Libor.LiborDerivatives
Class Swaption

java.lang.Object
  Libor.LiborDerivatives.LiborDerivative
      Libor.LiborDerivatives.Swaption

public class Swaption

extends LiborDerivative

Forward start payer swaption swpn(T,[T_p,T_q],k) pays off h=B_pq(T)*(S_pq(T)-k)^+ at exercise time T=T_tau, where k is the strike rate and S_pq(T) is the value of the [T_p,T_q]-swap rate at time T.

Here B_pq is the annuity sum_{k=p}^{q-1}delta_kB_{k+1} as usual. The time T=T_tau of swaption exercise is assumed to be a point on the Libor tenor structure.

Constructor Summary

Swaption(LiborProcess LP, int tau, int p, int q, double kappa)
Libors needed for forward payoff are L_j, j>=tau at time T_tau (forward transporting from time T_tau).

Method Summary

double	analyticForwardPrice(int t) Analytic approximation to the swaption price.
double	controlVariateMean(int t) Mean of the control variate conditioned on the state of the Libor path at time t.
double[]	currentControlledForwardPayoff() Control variate is (B_p(T)-B_q(T))/B_n(T), T=T_tau.
double	currentForwardPayoff() Payoff at time T_tau from current Libor path transported forward from time T_tau to time T_n.
double	lognormalForwardPayoffSample() The forward transported payoff (as seen from time t=0) computed from a new sample of the LiborVector object U=(X^0_tau(T_tau),...,X^0_{n-1}(T_tau)), a log-normal approximation to the vector of true Libors U=(X_tau(T_tau),...,X_{n-1}(T_tau)).
static void	main(java.lang.String[] args) Test program.
double	Sigma(int t) Conditionally deterministic approximation to the aggregate volatility (square root of the quadratic variation <log(S_pq)>_t^T) of the swap rate S_pq to expiration conditioned on the state of the Libor process at time t.
double	wpq(int j, int t) The weight w^{p,q}_j(t) in the representation of the swap rate S_pq(t) as a convex combination of Libors.
ColtVector	xpq(int t) The vector x_pq(t) used in the computation of the approximation of the swap rate volatility.

Methods inherited from class Libor.LiborDerivatives.LiborDerivative

controlledForwardPayoff, controlledMonteCarloForwardPrice, forwardPayoff, lognormalForwardPayoff, lognormalMonteCarloForwardPrice, monteCarloForwardPrice

Methods inherited from class java.lang.Object

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

Constructor Detail

Swaption

public Swaption(LiborProcess LP,
                int tau,
                int p,
                int q,
                double kappa)

Libors needed for forward payoff are L_j, j>=tau at time T_tau (forward transporting from time T_tau).

Parameters:

LP - underlying Libor process.

p - swap begins at T_p.

q - swap ends at T_q.

tau - swaption exercises at time T_tau<=T_p.

kappa - strike rate.

Method Detail

wpq

public double wpq(int j,
                  int t)

The weight w^{p,q}_j(t) in the representation of the swap rate S_pq(t) as a convex combination of Libors. See LiborProcess.ps.

Parameters:

j - Libor index in [p,q).

t - current discrete time.

xpq

public ColtVector xpq(int t)

The vector x_pq(t) used in the computation of the approximation of the swap rate volatility. See LiborProcess.ps.

Parameters:

t - current discrete time.

Sigma

public double Sigma(int t)

Conditionally deterministic approximation to the aggregate volatility (square root of the quadratic variation <log(S_pq)>_t^T) of the swap rate S_pq to expiration conditioned on the state of the Libor process at time t.

This quantity is used in the analytic approximation to the swaption price analyticForwardPrice(int). See LiborProcess.ps

Parameters:

t - current discrete time.

currentForwardPayoff

public double currentForwardPayoff()

Payoff at time T_tau from current Libor path transported forward from time T_tau to time T_n.

Specified by:

currentForwardPayoff in class LiborDerivative

controlVariateMean

public double controlVariateMean(int t)

Mean of the control variate conditioned on the state of the Libor path at time t. This is (B_p(t)-B_q(t))/B_n(t) (a P_n-martingale), see LiborProcess.ps.

Overrides:

controlVariateMean in class LiborDerivative

Parameters:

t - current discrete time t<=T.

currentControlledForwardPayoff

public double[] currentControlledForwardPayoff()

Control variate is (B_p(T)-B_q(T))/B_n(T), T=T_tau. See LiborProcess.ps

Overrides:

currentControlledForwardPayoff in class LiborDerivative

lognormalForwardPayoffSample

public double lognormalForwardPayoffSample()

The forward transported payoff (as seen from time t=0) computed from a new sample of the LiborVector object U=(X^0_tau(T_tau),...,X^0_{n-1}(T_tau)), a log-normal approximation to the vector of true Libors U=(X_tau(T_tau),...,X_{n-1}(T_tau)). Recall X_j(t)=delta_jL_j(t), see document LiborProcess.ps.

Overrides:

lognormalForwardPayoffSample in class LiborDerivative

analyticForwardPrice

public double analyticForwardPrice(int t)

Analytic approximation to the swaption price. Based on a log-normal distribution of the swap rate conditioned on the state of the Libor path at the time t of pricing.

Overrides:

analyticForwardPrice in class LiborDerivative

Parameters:

t - current discrete time.

main

public static void main(java.lang.String[] args)

Test program. Allocates a Libor process of dimension n=15 and prices the (at the money) swaption swpn(T_5,[T_5,T_15],0.04).

The Monte Carlo forward price of this swaption at time T_n is compared to the analytic price and to the Monte carlo price based on the log-normal Libor approximation X0 and the correlation of the payoff with the control variate computed. The forward transporting and discounting involves all Libors L_j, j>=5.