Package 'JumpDiffSim'

Diagnostic Plots for a JDSimResult Object

Description

Creates three diagnostic plots for a JDSimResult object: a simulated path fan chart, a histogram of log-returns with a normal density overlay, and the autocorrelation function of squared log-returns.

Usage

diagnosticPlots(object)

Arguments

object

A JDSimResult object.

Value

A named list of three ggplot objects:

• fan_chart: simulated path quantile fan chart.

• density: histogram of log-returns with a normal overlay.

• acf_sq: autocorrelation of squared log-returns with 95\

Examples

m    <- MertonModel()
sim  <- simulateMerton(m, n = 50, T_ = 1, steps = 100)
plts <- diagnosticPlots(sim)
print(plts$fan_chart)
print(plts$density)
print(plts$acf_sq)

---

Fit Model to Log-Returns via MLE

Description

Fit Model to Log-Returns via MLE

Usage

fit(object, log_returns, ...)

Arguments

object	A JumpDiffModel object.
log_returns	Numeric vector of observed log returns.
...	Passed to optim.

Value

A JDFitResult object.

---

Fit Merton Jump-Diffusion Model via MLE

Description

Maximises the log-likelihood of the Merton (1976) model using L-BFGS-B optimisation. Returns a JDFitResult object containing parameter estimates, standard errors, and convergence info.

Usage

fitMerton(
  log_returns,
  start = c(mu = 0.05, sigma = 0.2, lambda = 1, mu_j = -0.1, sigma_j = 0.15),
  dt = 1/252,
  N_max = 50L,
  verbose = FALSE
)

Arguments

log_returns	Numeric vector of observed log asset returns.
start	Named numeric vector of starting values. Defaults to c(mu=0.05, sigma=0.20, lambda=1, mu_j=-0.10, sigma_j=0.15).
dt	Numeric. Time step length (default 1/252).
N_max	Integer. Mixture truncation (default 50).
verbose	Logical. Print progress? (default FALSE).

Value

A JDFitResult object.

Examples

ret <- jdSampleData("merton", n = 500, seed = 42)
fit <- fitMerton(ret, verbose = TRUE)
print(fit)
confint(fit)

---

JDFitResult: MLE Estimation Output Class

Description

JDFitResult: MLE Estimation Output Class

Usage

## S4 method for signature 'JDFitResult'
show(object)

## S4 method for signature 'JDFitResult'
confint(object, parm, level = 0.95, ...)

Arguments

object	A JDFitResult object.
parm	Ignored (all parameters returned).
level	Confidence level (default 0.95).
...	Unused.

Methods (by generic)

• confint(JDFitResult): Wald-type 95\

Slots

estimates

Named numeric. Parameter estimates.

se

Named numeric. Standard errors.

loglik

Numeric. Log-likelihood at optimum.

converged

Logical. Did optimisation converge?

model_type

Character. Model name.

---

Generate Synthetic Log-Returns from a Jump-Diffusion Model

Description

Convenience function for generating reproducible synthetic log-returns for use in examples, tests, and vignettes. All package examples and tests use this function to avoid any dependency on live market data.

Usage

jdSampleData(
  model = "merton",
  n = 500,
  mu = 0.05,
  sigma = 0.2,
  lambda = 1,
  mu_j = -0.1,
  sigma_j = 0.15,
  dt = 1/252,
  seed = 42L
)

Arguments

model	Character. Model type: "merton" (default).
n	Integer. Number of log-returns to generate (default 500).
mu	Numeric. Drift (default 0.05).
sigma	Positive numeric. Diffusion vol (default 0.20).
lambda	Non-negative numeric. Jump intensity (default 1).
mu_j	Numeric. Mean log-jump (default -0.10).
sigma_j	Positive numeric. Std dev of log-jumps (default 0.15).
dt	Numeric. Time step (default 1/252).
seed	Integer. Random seed (default 42).

Value

Numeric vector of n log-returns.

Examples

ret <- jdSampleData("merton", n = 200, seed = 42)
head(ret)
hist(ret, breaks = 40, main = "Synthetic Merton Returns")

---

JDSimResult: Simulation Output Class

Description

JDSimResult: Simulation Output Class

Slots

paths

Matrix. Simulated price paths (n x steps+1).

times

Numeric vector. Time grid.

params

List. Model parameters used in simulation.

model_type

Character. Model name.

---

JumpDiffModel: Virtual Base Class for Jump-Diffusion Models

Description

Abstract base class. Do not instantiate directly. Concrete subclasses: MertonModel.

Slots

mu

Numeric. Drift parameter (annualised).

sigma

Positive numeric. Diffusion volatility.

lambda

Non-negative numeric. Jump intensity (jumps per year).

---

Theoretical Moments of the Log-Return Distribution

Description

Theoretical Moments of the Log-Return Distribution

Usage

jumpMoments(object)

Arguments

object

A JumpDiffModel object.

Value

Named numeric vector: mean, variance, skewness, kurtosis.

---

Theoretical Moments of the Merton Jump-Diffusion Log-Return

Description

Returns the theoretical mean, variance, skewness, and excess kurtosis of the log-return distribution under the Merton (1976) model.

Usage

## S4 method for signature 'MertonModel'
jumpMoments(object)

Arguments

object

A MertonModel object.

Value

Named numeric vector with elements mean, variance, skewness, kurtosis.

Examples

m <- MertonModel(mu = 0.05, sigma = 0.20, lambda = 1,
                 mu_j = -0.10, sigma_j = 0.15)
jumpMoments(m)

---

Compute Negative Log-Likelihood

Description

Compute Negative Log-Likelihood

Usage

loglik(object, log_returns, ...)

Arguments

object	A JumpDiffModel object.
log_returns	Numeric vector of log asset returns.
...	Passed to methods.

Value

Scalar negative log-likelihood value.

---

Negative Log-Likelihood for the Merton Jump-Diffusion Model

Description

Evaluates the negative log-likelihood of observing log_returns under the Merton (1976) model. The density is approximated by a Gaussian mixture truncated at N_max = 50 terms.

Usage

mertonLogLik(params, log_returns, dt = 1/252, N_max = 50L)

Arguments

params	Named numeric vector: c(mu, sigma, lambda, mu_j, sigma_j).
log_returns	Numeric vector of observed log asset returns.
dt	Numeric. Length of each time step (default 1/252).
N_max	Integer. Number of mixture terms (default 50).

Value

Scalar. Negative log-likelihood. Returns Inf for invalid parameter values.

Examples

ret <- jdSampleData("merton", n = 200, seed = 42)
p   <- c(mu = 0.05, sigma = 0.2, lambda = 1, mu_j = -0.1, sigma_j = 0.15)
mertonLogLik(p, ret)

---

Create a MertonModel Object

Description

Create a MertonModel Object

Usage

MertonModel(mu = 0.05, sigma = 0.2, lambda = 1, mu_j = -0.1, sigma_j = 0.15)

Arguments

mu	Numeric. Drift (default 0.05).
sigma	Positive numeric. Diffusion volatility (default 0.20).
lambda	Non-negative numeric. Jump intensity (default 1).
mu_j	Numeric. Mean log-jump size (default -0.10).
sigma_j	Positive numeric. Std dev of log-jumps (default 0.15).

Value

A validated MertonModel object.

Examples

m <- MertonModel(mu = 0.05, sigma = 0.20, lambda = 1,
                 mu_j = -0.10, sigma_j = 0.15)
show(m)

---

MertonModel: Merton (1976) Jump-Diffusion Model

Description

Extends JumpDiffModel with log-normal jump parameters.

Usage

## S4 method for signature 'MertonModel'
show(object)

Arguments

object

A MertonModel object.

Slots

mu_j

Numeric. Mean log-jump size.

sigma_j

Positive numeric. Std dev of log-jump sizes.

References

Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.

---

Price a European Option under the Merton Jump-Diffusion Model

Description

Computes the European call or put price under the Merton (1976) jump-diffusion model using the analytic series expansion. The price is expressed as a Poisson-weighted sum of Black-Scholes prices with jump-adjusted drift and volatility.

The formula is:

$C = \sum_{n=0}^{N} \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} \cdot BS(S, K, r_n, \sigma_n, T)$

where $\lambda' = \lambda(1 + \bar\mu_J)$, $r_n = r - \lambda\bar\mu_J + n\log(1+\bar\mu_J)/T$, and $\sigma_n^2 = \sigma^2 + n\sigma_J^2/T$.

Usage

priceEuropean(
  fit,
  S = 100,
  K = 100,
  r = 0.05,
  T_ = 1,
  type = "call",
  N_max = 50L
)

Arguments

fit	A JDFitResult object from fitMerton.
S	Positive numeric. Current asset price.
K	Positive numeric. Strike price.
r	Numeric. Continuously compounded risk-free rate.
T_	Positive numeric. Time to expiry in years.
type	Character. Either "call" or "put".
N_max	Integer. Number of series terms (default 50).

Value

A named list with elements:

price	Merton jump-diffusion option price.
bs_price	Black-Scholes benchmark price (no jumps).
jump_premium	Difference between Merton and BS price.
params	List of inputs and fitted parameters used.

References

Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.

Examples

ret <- jdSampleData("merton", n = 500, seed = 42)
fit <- fitMerton(ret)
price <- priceEuropean(fit, S = 100, K = 100,
                       r = 0.05, T_ = 1, type = "call")
print(price)

---

Simulate Asset Price Paths

Description

Simulate Asset Price Paths

Usage

simulate(object, n, T_, steps, seed = NULL, ...)

Arguments

object	A JumpDiffModel object.
n	Integer. Number of simulated paths.
T_	Positive numeric. Time horizon in years.
steps	Positive integer. Number of time steps.
seed	Optional integer seed for reproducibility.
...	Additional arguments passed to methods.

Value

A JDSimResult object.

---

Simulate Merton Jump-Diffusion Asset Paths

Description

Generates n exact compound-Poisson asset price paths under the Merton (1976) jump-diffusion model. Simulation is exact in the sense that jump arrivals are drawn directly from the Poisson process rather than approximated via Euler discretisation.

Usage

simulateMerton(object, n = 100, T_ = 1, steps = 252, S0 = 1, seed = NULL, ...)

Arguments

object	A MertonModel object.
n	Integer. Number of paths to simulate.
T_	Positive numeric. Time horizon in years (default 1).
steps	Positive integer. Number of time steps (default 252).
S0	Numeric. Initial asset price (default 1).
seed	Optional integer. Random seed for reproducibility.
...	Unused.

Value

A JDSimResult object.

Examples

m   <- MertonModel()
sim <- simulateMerton(m, n = 10, T_ = 1, steps = 252, seed = 42)
dim(sim@paths)   # should be 200 x 253

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