schwartz97distrfut: Schwartz two-factor Model: Distribution of Futures Prices

In schwartz97: A package on the Schwartz two-factor commodity model

Description

Density, distribution function, quantile function and random number generation of futures prices.

Usage

## S4 method for signature 'ANY,ANY,ANY,numeric'
dfutures(x, time = 0.1, ttm = 1, s0 = 50, delta0 = 0,
         mu = 0.1, sigmaS = 0.3, kappa = 1, alpha = 0,
         sigmaE = 0.5, rho = 0.75, r = 0.05, lambda = 0,
         alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f'
dfutures(x, time = 0.1, ttm = 1, s0, r = 0.05,
         lambda = 0, alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f.fit'
dfutures(x, time = 0.1, ttm = 1, s0, measure = c("P", "Q"), ...)


## S4 method for signature 'ANY,ANY,ANY,numeric'
pfutures(q, time = 0.1, ttm = 1, s0 = 50, delta0 = 0,
         mu = 0.1, sigmaS = 0.3, kappa = 1, alpha = 0,
         sigmaE = 0.5, rho = 0.75, r = 0.05, lambda = 0,
         alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f'
pfutures(q, time = 0.1, ttm = 1, s0, r = 0.05,
         lambda = 0, alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f.fit'
pfutures(q, time = 0.1, ttm = 1, s0, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,numeric'
qfutures(p, time = 0.1, ttm = 1, s0 = 50, delta0 = 0,
         mu = 0.1, sigmaS = 0.3, kappa = 1, alpha = 0,
         sigmaE = 0.5, rho = 0.75, r = 0.05, lambda = 0,
         alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f'
qfutures(p, time = 0.1, ttm = 1, s0, r = 0.05,
         lambda = 0, alphaT = NULL, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,schwartz2f.fit'
qfutures(p, time = 0.1, ttm = 1, s0, measure = c("P", "Q"), ...)

## S4 method for signature 'ANY,ANY,ANY,numeric'
rfutures(n, time = 0.1, ttm = 1, s0 = 50, delta0 = 0,
         mu = 0.1, sigmaS = 0.3, kappa = 1, alpha = 0, sigmaE = 0.5,
         rho = 0.75, r = 0.05, lambda = 0, alphaT = NULL, measure = c("P", "Q"))

## S4 method for signature 'ANY,ANY,ANY,schwartz2f'
rfutures(n, time = 0.1, ttm = 1, s0, r = 0.05,
         lambda = 0, alphaT = NULL, measure = c("P", "Q"))

## S4 method for signature 'ANY,ANY,ANY,schwartz2f.fit'
rfutures(n, time = 0.1, ttm = 1, s0, measure = c("P", "Q"))

Arguments

q, x	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
time	Time where the futures process is evaluated (relative to now).
ttm	Time to maturity (relative to now).
s0	Either a numeric representing the initial value of the commodity spot price or an object inheriting from class schwartz2f.
delta0	Initial value of the convenience yield.
mu	Drift term of commodity spot price.
sigmaS	Diffusion parameter of the spot price process.
kappa	Speed of mean-reversion of the convenience yield process.
alpha	Mean-level of the convenience yield process.
sigmaE	Diffusion parameter of the convenience yield process.
rho	Correlation coefficient between the Brownian motion driving the spot price and the convenience yield process.
lambda	Market price of convenience yield risk (see Details).
alphaT	Mean-level of the convenience yield process with respect to the equivalent martingale measure (see Details).
r	Instantaneous risk-free interest rate.
measure	under which the functions are computed. “P” denotes the objective measure, “Q” the risk-neutral measure (see Details).
...	Arguments to be passed to the functions d/p/q-norm.

Details

Futures prices depend on the spot-price and the convenience yield.

To get the real (i.e. the objective) distribution of futures prices at some date in the future the dynamics is considered under the objective measure P. The P-dynamics is

dS = (mu - delta) * S * dt + sigmaS * S * dW1,

d delta = kappa * (alpha - delta) * dt + sigmaE * dW2

dW1 * dW2 = rho dt,

where W1, W2 are Brownian motions under the objective measure, the measure P.

Options on futures are evaluated based on the risk-neutral dynamics of the spot-price and the convenience yield, i.e. under the measure Q. The Q-dynamics is

dS = (r - delta) * S * dt + sigmaS * S * dW1*

d delta = kappa * (alphaT - delta) * dt + sigmaE * dW2*,

where W1*, W2* are Brownian motions with respect to Q.

alphaT = alpha - lambda / kappa where lambda is the market price of convenience-yield risk. The market price of convenience yield risk can either be specified explicitly by lambda or implicitly by alphaT. The relation is alphaT = alpha - lambda / kappa. See the package vignette.

Value

Probabilities, densities, quantiles or samples of the log-normally distributed futures prices as numeric.

Note

Note that futures and forward prices coincide as the interest rate is assumed to be constant in the Schwartz two-factor model.

Author(s)

Philipp Erb, David Luethi, Juri Hinz

References

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging by Eduardo S. Schwartz
Journal of Finance 52, 1997, 923-973

Valuation of Commodity Futures and Options under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot by Jimmy E. Hilliard and Jorge Reis
Journal of Financial and Quantitative Analysis 33, 1998, 61-86

See Also

pricefutures, d/p/qstate, r/simstate

Examples

## Create a "schwartz2f"-object
model <- schwartz2f()

## Probability
pfutures(q = 10 * 3:9, time = 0.5, ttm = 2, model, lambda = 0.01)

## Density
dfutures(x = c(20, 40, 100), time = 0.5, ttm = 2, model, lambda = 0.01)

## Quantile
qfutures(p = 0.1 * 2:5, time = 0.5, ttm = 10, model, lambda = 0.01)

## Sample
sim <- rfutures(n = 1000, time = 0.5, ttm = 5, model, lambda = 0.01)

hist(sim, prob = TRUE)
lines(seq(30, 300, length = 100),
      dfutures(seq(30, 300, length = 100),
               time = 0.5, ttm = 5, model, lambda = 0.01), col = "red")

## At time 0 the futures price is a deterministic function of s0 and
## delta0. Therefore 3 times the same value is obtained:
rfutures(3, time = 0, ttm = 1, model, lambda = 0)

schwartz97 documentation built on May 2, 2019, 5:48 p.m.