R/LookbackOptions.R

In fExoticOptions: Rmetrics - Pricing and Evaluating Exotic Option

# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA  02111-1307  USA


################################################################################
# FUNCTION:                           DESCRIPTION:
# Lookback Options:
#   FloatingStrikeLookbackOption        Floating Strike Lookback Option
#   FixedStrikeLookbackOption           Fixed Strike Lookback Option
#   PTFloatingStrikeLookbackOption      Partial Floating Strike LB Option
#   PTFixedStrikeLookbackOption         Partial Fixed Strike LB Option
#   ExtremeSpreadOption                 Extreme Spread Option
################################################################################


FloatingStrikeLookbackOption =
    function(TypeFlag = c("c", "p"), S, SMinOrMax, Time, r, b, sigma,
             title = NULL, description = NULL)
{   # A function implemented by Diethelm Wuertz

    # Description:
    #   Floating strike lookback options

    # References:
    #   Haug, Chapter 2.9.1

    # FUNCTION:

    # Comute Settungs:
    TypeFlag = TypeFlag[1]
    if (TypeFlag == "c") m = SMinOrMax # Min
    if (TypeFlag == "p") m = SMinOrMax # Max
    a1 = (log(S / m) + (b + sigma^2 / 2) * Time) / (sigma * sqrt(Time))
    a2 = a1 - sigma * sqrt (Time)

    # Calculate Call and Put:
    if (TypeFlag == "c")
        FloatingStrikeLookback = (S * exp ((b - r) * Time) * CND(a1) -
                                  m * exp(-r * Time) * CND(a2) + exp
                                  (-r * Time) * sigma^2 / (2 * b) * S
                                  * ((S / m)^(-2 * b / sigma^2) *
                                  CND(-a1 + 2 * b / sigma *
                                  sqrt(Time)) - exp(b * Time) *
                                  CND(-a1)))

    if (TypeFlag == "p")
        FloatingStrikeLookback = (m * exp (-r * Time) * CND(-a2) - S *
                                  exp((b - r) * Time) * CND(-a1) + exp
                                  (-r * Time) * sigma^2 / (2 * b) * S
                                  * (-(S / m)^(-2 * b / sigma^2) *
                                  CND(a1 - 2 * b / sigma * sqrt(Time))
                                  + exp(b * Time) * CND(a1)))

    # Parameters:
    # TypeFlag = c("c", "p"), S, SMinOrMax, Time, r, b, sigma
    param = list()
    param$TypeFlag = TypeFlag
    param$S = S
    param$SMinOrMax = SMinOrMax
    param$Time = Time
    param$r = r
    param$b = b
    param$sigma = sigma

    # Add title and description:
    if (is.null(title)) title = "Floating Strike Lookback Option\n"
    if (is.null(description)) description = as.character(date())

    # Return Value:
    new("fOPTION",
        call = match.call(),
        parameters = param,
        price = FloatingStrikeLookback,
        title = title,
        description = description
        )
}


# ------------------------------------------------------------------------------


FixedStrikeLookbackOption =
    function(TypeFlag = c("c", "p"), S, SMinOrMax, X, Time, r, b, sigma,
             title = NULL, description = NULL)
{   # A function implemented by Diethelm Wuertz

    # Description:
    #   Fixed strike lookback options

    # References:
    #   Haug, Chapter 2.9.2

    # FUNCTION:

    # Compute Settings:
    TypeFlag = TypeFlag[1]
    if (TypeFlag == "c") m = SMinOrMax
    if (TypeFlag == "p") m = SMinOrMax
    d1 = (log(S / X) + (b + sigma^2 / 2) * Time) / (sigma * sqrt(Time))
    d2 = d1 - sigma * sqrt (Time)
    e1 = (log(S / m) + (b + sigma^2 / 2) * Time) / (sigma * sqrt(Time))
    e2 = e1 - sigma * sqrt (Time)

    # Calculate Call and Put:
    if (TypeFlag == "c" && X > m)
        FixedStrikeLookback = (S * exp ((b - r) * Time) * CND(d1) - X
                               * exp(-r * Time) * CND(d2) + S * exp
                               (-r * Time) * sigma^2 / (2 * b) * (-(S
                               / X)^(-2 * b / sigma^2) * CND(d1 - 2 *
                               b / sigma * sqrt(Time)) + exp(b * Time)
                               * CND(d1)))
    if (TypeFlag == "c" && X <= m)
        FixedStrikeLookback = (exp (-r * Time) * (m - X) + S *
                               exp((b-r) * Time) * CND(e1) - exp(-r *
                               Time) * m * CND(e2) + S * exp (-r *
                               Time) * sigma^2 / (2 * b) * (-(S /
                               m)^(-2 * b / sigma^2) * CND(e1 - 2 * b
                               / sigma * sqrt(Time)) + exp(b * Time) *
                                                            CND(e1)))
    if (TypeFlag == "p" && X < m)
        FixedStrikeLookback = (-S * exp ((b - r) * Time) * CND(-d1) +
                               X * exp(-r * Time) * CND(-d1 + sigma *
                               sqrt(Time)) + S * exp (-r * Time) *
                               sigma^2 / (2 * b) * ((S / X)^(-2 * b /
                               sigma^2) * CND(-d1 + 2 * b / sigma *
                               sqrt(Time)) - exp(b*Time) * CND(-d1)))
    if (TypeFlag == "p" && X >= m)
        FixedStrikeLookback = (exp (-r * Time) * (X - m) - S * exp((b
                               - r) * Time) * CND(-e1) + exp(-r *
                               Time) * m * CND(-e1 + sigma *
                               sqrt(Time)) + exp (-r * Time) * sigma^2
                               / (2 * b) * S * ((S / m)^(-2 * b /
                               sigma^2) * CND(-e1 + 2 * b / sigma *
                               sqrt(Time)) - exp(b*Time) * CND(-e1)))

    # Parameters:
    # TypeFlag = c("c", "p"), S, SMinOrMax, X, Time, r, b, sigma
    param = list()
    param$TypeFlag = TypeFlag
    param$S = S
    param$SMinOrMax = SMinOrMax
    param$X = X
    param$Time = Time
    param$r = r
    param$b = b
    param$sigma = sigma

    # Add title and description:
    if (is.null(title)) title = "Fixed Strike Lookback Option\n"
    if (is.null(description)) description = as.character(date())

    # Return Value:
    new("fOPTION",
        call = match.call(),
        parameters = param,
        price = FixedStrikeLookback,
        title = title,
        description = description
        )
}


# ------------------------------------------------------------------------------


PTFloatingStrikeLookbackOption =
    function(TypeFlag = c("c", "p"), S, SMinOrMax, time1, Time2, r, b,
             sigma, lambda, title = NULL, description = NULL)
{   # A function implemented by Diethelm Wuertz

    # Description:
    #   Partial-time floating strike lookback options

    # References:
    #   Haug, Chapter 2.9.3

    # FUNCTION:

    # Compute Settings:
    TypeFlag = TypeFlag[1]
    T2 = Time2
    t1 = time1
    if (TypeFlag == "c") m = SMinOrMax
    if (TypeFlag == "p") m = SMinOrMax
    d1 = (log(S / m) + (b + sigma^2 / 2) * T2) / (sigma * sqrt(T2))
    d2 = d1 - sigma * sqrt (T2)
    e1 = (b + sigma^2 / 2) * (T2 - t1) / (sigma * sqrt(T2 - t1))
    e2 = e1 - sigma * sqrt (T2 - t1)
    f1 = (log(S / m) + (b + sigma^2 / 2) * t1) / (sigma * sqrt(t1))
    f2 = f1 - sigma * sqrt (t1)
    g1 = log (lambda) / (sigma * sqrt(T2))
    g2 = log (lambda) / (sigma * sqrt(T2 - t1))

    # Calculate Call and Puts:
    if (TypeFlag == "c") {
        part1 = (S * exp ((b - r) * T2) * CND(d1 - g1) -
                 lambda * m * exp(-r * T2) * CND(d2 - g1))
        part2 = (exp (-r * T2) * sigma^2 / (2 * b) * lambda * S * ((S
                / m)^(-2 * b / sigma^2) * CBND(-f1 + 2*b*sqrt(t1) /
                sigma, -d1 + 2 * b * sqrt(T2) / sigma - g1, sqrt(t1 /
                T2)) - exp (b * T2) * lambda^(2 * b / sigma^2) *
                CBND(-d1 - g1, e1 + g2, -sqrt(1 - t1 / T2))) + S * exp
                ((b - r)*T2) * CBND(-d1 + g1, e1 - g2, -sqrt(1 - t1 /
                T2)))
        part3 = (exp (-r*T2) * lambda * m * CBND(-f2, d2 - g1,
                -sqrt(t1 / T2)) - exp (-b * (T2-t1)) * exp((b - r)*T2)
                * (1 + sigma^2 / (2 * b)) * lambda * S * CND(e2 - g2)
                * CND(-f1)) }
    if (TypeFlag == "p") {
        part1 = (lambda * m * exp (-r * T2) * CND(-d2 + g1) -
                 S * exp((b - r) * T2) * CND(-d1 + g1))
        part2 = (-exp (-r * T2) * sigma^2 / (2 * b) * lambda * S * ((S
                 / m)^(-2 * b / sigma^2) * CBND(f1 - 2 * b * sqrt(t1)
                 / sigma, d1 - 2 * b * sqrt(T2) / sigma + g1, sqrt(t1
                 / T2)) - exp (b * T2) * lambda^(2 * b / sigma^2) *
                 CBND(d1 + g1, -e1 - g2, -sqrt(1 - t1 / T2))) - S *
                 exp ((b - r)*T2) * CBND(d1 - g1, -e1 + g2, -sqrt(1 -
                 t1 / T2)))
        part3 = (-exp (-r*T2) * lambda*m * CBND(f2,
                  -d2 + g1, -sqrt(t1 / T2)) + exp (-b * (T2-t1)) *
                 exp((b - r)*T2) * (1 + sigma^2 / (2 * b)) * lambda *
                  S * CND(-e2 + g2) * CND(f1)) }

    PartialFloatLookback = part1 + part2 + part3
    # Parameters:
    # TypeFlag = c("c", "p"), S, SMinOrMax, time1, Time2, r, b, sigma, lambda
    param = list()
    param$TypeFlag = TypeFlag
    param$S = S
    param$SMinOrMax
    param$time1 = time1
    param$Time2 = Time2
    param$r = r
    param$b = b
    param$sigma = sigma
    param$lambda = lambda

    # Add title and description:
    if (is.null(title)) title = "Partial Time Floating Strike Lookback Option"
    if (is.null(description)) description = as.character(date())

    # Return Value:
    new("fOPTION",
        call = match.call(),
        parameters = param,
        price = PartialFloatLookback,
        title = title,
        description = description
        )
}


# ------------------------------------------------------------------------------


PTFixedStrikeLookbackOption =
    function(TypeFlag = c("c", "p"), S, X, time1, Time2, r, b, sigma,
             title = NULL, description = NULL)
{   # A function implemented by Diethelm Wuertz

    # Description:
    #   Partial Time Fixed Strike Lookback Option

    # References:
    #   Haug, Chapter 2.9.4

    # FUNCTION:

    # Compute Settings:
    TypeFlag = TypeFlag[1]
    d1 = ((log(S / X) + (b + sigma^2 / 2) * Time2) /
          (sigma * sqrt(Time2)))
    d2 = d1 - sigma * sqrt(Time2)
    e1 = (((b + sigma^2 / 2) * (Time2 - time1)) /
          (sigma * sqrt(Time2 - time1)))
    e2 = e1 - sigma * sqrt(Time2 - time1)
    f1 = (log(S / X) + (b + sigma^2 / 2) * time1) / (sigma * sqrt(time1))
    f2 = f1 - sigma * sqrt(time1)

    # Calculate Call and Put:
    if (TypeFlag == "c") {
        PartialFixedLB = (S * exp((b - r) * Time2) * CND(d1) - exp(-r
                          * Time2) * X * CND(d2) + S * exp(-r * Time2)
                          * sigma^2 / (2 * b) * (-(S / X)^(-2 * b /
                          sigma^2) * CBND(d1 - 2 * b * sqrt(Time2) /
                          sigma, -f1 + 2 * b * sqrt(time1) / sigma,
                          -sqrt(time1 / Time2)) + exp(b * Time2) *
                          CBND(e1, d1, sqrt(1 - time1 / Time2))) - S *
                          exp((b - r) * Time2) * CBND(-e1, d1, -sqrt(1
                          - time1 / Time2)) - X * exp(-r * Time2) *
                          CBND(f2, -d2, -sqrt(time1 / Time2)) + exp(-b
                          * (Time2 - time1)) * (1 - sigma^2 / (2 * b))
                          * S * exp((b - r) * Time2) * CND(f1) *
                          CND(-e2)) }
    if (TypeFlag == "p") {
        PartialFixedLB = (X * exp(-r * Time2) * CND(-d2) - S * exp((b
                          - r) * Time2) * CND(-d1) + S * exp(-r *
                          Time2) * sigma^2 / (2 * b) * ((S / X)^(-2 *
                          b / sigma^2) * CBND(-d1 + 2 * b *
                          sqrt(Time2) / sigma, f1 - 2 * b *
                          sqrt(time1) / sigma, -sqrt(time1 / Time2)) -
                          exp(b * Time2) * CBND(-e1, -d1, sqrt(1 -
                          time1 / Time2))) + S * exp((b - r) * Time2)
                          * CBND(e1, -d1, -sqrt(1 - time1 / Time2)) +
                          X * exp(-r * Time2) * CBND(-f2, d2,
                          -sqrt(time1 / Time2)) - exp(-b * (Time2 -
                          time1)) * (1 - sigma^2 / (2 * b)) * S *
                          exp((b - r) * Time2) * CND(-f1) * CND(e2)) }

    # Parameters:
    # TypeFlag = c("c", "p"), S, X, time1, Time2, r, b, sigma
    param = list()
    param$TypeFlag = TypeFlag
    param$S = S
    param$X = X
    param$time1 = time1
    param$Time2 = Time2
    param$r = r
    param$b = b
    param$sigma = sigma

    # Add title and description:
    if (is.null(title)) title = "Partial Time Fixed Strike Lookback Option"
    if (is.null(description)) description = as.character(date())

    # Return Value:
    new("fOPTION",
        call = match.call(),
        parameters = param,
        price = PartialFixedLB,
        title = title,
        description = description
        )
}


# ------------------------------------------------------------------------------


ExtremeSpreadOption =
    function(TypeFlag = c("c", "p", "cr", "pr"), S, SMin, SMax, time1, Time2,
             r, b, sigma, title = NULL, description = NULL)
{   # A function implemented by Diethelm Wuertz

    # Description:
    #   Extreme Spread Option

    # References:
    #   Haug, Chapter 2.9.5

    # FUNCTION:

    # Compute Settings:
    TypeFlag = TypeFlag[1]
    v = sigma
    Time = Time2
    if (TypeFlag == "c"  || TypeFlag == "cr") { eta = +1 }
    if (TypeFlag == "p"  || TypeFlag == "pr") { eta = -1 }
    if (TypeFlag == "c"  || TypeFlag == "p")  { kappa = +1 }
    if (TypeFlag == "cr" || TypeFlag == "pr") { kappa = -1 }
    if (kappa * eta == +1) { Mo = SMax }
    if (kappa * eta == -1) { Mo = SMin }
    mu1 = b - v^2 / 2
    mu = mu1 + v^2
    m = log(Mo/S)
    ExtremeSpread = NA

    # Extreme Spread Option:
    if (kappa == 1) {
        ExtremeSpread = (eta * (S * exp((b - r) * Time) * (1 + v^2 /
                         (2 * b)) * CND(eta * (-m + mu * Time) /
                         (v*sqrt(Time))) - exp(-r * (Time - time1)) *
                         S * exp((b - r) * Time) * (1 + v^2 / (2 * b))
                         * CND(eta * (-m + mu * time1) /
                         (v*sqrt(time1))) + exp(-r * Time) * Mo *
                         CND(eta * (m - mu1 * Time) / (v*sqrt(Time)))
                         - exp(-r * Time) * Mo * v^2 / (2 * b) * exp(2
                         * mu1 * m / v^2) * CND(eta * (-m - mu1 *
                         Time) / (v*sqrt(Time))) - exp(-r * Time) * Mo
                         * CND(eta * (m - mu1 * time1) /
                         (v*sqrt(time1))) + exp(-r * Time) * Mo * v^2
                         / (2 * b) * exp(2 * mu1 * m / v^2) * CND(eta
                         * (-m - mu1 * time1) / (v*sqrt(time1))))) }

    # Reverse Extreme Spread Option:
    if (kappa == -1) {
        ExtremeSpread = (-eta * (S * exp((b - r) * Time) * (1 + v^2 /
                         (2 * b)) * CND(eta * (m - mu * Time) /
                         (v*sqrt(Time))) + exp(-r * Time) * Mo *
                         CND(eta * (-m + mu1 * Time) / (v*sqrt(Time)))
                         - exp(-r * Time) * Mo * v^2 / (2 * b) * exp(2
                         * mu1 * m / v^2) * CND(eta * (m + mu1 * Time)
                         / (v*sqrt(Time))) - S * exp((b - r) * Time) *
                         (1 + v^2 / (2 * b)) * CND(eta * (-mu * (Time
                         - time1)) / (v*sqrt(Time - time1))) - exp(-r
                         * (Time - time1)) * S * exp((b - r) * Time) *
                         (1 - v^2 / (2 * b)) * CND(eta * (mu1 * (Time
                         - time1)) / (v*sqrt(Time - time1))))) }

    # Parameters:
    # TypeFlag = c("c", "p", "cr", "pr"), S, SMin, SMax, time1, Time2,
    #   r, b, sigma
    param = list()
    param$TypeFlag = TypeFlag
    param$S = S
    param$SMin = SMin
    param$SMax = SMax
    param$time1 = time1
    param$Time2 = Time2
    param$r = r
    param$b = b
    param$sigma = sigma

    # Add title and description:
    if (is.null(title)) title = "Extreme Spread Option"
    if (is.null(description)) description = as.character(date())

    # Return Value:
    new("fOPTION",
        call = match.call(),
        parameters = param,
        price = ExtremeSpread,
        title = title,
        description = description
        )
}


################################################################################