Source-linked AI summary
A Fourier transform method for spread option pricing
T. R. Hurd, Zhuowei Zhou
TL;DR
Spread-option pricing lacks a generally accurate, efficient, and flexible method across models, especially where closed formulas are unavailable. This paper derives a Fourier representation of the payoff and implements it with FFT-based numerical integration. The resulting method is reported as accurate, efficient, stable, and applicable across dimensions and a wide range of models.
Problem
Spread options have a long history of approximation methods, but no preferred approach is accurate, efficient, and flexible enough for general models.
Method
The paper derives an explicit Fourier transform formula for the spread-option payoff and combines it with analytically known joint-return characteristic functions for FFT pricing.
Results
The FFT implementation produces accurate and efficient spread-option prices across dimensions and can include models with explicit characteristic functions, including affine and Lévy jump models.
Takeaways & Limitations
The method can be applied to important energy and commodity-market problems and to a wide variety of asset-pricing models.
Takeaways & Limitations
Greeks require care in stochastic-volatility and variance-gamma models because differentiation can produce slowly decaying integrands.
Abstract
from arXiv · showhide
Spread options are a fundamental class of derivative contract written on multiple assets, and are widely used in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient and flexible enough to apply in general models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the spread option payoff function. Our detailed investigation proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of prices. It is found to be easy to implement, stable, efficient and applicable in a wide variety of asset pricing models.
1 Introduction
Spread options are widely used, but important positive-strike cases lack closed pricing formulas and existing approximations do not provide a generally preferred solution. The paper introduces a Fourier-based FFT method intended to combine model flexibility with competitive computational expense.
- Spread options also arise in energy markets, where crack and spark spreads require models with mean reversion and jumps beyond geometric Brownian motion.
- Positive-strike spread options under geometric Brownian motion have no explicit pricing formula, motivating a long history of approximation methods.
- The paper develops a new FFT numerical integration method for spread options in two or more dimensions when the joint-return characteristic function is analytically known.
- The approach begins with an explicit Fourier representation of the basic payoff P(x1, x2) = (e^x1 −e^x2 −1)+, reducing non-unit strikes through scaling and asset interchange.
- The pricing strategy combines the payoff representation with the characteristic function of log asset prices, under a simplifying homogeneous-dynamics assumption.
- Two-dimensional FFT approximations require controlling truncation and discretization through decay in Fourier space and decay of the computed price function in state space.
- The method is demonstrated in GBM, stochastic-volatility, and variance-gamma models, with accuracy compared against independent benchmark values.
2 Three kinds of stock models
The paper instantiates its characteristic-function framework across geometric Brownian motion, stochastic volatility, and exponential Lévy models. These specifications provide explicit bivariate characteristic functions suitable for the FFT pricing procedure.
- The case of geometric Brownian motion: In the two-asset Black–Scholes model, the joint characteristic function of the logarithms of terminal asset prices is expressed using volatilities and their constant correlation.
- The case of geometric Brownian motion: The geometric-Brownian-motion characteristic-function expression can be substituted into the spread-option pricing formula and computed numerically using the FFT.
- The case of stochastic volatility: The three-factor stochastic-volatility example specifies the spread-option problem through an SDE for the logarithms of the two asset prices.
- Exponential Lévy Models: Exponential Lévy models write asset prices as S_t = e^X_t, where X_t is a Lévy process with an explicitly known characteristic function.
- Exponential Lévy Models: The bivariate variance-gamma construction combines two idiosyncratic VG processes with a common VG process, while α controls dependence without changing marginal laws.
3 Numerical Integration by Fast Fourier Transform
The method approximates the spread-option Fourier integral with a two-dimensional FFT on reciprocal lattices. Accuracy depends on truncation, discretization, decay, and model-specific treatment of Greeks.
- FFT implementation: The double integral is approximated by a double sum over a lattice and evaluated using a two-dimensional FFT.The implementation uses reciprocal lattices for Fourier and log-price variables, with N commonly chosen as a power of 2.
- FFT implementation: The FFT grid introduces alternating phase factors that convert the lattice sum into a double inverse discrete Fourier transform.This transformation relies on N being even and on the reciprocal-lattice relationship between η and η∗.
- Error control: Truncation error is controlled by choosing a sufficiently large Fourier-domain square where the characteristic function has decayed below δ1.The payoff transform has an O(|u|^-2) bound, while characteristic-function decay determines the practical truncation range.
- Error control: Discretization accuracy depends on the decay of the transformed integrand and the spread payoff in log-price space.The combined truncation and discretization error is expected to be O(δ1 + δ2) under the stated choices of ¯u and η.
- Greeks: The same framework computes Greeks, but differentiation requires care in stochastic-volatility and variance-gamma models because the resulting integrand may decay slowly.For GBM, differentiation under the integral sign is always permissible, and vega follows directly from the FFT formula.
4 Numerical Results
Numerical experiments across GBM, stochastic-volatility, and variance-gamma models show that the FFT method is accurate and computationally efficient. It can generate price panels and use interpolation to reduce repeated computation, while benchmark quality affects apparent discrepancies.
- GBM experiments: 10^-14 is the approximate error floor for several GBM curves after truncation error becomes negligible.For fixed N, increasing ¯u reduces truncation error but enlarges discretization error.
- Benchmarking: 4 × 10^-4 is the lowest objective-function level reached with the stochastic-volatility Monte Carlo benchmark, reflecting benchmark inaccuracy.Each benchmark price used 1,000,000 simulations with 2000 time steps and no variance reduction.
- Variance-gamma experiments: 2 × 10^-5 is the truncation error at ¯u = 20 in the variance-gamma experiment, while other curves flatten near 7 × 10^-6.A benchmark computed with FFT using N = 2^12 and ¯u = 80 is consistent with round-off-level accuracy.
- Price accuracy: N = 256 produces very high accuracy across GBM, stochastic-volatility, and variance-gamma models.GBM discrepancies are attributed to round-off, variance-gamma discrepancies are below 1 basis point, and stochastic-volatility discrepancies are attributed to an inaccurate benchmark.
- Computational efficiency: A single FFT computes an N × N panel of prices, and interpolation across strikes gives competitive results while saving computational resources.The panel is indexed by reciprocal-lattice initial prices, and interpolation is especially useful when multiple strikes share the same underlying prices.
- Greeks and sensitivities: The option price is negatively correlated with underlying correlation, while first-order Greeks agree closely with finite-difference calculations.The paper also reports greater price sensitivity to S1 volatility than to S2 volatility.
- Computational efficiency: The FFT method generates a large price panel within a couple of seconds, although GBM is faster because its H-matrix entries admit a recursive calculation.The dominant H-matrix cost is O(N^2), exceeding the N log N FFT cost for large N.
5 High Dimensional Spread Options
The Fourier approach extends naturally from two-asset spread options to basket options with one asset offsetting a sum of multiple assets. The extension is based on a higher-dimensional payoff-transform proposition.
- Extension to basket options: The framework extends to basket options with payoff (˜S_T − S1_T − ··· − SM_T − 1)+ for M ≥ 2.The basket payoff generalizes the two-asset spread structure to multiple underlying assets.
- Fourier representation: The higher-dimensional analysis introduces log prices xm = log Sm and seeks a Fourier representation over u ∈ C^M and ˜u ∈ C.The derivation uses a proposition for the multidimensional payoff transform and interchanges the z integral with the x integrals.
6 Conclusion
The paper introduces a Fourier-transform formula for spread option payoffs and uses the Fast Fourier Transform to obtain accurate, efficient pricing algorithms. The method extends to important affine and Lévy jump models and supports systematic Greek evaluation.
- The method is based on an explicit gamma-function formula for the Fourier transform of the spread option payoff.
- The resulting algorithms are simple and transparent for spread options in all dimensions.
- The Fast Fourier Transform provides an accurate and efficient implementation with clear reliability and stability properties.
- Affine and Lévy jump models with explicit characteristic functions can be included with little difficulty.
- The method can be applied to important pricing problems in energy and commodity markets.
- Greeks can be evaluated systematically with similar performance and little extra work.
A Proof of Theorem 1 and Proposition 2
The proofs establish Fourier-transform properties of the spread option payoff by using exponential damping, variable changes, beta- and gamma-function identities, and induction. They also derive bounds on the transformed payoff that support the numerical method.
- Exponential damping places the payoff in L2(R2), enabling application of the Fourier inversion theorem.
- Restricting the integration domain to {x : x1 > 0, e^x2 < e^x1 − 1} is used in establishing the payoff-transform result.
- The proof uses beta-function bounds and gamma-function identities to derive formulas for the transformed payoff.
- The derivation yields an explicit bound on the Fourier transform, including a shifted bound involving B(ϵ, 2 + ϵ).
- A change of variables simplifies the proof of Proposition 2 by expressing the relevant quantities through exponentials.
- The proposition is proved by induction, beginning with M = 1 and extending the result from M = N to M = N + 1.