Hessian matrix.