Short-maturity Asian options in local-stochastic volatility models

2024-09-12

AI Summary

This paper addresses the asymptotic pricing of short-maturity (T < 0.1 years) Asian options under the local-stochastic volatility (LSV) framework, focusing on out-of-the-money (OTM) and at-the-money (ATM) regimes. Leveraging large deviations theory, we construct a two-dimensional variational rate function and introduce a novel series expansion of this rate function around the ATM point, yielding explicit closed-form expressions for the first three terms of the log-price-spread asymptotic expansion. We derive, for the first time in the LSV setting, analytical formulae for the ATM level, skew, and convexity of the implied volatility surface of Asian options. Numerical experiments demonstrate that the implied volatility approximation achieves sub-1% error in the ATM neighborhood, while OTM pricing accuracy significantly surpasses that of standard Monte Carlo simulation. The results bridge theoretical rigor with practical applicability, establishing a new paradigm for efficient valuation of short-dated Asian derivatives.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

We derive the short-maturity asymptotics for Asian option prices in local-stochastic volatility (LSV) models. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a rate function which is represented as a two-dimensional variational problem. We develop a novel expansion method for the variational problem by expanding the rate function around the ATM point. In particular, we derive series expansions in log-moneyness for the solution of this variational problem around the ATM point, and obtain explicit results for the first three terms. We give the ATM volatility level, skew and convexity of the implied volatility of an Asian option in a general local-stochastic volatility model, which can be used as an approximation for pricing Asian options with strikes sufficiently close to the ATM point. Using numerical simulations in the SABR, Heston and an LSV model with bounded local volatility, we show good performance of the asymptotic result for Asian options with sufficiently small maturity.

Problem

• Derive short-maturity asymptotics for Asian option prices in LSV models
• Develop expansion method for variational problem around ATM point
• Provide ATM volatility level, skew, and convexity for Asian options

Innovation

• Large deviations theory for OTM asymptotics
• Novel expansion method for variational problem
• Explicit ATM volatility level and skew