Options
Class AmericanOption

java.lang.Object
  Options.AmericanOption

Direct Known Subclasses:

AmericanBlackScholesPut

public abstract class AmericanOption

extends java.lang.Object

American option on a single asset. Limited functionality. Intended to support some experiments.

Constructor Summary

AmericanOption(Asset asset)
Constructor, does not initialize the option price path.

Method Summary

abstract double	currentDiscountedPayoff(int t) Option payoff discounted to time t=0 and computed from the current discounted price path of the underlying basket.
double	currentDiscountedPayoff(int t, Trigger exercise) The discounted option payoff h(rho_t) based on a given exercise strategy rho=(rho_t) computed from the current path of the underlying at time t, that is, the option has not been exercised before time t.
RandomVariable	deltaU(int t, Trigger exercise) The random variable U_{t+1}-U_t conditioned on F_t only, that is, it is assumed that a path of underlying is already computed up to time t.
double	discountedMonteCarloPrice(int t, int nPath, Trigger exercise) Monte Carlo price at time t dependent on a given exercise policy computed as a conditional expectation conditioned on information available at time t and computed from a sample of nPath (branches of) the price path of the underlying.
double	discountedMonteCarloPrice(int nPath, Trigger exercise) Monte Carlo option price at time t=0.
RandomVariable	discountedPayoff(int s) The discounted option payoff as a random variable when exercised at a fixed time s.
RandomVariable	discountedPayoff(Trigger exercise) The discounted option payoff based on a given exercise policy as a random variable.
double[]	get_C() Reference to the array C[ ] containing the discounted option price path.
double	get_dt() Size of time step.
double[]	get_S() Reference to the array C[ ] containing the discounted option price path.
int	get_T() Number of time steps to horizon.
Asset	get_underlying() Reference to the underlying asset.
RandomVariable	Kt(int nPath, Trigger exercise, int t) The random variable (E_t[U_{t+1}-U_t])^+ in the definition of the random variable K from AmericanOptions.tex (3.15).
Trigger	pureExercise(int nBranch) The naive exercise policy rho=(rho_t).
double	Q(int t, int nBranch) The approximation Q(t)=max{ E_t(h_{t+1}), E_t(h_{t+2}),..., E_t(h_T) } for the continuation value CV(t) computed from the current path.
double	upperBound(int nPath, Trigger exercise) This computes the upper bound U_0+Sum_{t where K_t=(E_t[U_{t+1}-U_t])^+) for the option price V_0.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

AmericanOption

public AmericanOption(Asset asset)

Constructor, does not initialize the option price path. This is left to the concrete subclasses which can decide wether analytic price formulas are available or not.

Parameters:

asset - the underlying asset.

Method Detail

get_T

public int get_T()

Number of time steps to horizon.

get_dt

public double get_dt()

Size of time step.

get_C

public double[] get_C()

Reference to the array C[ ] containing the discounted option price path.

get_S

public double[] get_S()

Reference to the array C[ ] containing the discounted option price path.

get_underlying

public Asset get_underlying()

Reference to the underlying asset.

currentDiscountedPayoff

public abstract double currentDiscountedPayoff(int t)

Option payoff discounted to time t=0 and computed from the current discounted price path of the underlying basket.

Parameters:

t - time of exercise

discountedPayoff

public RandomVariable discountedPayoff(int s)

The discounted option payoff as a random variable when exercised at a fixed time s.

Parameters:

s - fixed time of exercise.

currentDiscountedPayoff

public double currentDiscountedPayoff(int t,
                                      Trigger exercise)

The discounted option payoff h(rho_t) based on a given exercise strategy rho=(rho_t) computed from the current path of the underlying at time t, that is, the option has not been exercised before time t. The current path is given up to time t. From there the method computes a new path forward until the time of exercise.

This is the quantity h(rho_t) in the terminology of AmericanOption.tex.

Parameters:

t - current time.

exercise - exercise strategy rho=(rho_t).

discountedPayoff

public RandomVariable discountedPayoff(Trigger exercise)

The discounted option payoff based on a given exercise policy as a random variable.

Parameters:

exercise - exercise policy.

Q

public double Q(int t,
                int nBranch)

The approximation Q(t)=max{ E_t(h_{t+1}), E_t(h_{t+2}),..., E_t(h_T) } for the continuation value CV(t) computed from the current path.

Parameters:

t - current time

nBranch - number of path branches per conditional expectation

pureExercise

public Trigger pureExercise(int nBranch)

The naive exercise policy rho=(rho_t). See AmericanOption.tex. Given that the option has not been exercised before time t, the stopping time rho_t exercises at the first time t>=s where h_s>alpha*max{E_t(h(t+1),...,E_t(h(T))}. Here alpha>=1 is a parameter to be parametrized to a value close to one.

Current general implementation computes the conditional expectations by Monte Carlo simulation. Reimplement this in subclasses where analytic formulas are available.

Parameters:

nBranch - number of path branches spent on conditional expectations

discountedMonteCarloPrice

public double discountedMonteCarloPrice(int t,
                                        int nPath,
                                        Trigger exercise)

Monte Carlo price at time t dependent on a given exercise policy computed as a conditional expectation conditioned on information available at time t and computed from a sample of nPath (branches of) the price path of the underlying.

Parameters:

t - current time (determines information to condition on).

nPath - number of path branches used to compute the option price.

exercise - exercise policy

discountedMonteCarloPrice

public double discountedMonteCarloPrice(int nPath,
                                        Trigger exercise)

Monte Carlo option price at time t=0.

Parameters:

nPath - number of asset price paths used to compute the option price.

deltaU

public RandomVariable deltaU(int t,
                             Trigger exercise)

The random variable U_{t+1}-U_t conditioned on F_t only, that is, it is assumed that a path of underlying is already computed up to time t. See AmericanOption.tex

It is crucial that the trigger exercise provides an intelligent implementation of the mthod nextTime which moves the path of the underlying forward and does not use the default implementation in Triggers.Trigger.

Parameters:

t - current time

exercise - the exercise trigger

Kt

public RandomVariable Kt(int nPath,
                         Trigger exercise,
                         int t)

The random variable (E_t[U_{t+1}-U_t])^+ in the definition of the random variable K from AmericanOptions.tex (3.15). conditioning ignored.

Parameters:

exercise - exercise strategy rho_t.

t - current time.

nPath - number of paths spent on expectation.

upperBound

public double upperBound(int nPath,
                         Trigger exercise)

This computes the upper bound U_0+Sum_{t where K_t=(E_t[U_{t+1}-U_t])^+) for the option price V_0. Here the process U_t is defined as U_t=max{ h_t,h(rho_t) } where the exercise strategy rho_t is the parameter exercise below. See the file AmericanOptions.tex.

Parameters:

exercise - exercise strategy rho_t.

nPath - number of paths spent on expectation.