Source-linked AI summary
Smart expansion and fast calibration for jump diffusion
Eric Benhamou, Emmanuel Gobet, Mohammed Miri
TL;DR
The paper addresses the difficulty of pricing European options when market-implied volatility varies across strikes and maturities. It derives a Malliavin-calculus asymptotic expansion around a Merton proxy for local-volatility and Poisson-jump models, with accuracy improving in small-diffusion, small-jump, and related regimes. The resulting approximation is reported as highly accurate, while calibration can still face instability from insufficient information in call/put options alone.
Problem
Constant-volatility Black–Scholes pricing cannot reproduce the smile or skew formed by market-implied volatilities across strikes and maturities.
Method
The paper uses Malliavin calculus to derive correction terms and error bounds from an asymptotic expansion around an explicit Merton proxy for local-volatility and jump-diffusion models.
Results
The approximation becomes increasingly accurate as diffusion, jump activity, maturity, or coefficient variation decrease, and it can produce short-maturity smiles and long-maturity skews.
Takeaways & Limitations
The formulas provide a practical analytical route to numerical pricing and fast calibration for models with local volatility and jumps.
Takeaways & Limitations
Calibration can be unstable, depend on the initial guess, and encounter local minima because call/put options may not contain enough information about future volatility.
Abstract
from arXiv · showhide
Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.
1 Introduction
The paper develops an analytical approximation for European-option prices in local-volatility and jump-diffusion models, targeting volatility smiles and skews that constant-volatility Black–Scholes cannot capture. Its Malliavin-calculus expansion is most accurate for small diffusion, jump activity, maturity, or coefficient variation, and supports rapid pricing and calibration.
- Motivation: The approach addresses Black–Scholes limitations by combining local volatility with a jump process to model observed implied-volatility smiles and skews.The paper focuses on the Andersen–Andreasen model as an illustration for numerical pricing and fast calibration.
- Method: The proxy alone is too rough, so the paper derives correction terms to obtain a substantially more accurate approximation.The Merton proxy supplies an explicit starting price, while the expansion corrects its discrepancy from the original model.
- Method: The method expands European-option prices around an explicit Merton proxy using Malliavin calculus, with correction terms and error bounds for time-inhomogeneous coefficients and jumps.The expansion is a representation tool rather than a small-ε approximation at the target value ε = 1.
- Financial consequences: The approximation can generate a volatility smile at short maturities and a local-volatility-driven skew at long maturities.The short-maturity behavior is associated with the Merton proxy, while the long-maturity skew is attributed to local volatility.
- Scope: The framework handles arbitrary initial conditions, time-inhomogeneous coefficients, and jumps without extra effort relative to the stated setting.This is presented as a significant difference from previous research.
- Accuracy: The approximation becomes more accurate as diffusion and jump amplitudes, jump frequency, maturity, or coefficient derivatives decrease.These conditions are represented quantitatively through constants such as M0, M1, and MJ.
2 Smart Taylor Development
The paper constructs a smart expansion around the Merton model, using a parameterized process and Malliavin calculus to derive explicit correction terms for option prices. The approximation combines leading Merton pricing with volatility, drift, and jump corrections whose errors are rigorously estimated under payoff regularity assumptions.
- Approximation order: The first-order approximation yields the Merton model, while higher-order formulas remain explicit at the cost of introducing more random variables into the Greeks.The expansion can therefore be extended beyond the displayed approximation order.
- Error control: The main approximation theorem applies under process-data conditions and one of three payoff assumptions, with separate error estimates for each payoff case.The error analysis is tied to the regularity class of the payoff function.
- Malliavin calculus: Malliavin differentiation is performed with respect to the Brownian motion while holding the jump component fixed, followed by integration over the jumps.This produces random-weight representations of Greeks that make the correction terms explicit.
- Approximation structure: The approximation formula combines a leading Merton price with explicit volatility-drift and jump correction terms.The Merton term is available in closed form for calls and puts, while correction terms are expressed through Greeks.
3 Financial Modeling Consequences
The model combines local volatility with jumps to reproduce volatility-smile and skew patterns across maturities. Its CEV-type volatility parameterization can generate the model’s attainable Black-Scholes smiles, while Merton behavior dominates at short maturities and local volatility shapes long-maturity skew.
- Model choice: The paper focuses on the Andersen-Andreasen model on the log asset, combining local volatility with a jump process to fit the smile.This model was introduced because pure local volatility was not compatible with cited empirical evidence.
- Volatility parameterization: The CEV-type volatility parameterization can generate all attainable prices and Black-Scholes smiles within this class of models.Its two time-dependent degrees of freedom determine both σ(t,x0) and σ(1)(t,x0).
- Attainable smiles: At short maturities the model is close to Merton, producing a smile, whereas at long maturities the smile becomes a local-volatility-driven skew.The Merton smile flattens with maturity, leaving the local volatility function responsible for the long-maturity skew.
- Smile dynamics: The model uses the Merton model as a proxy, so implied volatilities should move in the same direction as the forward.The stated rationale is that Merton implied volatilities depend only on the forward-to-strike ratio.
4 Numerical Experiments
The numerical implementation evaluates the approximation for smooth or piecewise-constant inputs and supports recursive calibration across maturities. Tests report sub-2 bp implied-volatility errors against PIDE, millisecond pricing, and sub-second calibration, while calibration remains unstable in some cases.
- 4.2 Accuracy of the approximation: Errors in implied Black-Scholes volatility do not exceed 2 bp across a large range of strikes and maturities against a PIDE benchmark.The comparison is reported for the approximation formula and PIDE method in Table 4.1.
- 4.2 Accuracy of the approximation: The formula takes less than four milliseconds to compute on a 2.6 GHz Pentium PC.This pricing speed underpins the reported reduction in calibration time.
- 4.3 Calibration issues: The bootstrap calibration fits implied volatilities maturity by maturity, recursively updating coefficients after optimizing the piecewise-constant volatility parameters.A local minimization algorithm is applied across strikes at each maturity.
- 4.3 Calibration issues: Calibration can be unstable because final parameters depend on the initial guess and the objective has many local minima.The authors suggest regularization or adding volatility options because call and put prices may not contain enough information about future volatility.
- 4.3 Calibration issues: The approach calibrates a 4×4 EUR/USD quoted-price grid in less than one second.The calibrated model uses jump parameters λ = 1.21%, ηJ = −19.07%, and γJ = 40.30%.
5 Error Analysis
The paper derives error bounds for its approximation formula across smooth, vanilla, and binary payoffs, using estimates of model amplitudes, jump frequency, and maturity. The approximation is second order for smooth and vanilla payoffs but only first order for binary payoffs because of payoff nonsmoothness.
- Error bounds: The error bounds explicitly track constants M0, M1, MJ, jump frequency λ, and maturity T.These dependencies support the heuristic choice of the model proxy and quantify how model features affect approximation accuracy.
- Smooth payoff: For smooth payoffs, accuracy improves when local coefficients and jump-size parameters are small or vary weakly with the state.If coefficients depend only on time, the approximation is exact; small spatial variation and small model amplitudes also yield high accuracy.
- Smooth and vanilla payoffs: The approximation is second order in model-data amplitudes, with third-order error terms, for smooth and vanilla payoffs.The error analysis establishes this order for smooth payoffs and states the analogous result for vanilla payoffs.
- Proof strategy: The proofs combine ε-derivative estimates, tight moment bounds, Taylor expansion around the Merton proxy, and Malliavin calculus for nonsmooth payoffs.For vanilla and binary options, Malliavin integration by parts avoids relying on a second payoff derivative.
- Binary payoff: For binary payoffs, the formula is only first order in model-data amplitudes, with second-order error terms because the payoff lacks regularity.This case requires a separate error estimate under the binary-payoff assumption.
6 Appendix
The appendix supplies auxiliary stochastic-calculus results used in the approximation analysis. These include integration-by-parts identities, semimartingale formulas, Gaussian jump calculations, and moment bounds for the compound Poisson process.
- Stochastic-calculus identities: The appendix develops integration-by-parts identities for continuous semimartingales and deterministic or predictable integrands.The proofs use Itô's formula and Malliavin duality between Itô and Skorohod integrals for adapted integrands.
- Jump-process estimates: The appendix bounds L^p moments of the compound Poisson process by standardizing Gaussian jump sizes and analyzing Poisson jump counts.The argument treats even and odd moments separately using the characteristic function and elementary inequalities.