Jump Diffusion Simulation and Calibration for Merton and Kou Models.

JumpDiffSim

JumpDiffSim is an R package that implements the Merton (1976) and Kou (2002) jump-diffusion models through a unified S4 object-oriented interface. It provides exact compound-Poisson asset price simulation, maximum-likelihood parameter estimation with Hessian-based standard errors, Wald-type confidence intervals, theoretical moment calculations, and publication-quality diagnostic plots — all designed to run entirely offline without any dependency on live market data.

Installation

Install the development version from GitHub:

# install.packages("devtools")
devtools::install_github("kennedy2244/JumpDiffSim")

Install a specific release version:

devtools::install_github("kennedy2244/[email protected]")

Quick Start

The core workflow is three steps: create a model → simulate paths → fit to data.

library(JumpDiffSim)

# ── Step 1: Create a Merton model object ─────────────────────
m <- MertonModel(
  mu      =  0.05,   # drift
  sigma   =  0.20,   # diffusion volatility
  lambda  =  1.00,   # average jumps per year
  mu_j    = -0.10,   # mean log-jump size
  sigma_j =  0.15    # std dev of log-jumps
)

show(m)
#> Merton Jump-Diffusion Model
#> ---------------------------
#>   mu      : 0.0500
#>   sigma   : 0.2000
#>   lambda  : 1.0000
#>   mu_j    : -0.1000
#>   sigma_j : 0.1500

# ── Step 2: Simulate 200 asset price paths ───────────────────
sim  <- simulateMerton(m, n = 200, T_ = 1, steps = 252, seed = 42)
plts <- diagnosticPlots(sim)

print(plts$fan_chart)   # path quantile fan (5/25/50/75/95th percentiles)
print(plts$density)     # empirical return density vs Normal
print(plts$acf_sq)      # ACF of squared log-returns

# ── Step 3: Fit model to synthetic data via MLE ──────────────
ret <- jdSampleData("merton", n = 500, seed = 42)
fit <- fitMerton(ret)

print(fit)
#> Merton MLE Fit Result
#> ---------------------
#>   Converged : TRUE
#>   Log-lik   : 487.2341
#>   Estimates (SE):
#>     mu       :  0.0489  (0.0021)
#>     sigma    :  0.1987  (0.0045)
#>     lambda   :  0.9823  (0.1234)
#>     mu_j     : -0.0998  (0.0187)
#>     sigma_j  :  0.1502  (0.0134)

confint(fit)
#>              2.5 %    97.5 %
#> mu       0.044710  0.053090
#> sigma    0.189896  0.207504
#> lambda   0.740434  1.224166
#> mu_j    -0.136443 -0.063157
#> sigma_j  0.123890  0.176510

Parameters

A negativemu_j with a positive lambda implies that jumps on average reduce the asset price — consistent with the crash-risk interpretation in Merton (1976).

Theoretical Moments

jumpMoments(m)
#>       mean   variance   skewness   kurtosis
#>   0.000489   0.046225  -0.003012   0.000451

Excess kurtosis greater than zero confirms that the Merton model generates heavier tails than standard Geometric Brownian Motion.

Package Structure

JumpDiffSim/
├── R/
│   ├── generics.R        # S4 generic function definitions
│   ├── merton_model.R    # MertonModel class + JDSimResult + JDFitResult
│   ├── merton_sim.R      # simulateMerton(), jdSampleData(), diagnosticPlots()
│   ├── merton_fit.R      # mertonLogLik(), fitMerton(), confint()
│   ├── utils.R           # jumpMoments()
│   ├── kou_model.R       # KouModel placeholder (v0.2.0)
│   ├── kou_sim.R         # simulateKou() placeholder (v0.2.0)
│   └── kou_fit.R         # fitKou() placeholder (v0.2.0)
├── tests/testthat/       # 36 unit tests (T1-T12 + TU1-TU2)
├── vignettes/            # Getting Started with JumpDiffSim
└── scaffolds/            # Team development scaffold files

Test Coverage

R CMD check result: 0 errors | 0 warnings | 0 notes

Vignette

A full introduction to the package is available in the vignette:

# After installation
vignette("JumpDiffSim-intro", package = "JumpDiffSim")

The vignette covers:

1. Why jump-diffusion? Motivation and model equation
2. Simulating asset price paths with simulateMerton()
3. Fitting the model to data with fitMerton()
4. Interpreting parameters
5. Theoretical moments with jumpMoments()

Roadmap

Team

This package was developed collaboratively by students in the Department of Statistics, Yeungnam University, under the supervision of Professor Lee Kyungsub.

References

• Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1–2), 125–144.
• Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science, 48(8), 1086–1101.
• Wickham, H. and Bryan, J. (2023). R Packages (2nd ed.). O'Reilly Media. https://r-pkgs.org/

License

MIT © 2026 Kennedy Kayaki, Yeungnam University.