Getting Started with JumpDiffSim

1. Why Jump-Diffusion?

Standard Geometric Brownian Motion (GBM) assumes that log-returns are normally distributed. Empirical equity and cryptocurrency returns, however, exhibit well-documented departures from normality:

• Leptokurtosis (excess kurtosis > 0): more mass in the centre and tails than a Normal distribution predicts.
• Negative skewness: large negative moves are more common and more severe than large positive moves.

The Merton (1976) jump-diffusion model adds a compound Poisson process to GBM, allowing the price to jump discontinuously at random times:

\[ dS_t = S_t \left[ (\mu - \lambda \bar{\mu}_J)\,dt + \sigma\,dW_t + dJ_t \right] \]

where \(J_t = \sum_{i=1}^{N(t)}(Y_i - 1)\), \(N(t) \sim \text{Poisson}(\lambda t)\), and \(\log Y_i \sim \mathcal{N}(\mu_J, \sigma_J^2)\).

JumpDiffSim provides a clean, unified S4 interface for simulating paths and calibrating parameters under this model, with all examples running entirely offline.

2. Simulate Paths

library(JumpDiffSim)

# Create a MertonModel S4 object with default parameters
m <- MertonModel(
  mu      =  0.05,   # drift
  sigma   =  0.20,   # diffusion volatility
  lambda  =  1,      # average jumps per year
  mu_j    = -0.10,   # mean log-jump size
  sigma_j =  0.15    # std dev of log-jumps
)

# Display the model
show(m)
#> Merton Jump-Diffusion Model
#> ---------------------------
#>   mu      : 0.0500
#>   sigma   : 0.2000
#>   lambda  : 1.0000
#>   mu_j    : -0.1000
#>   sigma_j : 0.1500
#>   Persist : 0.2500  [alpha+beta not applicable to raw model]

# Simulate 200 paths over 1 year with 252 daily steps
sim <- simulateMerton(m, n = 200, T_ = 1, steps = 252, seed = 42)

# Diagnostic plots
plts <- diagnosticPlots(sim)
print(plts$fan_chart)

print(plts$density)

The fan chart shows the 5th, 25th, 50th, 75th, and 95th percentile bands of the 200 simulated paths. The density plot overlays the empirical return histogram against a Normal curve, illustrating the heavy-tailed character of Merton returns.

3. Fit Model to Data

# Generate reproducible synthetic log-returns
# (all examples use jdSampleData() -- no internet required)
ret <- jdSampleData("merton", n = 500, seed = 42)

# Fit the Merton model via Maximum Likelihood Estimation
fit <- fitMerton(ret, verbose = FALSE)

# Parameter estimates and convergence
print(fit)
#> Merton MLE Fit Result
#> ---------------------
#>   Converged : TRUE
#>   Log-lik   : 1464.8117
#>   Estimates (SE):
#>     mu       :   0.0889  (     NaN)
#>     sigma    :   0.2022  (     NaN)
#>     lambda   :   0.5040  (     NaN)
#>     mu_j     :   0.0801  (     NaN)
#>     sigma_j  :   0.0000  (     NaN)

# 95% Wald confidence intervals
confint(fit)
#>         2.5 % 97.5 %
#> mu        NaN    NaN
#> sigma     NaN    NaN
#> lambda    NaN    NaN
#> mu_j      NaN    NaN
#> sigma_j   NaN    NaN

The fitMerton() function uses L-BFGS-B optimisation and computes Hessian-based standard errors via numDeriv::hessian(). The converged slot confirms whether the optimiser reached a solution.

4. Interpreting Parameters

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

5. Theoretical Moments

# Theoretical mean, variance, skewness, and excess kurtosis
# contributed by the jump component
jumpMoments(m)
#>       mean   variance   skewness   kurtosis 
#>  0.0300000  0.0725000 -0.3970038  0.5648038

Excess kurtosis greater than zero confirms that the Merton model generates heavier tails than GBM, which is the key motivation for using a jump-diffusion specification.

References

• Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.
• Wickham, H. and Bryan, J. (2023). R Packages (2nd ed.). O’Reilly Media. https://r-pkgs.org/