[Paper Review] Pricing Spread Options under Stochastic Correlation and Jump-Diffusion Models

Olivares, Pablo, Cane, Matthew|arXiv (Cornell University)|Sep 3, 2014

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper proposes a fast, accurate method for pricing spread options under multivariate models with stochastic correlation, stochastic volatility (CIR), and jump-diffusion processes using the bivariate Fast Fourier Transform (FFT). By deriving characteristic functions for two joint-normal jump and stochastic volatility models, the authors achieve computational efficiency orders of magnitude faster than Monte Carlo while capturing complex market features like volatility and correlation smiles.

ABSTRACT

This paper examines the problem of pricing spread options under some models with jumps driven by Compound Poisson Processes and stochastic volatilities in the form of Cox-Ingersoll-Ross(CIR) processes. We derive the characteristic function for two market models featuring joint normally distributed jumps, stochastic volatility, and different stochastic dependence structures. With the use of Fast Fourier Transform(FFT) we accurately compute spread option prices across a variety of strikes and initial price vectors at a very low computational cost when compared to Monte Carlo pricing methods. We also look at the sensitivities of the prices to the model specifications and find strong dependence on the selection of the jump and stochastic volatility parameters. Our numerical implementation is based on the method developed by Hurd and Zhou (2009).

Motivation & Objective

• To address the limitations of the Black-Scholes model in capturing empirical market features such as volatility smiles and jumps.
• To extend existing models by incorporating both stochastic correlation and jump-diffusion dynamics in a multivariate setting.
• To develop an efficient computational framework for pricing spread options under these complex dynamics.
• To compare the performance and accuracy of the FFT-based method against traditional Monte Carlo simulations.
• To analyze the sensitivity of option prices to key model parameters, especially jump and stochastic volatility components.

Proposed method

• Derives the characteristic function for two multivariate market models featuring jointly normally distributed jumps, CIR stochastic volatility, and different stochastic dependence structures.
• Adapts the bivariate inverse Fourier transform method of Hurd and Zhou (2009) to handle models with jumps and stochastic correlation.
• Employs the Fast Fourier Transform (FFT) on a two-dimensional grid to compute spread option prices efficiently.
• Uses damping parameters and optimal step-size algorithms to control truncation and discretization errors in the FFT implementation.
• Calibrates the model using a benchmark parameter set and validates results against Monte Carlo simulations.
• Implements numerical checks on sensitivity to FFT parameters (e.g., damping, step size, truncation interval) to ensure robustness.

Experimental results

Research questions

• RQ1 How does the inclusion of stochastic correlation and jump-diffusion dynamics affect the pricing of spread options compared to standard models?
• RQ2 Can the FFT-based method achieve high accuracy and computational speed for spread options under complex multivariate models with jumps and stochastic volatility?
• RQ3 How sensitive are option prices to variations in jump size variance, jump intensity, and stochastic volatility parameters?
• RQ4 What are the numerical stability and error characteristics of the bivariate FFT implementation under these models?
• RQ5 How do the model’s implied volatility and correlation smiles compare to observed market patterns?

Key findings

• The FFT-based method computes spread option prices with high accuracy and significantly lower computational cost than Monte Carlo simulations.
• Option prices increase non-linearly with jump size variance, with a 40% price increase observed when jump variance rose from 0.02 to 0.2.
• The long asset’s jump variance has a greater impact on option prices than the short asset’s, due to its greater influence on the spread distribution.
• The model shows strong sensitivity to jump and stochastic volatility parameters, highlighting the need for careful calibration.
• The FFT method is robust to changes in step size and truncation interval, with minimal price variation across different N and ūmin values.
• Damping parameters (ε) have a small effect overall, but errors emerge when ε₁ - ε₂ - 1 < 0.2, indicating a narrow range of instability.