Class TouchOption

java.lang.Object

com.quantfinlib.pricing.TouchOption

public final class TouchOption
extends Object

One-touch and no-touch options (pay-at-expiry) under continuously monitored geometric Brownian motion — with the barrier-hitting probability itself exposed, since desks quote one-touches as (roughly) discounted hit probabilities.

With log-drift m = r − q − σ²/2 and barrier log-distance h = ln(H/S), the reflection principle gives the probability of touching an upper barrier (h > 0) by expiry:

  P = N((−h + mT)/σ√T) + e^{2mh/σ²} · N((−h − mT)/σ√T)

and symmetrically for a lower barrier. A one-touch paying at expiry is payout·e^{−rT}·P; a no-touch is the complement. Conventions match BlackScholes: carry is the continuous yield (foreign rate for FX).

Pay-at-hit one-touches (rebate paid immediately on touch) discount to the hitting time instead: oneTouchPayAtHit(double, double, double, double, double, double, double). Double no-touch and double one-touch (two barriers, pay at expiry) are doubleNoTouch(double, double, double, double, double, double, double, double) and doubleOneTouch(double, double, double, double, double, double, double, double).

Field Summary

Fields

Modifier and Type	Field	Description
static final int	DOUBLE_TOUCH_SERIES_TERMS	Number of image terms on each side used by doubleNoTouchProbability(double, double, double, double, double, double, double): the series runs over n = -N..N.

Method Summary

Modifier and Type	Method	Description
static double	doubleNoTouch(double spot, double lower, double upper, double rate, double carry, double vol, double timeYears, double payout)	Double no-touch: pays payout at expiry if spot never touches lower or upper.
static double	doubleNoTouchProbability(double spot, double lower, double upper, double rate, double carry, double vol, double timeYears)	Probability that spot stays strictly inside (lower, upper) until timeYears under GBM with continuous monitoring.
static double	doubleOneTouch(double spot, double lower, double upper, double rate, double carry, double vol, double timeYears, double payout)	Double one-touch paying at expiry: pays payout if either barrier is touched before expiry.
static double	hitProbability(double spot, double barrier, double rate, double carry, double vol, double timeYears)	Probability that spot touches barrier at least once before timeYears, under GBM with the given rate/carry/vol.
static double	noTouch(double spot, double barrier, double rate, double carry, double vol, double timeYears, double payout)	Pays payout at expiry if the barrier never traded.
static double	oneTouch(double spot, double barrier, double rate, double carry, double vol, double timeYears, double payout)	Pays payout at expiry if the barrier traded at any point.
static double	oneTouchPayAtHit(double spot, double barrier, double rate, double carry, double vol, double timeYears, double payout)	One-touch paying payout at the moment the barrier is first touched (American digital, "rebate at hit").

Methods inherited from class Object

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

Field Details

DOUBLE_TOUCH_SERIES_TERMS

public static final int DOUBLE_TOUCH_SERIES_TERMS

Number of image terms on each side used by doubleNoTouchProbability(double, double, double, double, double, double, double): the series runs over n = -N..N. Term n is bounded by exp(-2 n^2 l^2 / (vol^2 T)) times a factor of order exp(2 |mu| n l), with l = ln(U/L); for l / (vol sqrt T) >= 0.1 the omitted tail is below 1e-15 of the payout (checked against the eigenfunction series in the tests).

See Also:

• Constant Field Values

Method Details

hitProbability

public static double hitProbability(double spot,
 double barrier,
 double rate,
 double carry,
 double vol,
 double timeYears)

Probability that spot touches barrier at least once before timeYears, under GBM with the given rate/carry/vol.

oneTouch

public static double oneTouch(double spot,
 double barrier,
 double rate,
 double carry,
 double vol,
 double timeYears,
 double payout)

Pays payout at expiry if the barrier traded at any point.

noTouch

public static double noTouch(double spot,
 double barrier,
 double rate,
 double carry,
 double vol,
 double timeYears,
 double payout)

Pays payout at expiry if the barrier never traded.

oneTouchPayAtHit

public static double oneTouchPayAtHit(double spot,
 double barrier,
 double rate,
 double carry,
 double vol,
 double timeYears,
 double payout)

One-touch paying payout at the moment the barrier is first touched (American digital, "rebate at hit"). Closed form of Reiner and Rubinstein (1991, "Breaking down the barriers", Risk 4(8)), as printed in Haug (2007, The Complete Guide to Option Pricing Formulas, 2nd ed., section 4.17.3, rebate term F). With b = rate - carry,

  mu     = (b - vol^2/2) / vol^2
  lambda = sqrt(mu^2 + 2 rate / vol^2)
  z      = ln(H/S) / (vol sqrt T) + lambda vol sqrt T
  price  = payout [ (H/S)^(mu+lambda) N(eta z) + (H/S)^(mu-lambda) N(eta z - 2 eta lambda vol sqrt T) ]

with eta = +1 for a lower barrier (H < S) and eta = -1 for an upper one. The formula is the Laplace transform E[e^{-rate tau} 1{tau <= T}] of the first-passage time tau, so rate discounts from the hitting time, not from expiry. For FX, rate is the domestic rate and carry the foreign rate, and the payout is in domestic currency. With rate = 0 it equals oneTouch(double, double, double, double, double, double, double) (tested). Requires mu^2 + 2 rate / vol^2 >= 0; a more negative domestic rate has no real lambda and is rejected.

Parameters:

barrier - the touch level; above spot is an up-touch, below a down-touch

Returns:

present value in the payout currency; payout if spot is at the barrier

doubleNoTouchProbability

public static double doubleNoTouchProbability(double spot,
 double lower,
 double upper,
 double rate,
 double carry,
 double vol,
 double timeYears)

Probability that spot stays strictly inside (lower, upper) until timeYears under GBM with continuous monitoring. Image-sum series of Ikeda and Kunitomo (1992, "Pricing options with curved boundaries", Mathematical Finance 2(4)) with flat boundaries, written in log space with x = ln(S/L), l = ln(U/L), m = rate - carry - vol^2/2, a = m / vol^2, s = vol sqrt T:

  P = sum_{n=-N..N} e^{-2 a n l}     [N((l - x + 2nl - mT)/s) - N((2nl - x - mT)/s)]
                  - e^{-2 a (x + n l)} [N((l + x + 2nl - mT)/s) - N((x + 2nl - mT)/s)]

with N = DOUBLE_TOUCH_SERIES_TERMS. Returns 0 when spot is on or outside a barrier.

doubleNoTouch

public static double doubleNoTouch(double spot,
 double lower,
 double upper,
 double rate,
 double carry,
 double vol,
 double timeYears,
 double payout)

Double no-touch: pays payout at expiry if spot never touches lower or upper. Equals payout e^{-rate T} doubleNoTouchProbability(...); rate is the domestic rate and carry the foreign rate for FX.

doubleOneTouch

public static double doubleOneTouch(double spot,
 double lower,
 double upper,
 double rate,
 double carry,
 double vol,
 double timeYears,
 double payout)

Double one-touch paying at expiry: pays payout if either barrier is touched before expiry. The complement of doubleNoTouch(double, double, double, double, double, double, double, double).