Pricing Options on Forwards in Function-Valued Affine Stochastic Volatility Models

ArXiv ID: 2508.14813 “View on arXiv”

Authors: Jian He, Sven Karbach, Asma Khedher

Abstract

We study the pricing of European-style options written on forward contracts within function-valued infinite-dimensional affine stochastic volatility models. The dynamics of the underlying forward price curves are modeled within the Heath-Jarrow-Morton-Musiela framework as solution to a stochastic partial differential equation modulated by a stochastic volatility process. We analyze two classes of affine stochastic volatility models: (i) a Gaussian model governed by a finite-rank Wishart process, and (ii) a pure-jump affine model extending the Barndorff–Nielsen–Shephard framework with state-dependent jumps in the covariance component. For both models, we derive conditions for the existence of exponential moments and develop semi-closed Fourier-based pricing formulas for vanilla call and put options written on forward price curves. Our approach allows for tractable pricing in models with infinitely many risk factors, thereby capturing maturity-specific and term structure risk essential in forward markets.

Keywords: Function-Valued Stochastic Models, Heath-Jarrow-Morton-Musiela (HJM-M), Affine Stochastic Volatility, Wishart Process, Fourier-Based Pricing, Fixed Income/Forward Contracts

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematically dense, featuring infinite-dimensional stochastic partial differential equations, affine processes, operator-valued calculus, and Fourier-based pricing derivations. However, it presents no backtested strategies, implementation details, or statistical validation, focusing instead on theoretical derivations and numerical consistency checks.

  flowchart TD
    A["Research Goal: Price options on forwards<br>in function-valued affine SV models"] --> B["Model Selection & Setup<br>HJM-M framework + SPDE dynamics"]
    
    B --> C{"Two Affine SV Class Types"}
    
    C --> D["Model 1: Gaussian<br>Finite-rank Wishart Process"]
    C --> E["Model 2: Pure-Jump<br>Barndorff-Nielsen-Shephard"]

    D --> F["Computational Process:<br>Derive Exponential Moment Conditions"]
    E --> F

    F --> G["Semi-Closed Pricing Formulas<br>via Fourier Transform Methods"]

    G --> H["Key Outcomes:"]
    H --> I["Tractable pricing in<br>infinite-dimensional models"]
    H --> J["Captures maturity-specific<br>and term structure risk"]
    H --> K["Explicit formulas for<br>vanilla calls/puts on forwards"]