spot_price_simulate {NFCP}

Simulate spot prices of an N-factor model through Monte Carlo simulation

Description

Simulate risk-neutral price paths of an an N-factor commodity pricing model through Monte Carlo Simulation.

Usage

spot_price_simulate(
  x_0,
  parameters,
  t = 1,
  dt = 1,
  N_simulations = 2,
  antithetic = TRUE,
  verbose = FALSE
)

Arguments

x_0

vector. Initial values of the state variables, where the length must correspond to the number of factors specified in the parameters.

parameters

vector. A named vector of parameter values of a specified N-factor model. Function NFCP_parameters is recommended.

t

numeric. Number of years to simulate.

dt

numeric. Discrete time step, in years, of the Monte Carlo simulation.

N_simulations

numeric. The total number of Monte Carlo simulations.

antithetic

logical. Should antithetic price paths be simulated?

verbose

logical. Should simulated state variables be output?

Details

The spot_price_simulate function is able to quickly and efficiently simulate a large number of state variables and risk-neutral price paths of a commodity following the N-factor model. Simulating risk-neutral price paths of a commodity under an N-factor model through Monte Carlo simulations allows for the valuation of commodity related investments and derivatives, such as American options and real Options through dynamic programming methods. The spot_price_simulate function quickly and efficiently simulates an N-factor model over a specified number of years, simulating antithetic price paths as a simple variance reduction technique. The spot_price_simulate function uses the mvrnorm function from the MASS package to draw from a multivariate normal distribution for the correlated simulation shocks of state variables.

The N-factor model stochastic differential equation is given by:

Brownian Motion processes (ie. factor one when GBM = T) are simulated using the following solution:

\[x_{1,t+1} = x_{1,t} + \mu^*\Delta t + \sigma_1 \Delta t Z_{t+1}\]

Where \(\Delta t\) is the discrete time step, \(\mu^*\) is the risk-neutral growth rate and \(\sigma_1\) is the instantaneous volatility. \(Z_t\) represents the independent standard normal at time \(t\).

Ornstein-Uhlenbeck Processes are simulated using the following solution:

\[x_{i,t} = x_{i,0}e^{-\kappa_it}-\frac{\lambda_i}{\kappa_i}(1-e^{-\kappa_it})+\int_0^t\sigma_ie^{\kappa_is}dW_s\]

Where a numerical solution is obtained by numerically discretising and approximating the integral term using the Euler-Maruyama integration scheme: \[\int_0^t\sigma_ie^{\kappa_is}dW_s = \sum_{j=0}^t \sigma_ie^{\kappa_ij}dW_s\]

Finally, deterministic seasonality is considered within the spot prices of simulated price paths.

Value

spot_price_simulate returns a list when verbose = T and a matrix of simulated price paths when verbose = F. The returned objects in the list are:

State_Variables	A matrix of simulated state variables for each factor is returned when verbose = T. The number of factors returned corresponds to the number of factors in the specified N-factor model.
Prices	A matrix of simulated price paths. Each column represents one simulated price path and each row represents one simulated observation.

References

Schwartz, E. S., and J. E. Smith, (2000). Short-Term Variations and Long-Term Dynamics in Commodity Prices. Manage. Sci., 46, 893-911.

Cortazar, G., and L. Naranjo, (2006). An N-factor Gaussian model of oil futures prices. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 26(3), 243-268.

Examples

# Example 1
## Simulate a geometric Brownian motion (GBM) process:
simulated_spot_prices <- spot_price_simulate(
 x_0 = log(20),
 parameters = c(mu_rn = (0.05 - (1/2) * 0.2^2), sigma_1 = 0.2),
 t = 1,
 dt = 1/12,
 N_simulations = 1e3)

# Example 2
## Simulate the Short-Term/Long-Term model:

### Step 1 - Obtain contemporary state variable estimates through the Kalman Filter:
SS_2F_filtered <- NFCP_Kalman_filter(parameter_values = SS_oil$two_factor,
                                    parameter_names = names(SS_oil$two_factor),
                                    log_futures = log(SS_oil$stitched_futures),
                                    dt = SS_oil$dt,
                                    futures_TTM = SS_oil$stitched_TTM,
                                    verbose = TRUE)

### Step 2 - Use these state variable estimates to simulate futures spot prices:
simulated_spot_prices <- spot_price_simulate(
 x_0 = SS_2F_filtered$x_t,
 parameters = SS_oil$two_factor,
 t = 1,
 dt = 1/12,
 N_simulations = 1e3,
 antithetic = TRUE,
 verbose = TRUE)