Option

Overview

Package

Class

Tree

Deprecated

Index

Help

PREV CLASS NEXT CLASS

FRAMES NO FRAMES

SUMMARY: NESTED | FIELD | CONSTR | METHOD

DETAIL: FIELD | CONSTR | METHOD

Options
Class Option

java.lang.Object
  Options.Option

Direct Known Subclasses:: CallAsPathDependent, DigitalOption, PathIndependentOption

public abstract class Option
extends java.lang.Object

Interface and default methods to price and hedge a possibly path * path dependent European option on a single underlying asset.

* *

We define a variety of methods by providing an implementation * which prints an error message and aborts the program. Admittedly * this is less elegant than throwing an exception but sorry I am too lazy * to write all these try-catch blocks. Mostly a call to such a method * when it is undefined is a fatal error in any case.

* *

It would be more elegant to define these methods as abstract but then * each concrete subclass has to implement them in some way. Proceeding * as we do we only have to implement what we need.

* *

Fundamental methods which must have nontrivial implementations in each * concrete subclass are declared as abstract if they cannot be implemented * at this level of generality.

* *

Pricing: analytic option prices (if analytic formulas exist) and * Monte Carlo prices with and without control variates are computed.

* * Default control variate: as default control * variate for the discounted option payoff we use the discounted * price S[T]=S^B(t) of the underlying at expiration simulated under the * risk neutral probability.

* *

Note that exp(qt)S^B(t) is a martingale in the risk neutral probability * P_B. Thus the conditional control variate mean is given as * E^{P_B}_t[S^B(T)]=exp(-q(T-t))S^B(t). * See book, chapter 3, section 1, equation 3.15.

* *

Hedging: The profit and loss of * hedging the option using various hedge deltas:

analytic deltas
analytic quotient deltas
analytic market deltas
minimum variance deltas
Monte Carlo deltas
quotient deltas
market deltas

is analyzed by computing mean and standard deviation from a sample of * price paths of the underlying asset. See the file "HedgeWeights.ps" * for the details.

* *

Information: the information available * at time t for conditioning is the history of the price path of the * underlying asset on the interval [0,t] (up to the present). * Conditioning on this information means to limit simulations of the price * of the underlying asset to paths which are branches of the current path where * branching occurs at time t.

* *

Warning: the abstract class Option 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.

Field Summary
`static int`	`ANALYTIC_DELTA` Flag identifying hedge weights.
`static int`	`ANALYTIC_MINIMUM_VARIANCE_DELTA` Flag identifying hedge weights.
`static int`	`ANALYTIC_QUOTIENT_DELTA` Flag identifying hedge weights.
`static int`	`MARKET_PROBABILITY` Flag indicating probability controlling paths of the underlying.
`static int`	`MINIMUM_VARIANCE_DELTA` Flag identifying hedge weights.
`static int`	`MONTE_CARLO_DELTA` Flag identifying hedge weights.
`static int`	`QUOTIENT_DELTA` Flag identifying hedge weights.
`static int`	`RISK_NEUTRAL_PROBABILITY` Flag indicating probability controlling paths of the underlying.

Constructor Summary
`Option(Asset asset, java.lang.String name)` Constructor, does not initialize the option price path.

Method Summary
`double`	`analyticDelta(int t)` Delta `dC/dS` computed from discounted analytic price * `C(t)` Implementation at this level * is an error message and program exit.
`double`	`analyticGamma(int t)` Gamma `d^2C/dS^2` computed from discounted analytic price * `C(t)`.
`double`	`analyticMinimumVarianceDelta(int whichProbability, int t)` Analytic approximation to the `minimumVarianceDelta(int, int, int)` hedge * weights.
`double`	`analyticQuotientDelta(int t)` Analytic approximation to the `quotientDelta(int, int)` hedge * weights.
`double`	`analyticSGamma(int t)` `S(t)*Gamma(t)`.
`double`	`analyticTheta(int t)` Theta `dC/dt` computed from discounted analytic price * `C(t)`.
`double`	`analyticThetaDelta(int t)` Theta of delta `dDelta/dt` computed from discounted analytic * price `C(t)`.
`double`	`analyticVega(int t)` Vega `dC/dsigma` computed from discounted analytic price * `C(t)`.
`double`	`conditionalControlVariateCorrelation(int t, int nPaths)` Correlation of the discounted payoff with the control * variate conditioned on information * available at time t.
`double`	`controlledDiscountedMonteCarloPrice(double precision, double confidence, int nSignChange)` Monte Carlo price at time t=0 using the * default control variate * with price paths of the underlying simulated until desired precision is * reached with desired confidence.
`double`	`controlledDiscountedMonteCarloPrice(int nPath)` Monte Carlo price at time t=0 computed using the * default control variate.
`double`	`controlledDiscountedMonteCarloPrice(int t, double precision, double confidence, int nSignChange)` Monte Carlo price at time t using the * default control variate * with price paths of the underlying simulated until desired precision is * reached with desired confidence.
`double`	`controlledDiscountedMonteCarloPrice(int t, int nPath)` Monte Carlo price at time t computed using the * default control variate.
`ControlledRandomVariable`	`controlledDiscountedPayoff()` The discounted payoff as a controlled random variable using the * default control variate.
`double`	`controlVariateCorrelation(int nPaths)` Correlation of the discounted payoff with the control variate at * time t=0 (no information to condition on).
`double`	`currentDiscountedHedgeGain(int whichDelta, int whichProbability, int nBranch, double fixed_trc, double prop_trc)` HEDGE ERROR: The discounted profit and loss of hedging a short * position in one call on one share of the underlying along one path of the * underlying.
`abstract double`	`currentDiscountedPayoff()` Option payoff discounted to time t=0 and computed from the current * discounted price path of the underlying.
`double`	`discountedAnalyticPrice(int t)` Discounted option price `C(t)`.
`RandomVariable`	`discountedHedgeGain(int whichDelta, int whichProbability, int nBranch, double fixed_trc, double prop_trc)` The hedge gain as a random variable.
`double`	`discountedMonteCarloPrice(double precision, double confidence, int nSignChange)` Monte Carlo price at time t=0 * with price paths of the underlying simulated until desired precision is * reached with desired confidence.
`double`	`discountedMonteCarloPrice(int nPath)` Monte Carlo option price at time t=0.
`double`	`discountedMonteCarloPrice(int t, double precision, double confidence, int nSignChange)` Monte Carlo price at time t computed as a conditional expectation * conditioned on information available at time t * with price paths of the underlying simulated until desired precision is * reached with desired confidence.
`double`	`discountedMonteCarloPrice(int t, int nPath)` Monte Carlo price at time t 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.
`RandomVariable`	`discountedPayoff()` The discounted option payoff 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.
`int`	`get_T()` Number of time steps to horizon.
`Asset`	`get_underlying()` Reference to the underlying asset.
`java.lang.String`	`getName()` The name of the option (a string).
`boolean`	`hasAnalyticPrice()` `True` if and analytic formula for the option price * is implemented, `false` otherwise.
`double`	`minimumVarianceDelta(int whichProbability, int t, int nPath)` The minimum variance delta at time t (see file "HedgeWeights.ps") * minimizing the squared hedge error over the next time step.
`double`	`minimumVarianceDelta(int whichProbability, int t, int nPath, Trigger rebalance)` Same as `minimumVarianceDelta(int,int,int)` but the * hedge is rebalanced when the next hedge trade is triggered and the * squared error minimized over the stochastic time interval until * the next hedge trade.
`double`	`monteCarloDelta(int t, int nPath)` The Monte Carlo delta at time t (see file "HedgeWeights.ps").
`void`	`newDiscountedPricePath(int whichProbability, int nPath)` Computes one path of the underlying S and a corresponding path C of the * option (discounted prices).
`double`	`quotientDelta(int t, int nPath)` Weight w=E^P_t(\DeltaC(t))/E^P_t(\DeltaS(t)) mich makes the * conditional mean of the hedge error between hedge trades equal to zero.
`double`	`quotientDelta(int t, int nPath, Trigger rebalance)` Same as `quotientDelta(int,int)` but the * hedge is rebalanced when the next hedge trade is triggered.
`boolean`	`underlyingIsCVA()` Returns `true` if the underlying is a * `ConstantVolatilityAsset`, `false` otherwise.
`boolean`	`underlyingIsDividendFreeCVA()` Returns `true` if the underlying is a dividend free * `ConstantVolatilityAsset`, `false` otherwise.

Methods inherited from class java.lang.Object

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

Field Detail

MARKET_PROBABILITY

public static final int MARKET_PROBABILITY

Flag indicating probability controlling paths of the underlying.

See Also:: Constant Field Values

RISK_NEUTRAL_PROBABILITY

public static final int RISK_NEUTRAL_PROBABILITY

Flag indicating probability controlling paths of the underlying.

See Also:: Constant Field Values