Description Usage Arguments Details Value References Examples
View source: R/LSM_american_option.R
Given a set of state variables and associated payoffs simulated through Monte Carlo simulation, solve for the value of an Americanstyle call or put option through the leastsquares Monte Carlo simulation method.
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state_variables 

payoff 

K 
the exercise price of the Americanstyle option 
dt 
Constant, discrete time step of simulated observations 
rf 
The annual riskfree interest rate 
call 

orthogonal 

degree 
The number of orthogonal polynomials used in the leastsquares fit. See details. 
cross_product 

verbose 

The LSM_american_option
function provides an implementation of the leastsquares Monte Carlo (LSM) simulation approach to numerically approximate
the value of Americanstyle options (options with early exercise opportunities). The function provides flexibility in the stochastic process followed by the underlying asset, with simulated values
of stochastic processes provided within the state_variables
argument. It also provides flexibility in the payoffs of the option, allowing for vanilla as well as more exotic options to be considered.
LSM_american_option
also provides analysis into the exercise timing and probability of early exercise of the option.
LeastSquares Monte Carlo Simulation:
The leastsquares Monte Carlo (LSM) simulation method is a numeric approach first presented by Longstaff and Schwartz (2001) that approximates the value of options with early exercise opportunities. The LSM simulation method is considered one of the most efficient methods of valuing Americanstyle options due to its flexibility and computational efficiency. The approach can feature multiple stochastically evolving underlying uncertainties, following both standard and exotic stochastic processes.
The LSM method first approximates stochastic variables through a stochastic process to develop crosssectional information, then directly estimates the continuation value of inthemoney simulation paths by "(regressing) the expost realized payoffs from continuation on functions of the values of the state variables" (Longstaff and Schwartz, 2001).
The 'LSM_american_option' function at each discrete time period, for each simulated price path, compares the payoff that results from immediate exercise of
the option with the expected value of continuing to hold the option for subsequent periods. The payoff of immediate exercise is provided in the payoff
argument and
could take several different meanings depending upon the type of Americanstyle option being valued (e.g. the current stock price, the maximum price between multiple assets, etc.).
The immediate profit resulting from exercise of the option is dependent upon the type of option being calculated. The profit of price path i and time t is given by:
When call = TRUE
:
profit[t,i] = max(payoff[t,i]  K, 0)
When call = FALSE
:
profit[t,i] = max(K  payoff[t,i], 0)
Orthogonal Polynomials:
To improve the accuracy of estimation of continuation values, the economic values in each period are regressed on a linear combination of a set of basis functions of the stochastic variables. These estimated regression parameters and the simulated stochastic variables are then used to calculate the estimator for the expected economic values.
Longstaff and Schwartz (2001) state that as the conditional expectation of the continuation value belongs to a Hilbert space,
it can be represented by a combination of orthogonal basis functions. Increasing the number of stochastic state variables
therefore increases the number of required basis functions exponentially. The orthogonal polynomials available in the
LSMRealOptions
package are: Laguerre, Jacobi, Legendre (spherical), Hermite (probabilistic), Chebyshev (of the first kind).
The simple powers of state variables is further available. Explicit expressions of each of these orthogonal polynomials are
available within the textbook of Abramowitz and Stegun (1965).
The 'LSM_american_option' function by default returns a numeric
object corresponding to the calculated value of the Americanstyle option.
When verbose = T
, 6 objects are returned within a list
class object. The objects returned are:
Value  The calculated option value. 
Standard Error  The standard error of the option value. 
Expected Timing  The expected time of early exercise. 
Expected Timing SE  The standard error of the expected time of early exercise. 
Exercise Probability  The probability of early exercise of the option being exercised. 
Cumulative Exercise Probability  vector . The cumulative probability of option exercise at each discrete observation point 
Abramowitz, M., and I. A. Stegun, (1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Corporation.
Longstaff, F. A., and E. S. Schwartz, (2001). "Valuing American options by simulation: a simple leastsquares approach." The review of financial studies, 14(1), 113147.
1 2 3 4 5 6 7 8 9 10 11 12 13  # Price a vanilla American put option on an asset that follows
# Geometric Brownian Motion
## Step 1  simulate stock prices:
stock_prices < GBM_simulate(n = 100, t = 1, mu = 0.05,
sigma = 0.2, S0 = 100, dt = 1/2)
## Step 2  Value the American put option:
option_value < LSM_american_option(state_variables = stock_prices,
payoff = stock_prices,
K = 100,
dt = 1/2,
rf = 0.05)

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