LookbackOptions           package:fOptions           R Documentation

_V_a_l_u_a_t_i_o_n _o_f _L_o_o_k_b_a_c_k _O_p_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description of functions to valuate  lookback
     options. The payoff from a pathdependent  lookback call (put)
     depends on the exercise price  being set to the minimum (maximum)
     asset price achieved  during the life of the option. Thus, a
     lookback call  (put) allows the purchaser to buy (sell) the asset
     at  its minimum (maximum) price. 

     The functions are:

       'FloatingStrikeLookbackOption'    Floating Strike Lookback Option,
       'FixedStrikeLookbackOption'       Fixed Strike Lookback Option,
       'PTFloatingStrikeLookbackOption'  PT Floating Strike Lookback Option,
       'PTFixedStrikeLookbackOption'     PT Fixed Strike Lookback Option,
       'ExtremeSpreadOption'             Extreme Spread Options.

_U_s_a_g_e:

     FloatingStrikeLookbackOption(TypeFlag, S, SMinOrMax, Time, r, b, sigma)
     FixedStrikeLookbackOption(TypeFlag, S, SMinOrMax, X, Time, r, b, sigma)
     PTFloatingStrikeLookbackOption(TypeFlag, S, SMinOrMax, time1, Time2, r, 
         b, sigma, lambda) 
     PTFixedStrikeLookbackOption(TypeFlag, S, X, time1, Time2, r, b, sigma)
     ExtremeSpreadOption(TypeFlag, S, SMin, SMax, time1, Time2, r, b, sigma)

_A_r_g_u_m_e_n_t_s:

       b: the annualized cost-of-carry rate, a numeric value;  e.g. 0.1
          means 10% pa. 

  lambda: The 'lambda' factor enables the creation of so-called
          "fractional" lookback options where the strike is fixed at
          some percentage or below the extremum, i.e. 'lambda'  is
          greater than 1 for calls, and between 0 and 1 for puts. 

       r: the annualized rate of interest, a numeric value;  e.g. 0.25
          means 25% pa. 

       S: the asset price, a numeric value. 

   sigma: the annualized volatility of the underlying security,  a
          numeric value; e.g. 0.3 means 30% volatility pa. 

SMax, SMin: [ExtremeSpread*] - 
           the maximum (minimum) value of the underlying asset. Note,
          the payoff at maturity of the extreme spread call (put)
          equals the  positive part of the difference between the
          maximum (minimum) value of the underlying asset, 'SMax', of
          the second (first) period and the maximum (minimum) of the
          underlying asset of the first (second) period. Likewise,
          reverse conditions are valid for the reverse extreme spread
          option. 

SMinOrMax: the lowest price observed of the underlying in the case of
          the coll, or the highest price in the case of the put. A 
          numeric value. 

    Time: the time to maturity measured in years, a numeric value; 
          e.g. 0.5 means 6 months. 

time1, Time2: [PTFloatingStrikeLookback*] - 
           the time to the end of the lookback period 'time1', and the
          time to expiry 'Time2' where 'time1<Time2', 
           [PTFixedStrikeLookback*] - 
           the predetermined time 'time1' where the lookback  period
          starts, and the time to expiry 'Time2', 
           [ExtremeSpread*] - 
           the two time periods, one starting today and ending at 
          'time1', and the other starting at 'time1' and ending at the
          maturity time 'Time2' of the option. 

TypeFlag: usually a character string either '"c"' for a call option  or
          a '"p"' for a put option, except for 
           [ExtremeSpread*] - 
           a character string either, 
           '"c"' for the extreme call, 
           '"p"' for the extreme put, 
           '"cr"' for the reverse extreme call, 
           '"pr"' for the revers extreme put. 

       X: the exercise price, a numeric value. 

_D_e_t_a_i_l_s:

     *Floating Strike Lookback Options:* 

      The lookback call (put) option gives the holder the right to buy
     (sell)  an asset at its lowest (highest) price observed during the
     life of the  option. This observed price is applied as the strike
     price. The payout  for a call option is essentially the asset
     price minus the minimum spot  price observed during the life of
     the option. The payout for a put option  is essentially the
     maximum spot price observed during the life of the  option minus
     the asset price. Therefore, a floating strike lookback  option is
     always in the money and should always be exercised. Floating 
     strike options can be priced analytically using a model introduced
     by  Goldman, Sosin, and Gatto (1979). Monte Carlo simulation is
     used for  the numerical calculation of a European style floating
     strike options. 
      [Haug's Book, Chapter 2.9.1] 

     *Fixed Strike Lookback Options:* 

      For a fixed strike lookback option, the strike price is known in
     advance.  The call option payoff is given by the difference
     between the maximum  observed price of the underlying asset during
     the life of the option and  the fixed strike price. The put option
     payoff is given by the difference  between the fixed strike price
     and the minimum observed price of the  underlying asset during the
     life of the option. A fixed strike lookback  call (put) option
     payoff is equal to that of a standard plain call (put)  option
     when the final asset price is the maximum (minimum) observed value
      during the options life. Fixed strike lookback options can be
     priced  analytically using a model introduced by Conze and
     Viswanathan (1991). 
      [Haug's Book, Chapter 2.9.2] 

     *Partial-Time Floating Strike Options:* 

      For a partial-time floating strike lookback option, the lookback
     period  starts at time zero and ends at an arbitrary date before
     expiration.  Except for the partial lookback period, the option is
     similar to a  floating strike lookback option. The partial-time
     floating strike  lookback option is cheaper than a similar
     standard floating strike  lookback option. Partial-time floating
     strike lookback options can be  priced analytically using a model
     introduced by Heynen and Kat (1994). 
      [Haug's Book, Chapter 2.9.3] 

     *Partial-Time Fixed Strike Options:* 

      For a partial-time fixed strike lookback option, the lookback
     period  starts at a predetermined date after the initialization
     date of the  option. The partial-time fixed strike lookback call
     option payoff is  given by the difference between the maximum
     observed price of the  underlying asset during the lookback period
     and the fixed strike  price. The partial-time fixed strike
     lookback put option payoff is  given by the difference between the
     fixed strike price and the  minimum observed price of the
     underlying asset during the lookback  period. The partial-time
     fixed strike lookback option is cheaper than  a similar standard
     fixed strike lookback option. Partial-time fixed  strike lookback
     options can be priced analytically using a model  introduced by
     Heynen and Kat (1994). 
      [Haug's Book, Chapter 2.9.4] 

     *Extreme Spread Options:* 

      The time to maturity of an extreme spread option is divided into
     two  periods: one period starting at time zero and ending at some
     arbitrary  date, and another starting at that arbitrary date and
     ending at the  expiration date. A payoff at maturity of an extreme
     spread call (put)  option equals the positive part of the
     difference between the maximum  (minimum) value of the underlying
     asset of the second (first) period  and the maximum (minimum)
     value of the underlying asset of the first  (second) period.[1]
     The payoff at expiration of a reverse extreme  spread call (put)
     option equals the positive part of the difference  between the
     minimum (maximum) of the underlying asset of the second  (first)
     period and the minimum (maximum) value of the underlying asset  of
     the first (second) period. Extreme spread options can be priced 
     analytically using a model introduced by Bermin (1996). 
      [Haug's Book, Chapter 2.9.5]

_V_a_l_u_e:

     The option price, a numeric value.

_N_o_t_e:

     The functions implement the algorithms to valuate plain vanilla 
     options as described in Chapter 2.9 of Haug's Book (1997).

_A_u_t_h_o_r(_s):

     Diethelm Wuertz for this R-Port.

_R_e_f_e_r_e_n_c_e_s:

     Bermin H.P. (1996b); _Exotic Lookback Options: The case of Extreme
     Spread Options_, Department of Economics, Lund University, Sweden.

     Conze A., Viswanathan R. (1991); _Path Dependent Options: The Case
     of Lookback Options_, Journal of Finance 46, 1893-1907.

     Goldmann B.M., Sosin H.B., Gatto M.A. (1993); _Path Dependent
     Options: Buy at the Low, Sell at the High_, Journal of Finance 34,
     1111.

     Haug E.G. (1997);  _The Complete Guide to Option Pricing
     Formulas_,  McGraw-Hill, New York.

     Heynen R.C., Kat H.M. (1994); _Selective Memory_, Risk Magazine 7,
     1994.

_E_x_a_m_p_l_e_s:

     ## Examples from Chapter 2.9 in E.G. Haug's Option Guide (1997)

     ## Floating Strike Lookback Option [2.9.1]:
        xmpOptions("\nStart: Floating Strike Lookback Option > ")
        FloatingStrikeLookbackOption(TypeFlag = "c", S = 120, 
          SMinOrMax = 100, Time = 0.5, r = 0.10, b = 0.10-0.06, 
          sigma = 0.30)
       
     ## Fixed Strike Lookback Option [2.9.2]:
        xmpOptions("\nNext: Fixed Strike Lookback Option > ")
        FixedStrikeLookbackOption(TypeFlag = "c", S = 100, 
          SMinOrMax = 100, X = 105, Time = 0.5, r = 0.10, b = 0.10, 
          sigma = 0.30)
        
     ## Partial Time Floating Strike Lookback Option [2.9.3]:
        xmpOptions("\nNext: PT Floating Strike Option > ")
        PTFloatingStrikeLookbackOption(TypeFlag = "p", S = 90, 
          SMinOrMax = 90, time1 = 0.5, Time2 = 1, r = 0.06, b = 0.06, 
          sigma = 0.20, lambda  = 1)
        
     ## Partial Time Fixed Strike Lookback Option [2.9.4]:
        xmpOptions("\nNext: PT Fixed Strike Lookback Option > ")
        PTFixedStrikeLookbackOption(TypeFlag = "c", S = 100, X = 90, 
          time1 = 0.5, Time2 = 1, r = 0.06, b = 0.06, sigma = 0.20)
          
     ## Extreme Spread Option [2.9.5]:
        xmpOptions("\nNext: PT Fixed Strike Lookback Option > ")
        ExtremeSpreadOption(TypeFlag = "c", S = 100, SMin = NA, 
          SMax = 110, time1 = 0.5, Time2 = 1, r = 0.1, b = 0.1, 
          sigma = 0.30)
        ExtremeSpreadOption(TypeFlag = "cr", S = 100, SMin = 90, 
          SMax = NA, time1 = 0.5, Time2 = 1, r = 0.1, b = 0.1, 
          sigma = 0.30)
             

