MultExercisesOptions        package:fOptions        R Documentation

_V_a_l_u_a_t_i_o_n _o_f _M_u_t_i_p_l_e _E_x_e_r_c_i_s_e_s _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  multiple
     exercise options. Multiple exercises options,  as the name
     implies, are options whose payoff is based  on multiple exercise
     dates. 

     The functions are:

       'ExecutiveStockOption'    Executive Stock Option,
       'ForwardStartOption'      Forward Start Option,
       'RatchetOption'           Ratchet Option,
       'TimeSwitchOption'        Time Switch Option,
       'SimpleChooserOption'     Simple Chooser Option,
       'ComplexChooserOption'    Complex Chooser Option,
       'OptionOnOption'          Option On Option,
       'WriterExtendibleOption'  Writer Extendible Option,
       'HolderExtendibleOption'  Holder Extendible Option.

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

     ExecutiveStockOption(TypeFlag, S, X, Time, r, b, sigma, lambda)
     ForwardStartOption(TypeFlag, S, alpha, time1, Time2, r, b, sigma) 
     RatchetOption(TypeFlag, S, alpha, time1, Time2, r, b, sigma)
     TimeSwitchOption(TypeFlag, S, X, Time, r, b, sigma, A, m, dt)
     SimpleChooserOption(S, X, time1, Time2, r, b, sigma) 
     ComplexChooserOption(S, Xc, Xp, Time, Timec, Timep, r, b, sigma, doprint = FALSE)
     OptionOnOption(TypeFlag, S, X1, X2, time1, Time2, r, b, sigma, doprint = FALSE)
     WriterExtendibleOption(TypeFlag, S, X1, X2, time1, Time2, r, b, sigma)
     HolderExtendibleOption(TypeFlag, S, X1, X2, time1, Time2, r, b, sigma, A)

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

       A: [HolderExtendible*] - 
           defined by the amount 'A*dt' the investor receives at
          maturity time 'Time' for each time interval 'deltat' the
          corresponding asset price has exceeded  the exercise price
          'X', in the case of a call option, or the corresponding asset
          price has been below  the exercise price 'X', in the case of
          a put option. A numeric value. 

   alpha: [Ratchet*] - 
           the exercise price is 'alpha' times the asset price 'S'
          after the known time 'time'. 'alpha' is a numeric value. If
          'alpha' is less than unity, the call (put) will start
          '100*(1-alpha)' percent in  the money (out-of-the-money); if 
          'alpha' is unity,  the option will start at the money; and if
          'alpha'  is larger than unity, the call (put) will start 
          '100*(alpha-1)' percentage out of the money  (in-the-money). 

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

 doprint: a logical. Should the critical value 'I' be printed? By
          defaut 'FALSE'. 

      dt: the time interval; a numeric value. 

  lambda: the jump rate pa. 

       m: defined by the number of time units where the option has
          already fulfilled the thresold condition. This applies to
          cases, for which some of the option's total lifetime has
          already passed. An integer value. 

       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. 

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

Timec, Timep: [ComplexChooser*] - 
           decision time measured in years, e.g. 0.5 means 6 months.  
          'Timec', is the time to maturity of the call option, 
          'Timep', is the time to maturity of the put option,  both
          also measured in years. Numeric values. 

time1, Time2: the time to maturity, 'Time1', measured in years,  e.g.
          0.5 means 6 months, and the elapsed time in the  future,
          'Time2'. In detail, the forward start option  with time to
          maturity 'Time1' starts at-the-money or  proportinally
          in-the-money or out-of-the-money after this elapsed time
          'Time2' in the future. 

TypeFlag: usually a character string either '"c"' for a call option  or
          a '"p"' for a put option;
           [OptionOnOption] - 
           a character string either  '"cc"' for a call-on-call option,
          or '"cp"' for a call-on-put option, or '"pc"' for a
          put-on-call option, or  '"pp"' for a put-on-put option. 

       X: the exercise price, a numeric value. 

  Xc, Xp: [ComplexChooser*] - 
           the exercise price of the call option, 'Xc', and the 
          exercise price of the put option, 'Xp', numeric  values. 

  X1, X2: the exercise price of the underlying option, 'X1', and the
          exercise price of the option on the option, 'X2', numeric
          values. 

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

     *Executive Stock Options:* 

      Executive stock options are usually at-the-money options that are
      issued to motivate employees to act in the best interest of the 
     company. They cannot be sold and often last as long as 10 or 15 
     years. The executive model takes into account that an employee
     often  looses their options when they leave the company before
     expiration.  The value of an executive option equals the standard
     Black-Scholes  model multiplied by the probability that the
     employee will stay with  the firm until the option expires.
     Executive stock options can be priced analytically using a model
     published by Jennergren and Naslund (1993).  
      [Haug's Book, Chapter 2.1] 

     *Forward Start Options:* 

      A forward start option is an option which is paid for today, but 
     will start at some determined time in the future known as the
     issue  date. The option usually starts at-the-money or
     proportionally in  or out-of-the-money at a future date. The
     strike is set to a positive  constant a times the asset price S at
     a future date. If a is less  than one, the call (put) will start 1
     - a percent in-the-money  (out-of-the-money); if a is one, the
     option will start at-the-money;  and if a is larger than one, the
     call (put) will start a - 1 percent  out-of-the-money
     (in-the-money).[1] Forward start options can be  priced
     analytically using a model published by Rubinstein (1990). 
      [Haug's Book, Chapter 2.2] 

     *Ratchet [Compound] Options:* 

      A compound option is an option on an option. Therefore, when one 
     option is exercised, the underlying security is another option. 
     There are four types of possible compound options: a call on a
     call,  a call on a put, a put on a call, and a put on a put. The
     owner of  a compound option has until the expiration date of the
     compound  option to determine whether to exercise the compound
     option. If  exercised, the owner will receive the underlying
     option with its  own exercise price and time until expiration. If
     the underlying  option is exercised, the owner will receive the
     underlying security.  European compound options can be priced
     analytically using a model  published by Rubinstein (1991). A
     binomial lattice is used for the  numerical calculation of an
     American or European style exchange option. A ratchet option is
     also called sometimes a "moving strike option"  or "cliquet
     option". 
      [Haug's Book, Chapter 2.3] 

     *Time-Switch Options:* 

      For a discrete time-switch call (put) option, the holder receives
     an  amount ADt at expiration for each time interval, Dt, the
     corresponding  asset price has been above (below) the strike
     price. If some of the  option's total lifetime has passed, it is
     required to add a fixed  amount to the pricing formula. Discrete
     time-switch options can be  priced analytically using a model
     published by Pechtl (1995). 
      [Haug's Book, Chapter 2.4] 

     *Simple Chooser Options:* 

      A chooser option allows the holder to determine at some date,
     after the  trade date, whether the option becomes a plain vanilla
     call or put.  Chooser options are also called "as you like it"
     options. Chooser  options are useful for hedging a future event
     that might not occur.  Due to their increased flexibility, chooser
     options are more expensive  than plain vanilla options. It is
     assumed at the options expiration  date that a holder of the
     chooser option will choose the more valuable  of the put or call
     option. The less valuable option that was not chosen  will become
     worthless. Chooser options can be priced analytically using  a
     model introduced by Rubinstein (1991). 
      [Haug's Book, Chapter 2.5.1] 

     *Complex Chooser Options:* 

      A complex chooser option allows the holder to determine at some
     date,  after the trade date, whether the option is to be a
     standard call  chooser model, a complex chooser option will be
     more expensive than  a plain vanilla option. Complex chooser
     options can be priced analytically  using a model introduced by
     Rubinstein (1991). 
      [Haug's Book, Chapter 2.5.2] 

     *Option On Options:* 

      This derivative prices options on options. An option on an option
     is more  expensive to purchase than the underlying option itself,
     as the purchaser  has received a price guarantee and effectively
     extended the life of the  option. These options provide the
     benefit of a guaranteed price for the  option at a date in the
     future. Options on Options can be prices as published by Geske
     (1977). His model was later extended and discussed  by Geske
     (1979), Hodges and Selby (1987), and Rubinstein (1991).  
      [Haug's Book, Chapter 2.6] 

     *Writer [Holder] Extendible Options:* 

      Writer extendible options can be found embedded in various
     financial  contracts. For example, corporate warrants often give
     the issuing  firm the right to extend the life of the warrants.
     These options can  be exercised at their initial maturity, but are
     extended to a new  maturity if they are out-of-the-money at
     initial maturity. Discrete  time-switch options can be priced
     analytically using a model published  by Longstaff (1995). 
      [Haug's Book, Chapter 2.6]

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

     The option price, a numeric value.

_N_o_t_e:

     Options on options are also known as compound options or as 
     mother-and-daughter options.

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

     Diethelm Wuertz for this R-Port.

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

     Geske R. (1977); _The Valuation of Corporate Liabilities as
     Compound Options_, Journal of Financial and Quantitative Analysis,
     541-552.

     Geske R. (1979); _The Valuation of Compound Options_, Journal of
     Financial Economics 7, 63-81.

     Haug E.G. (1997); _The complete Guide to Option Pricing Formulas_,
      Chapter 2.8.1, McGraw-Hill, New York.

     Hodges S.D., Selby J.P. (1987); _On the Evaluation of Compound
     Options_; Management Science 33, 347-355.

     Jennergren L.P., Naslund B. (1993);  _A Comment on Valuation of
     Executive Stock Options and the  FASB Proposal_,  The Accounting
     Review 68, 179, 1993.

     Longstaff F.A. (1990); _Pricing Options with Extendible
     Maturities: Analysis and Applications_, Journal of Finance 45,
     474-491.

     Pechtl A. (1990);  _Classified Information_, Risk Magazine 8.

     Rubinstein, M. (1990);  _Pay Now, Choose Later_,  Risk Magazine 3.

     Rubinstein M. (1991);  _Options for the Undecide_,  Risk Magazine
     4.

     Rubinstein M. (1991); _Double Trouble_; Risk Magazine 5.

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

     ## Examples from Chapter 2.1 - 2.7 in E.G. Haug's Option Guide (1997)

     ## ExecutiveStockOption [2.1]:
        xmpOptions("\nStart: Executive Stock Option > ")
        ExecutiveStockOption(TypeFlag = "c", S = 60, X = 64, Time = 2, 
         r = 0.07, b = 0.07-0.03, sigma = 0.38, lambda = 0.15) 
         
     ## ForwardStartOption [2.2]:
        xmpOptions("\nNext: Forward Start Option > ")
        ForwardStartOption(TypeFlag = "c", S = 60, alpha = 1.1, 
          time1 = 1, Time2 = 1/4, r = 0.08, b = 0.08-0.04, sigma = 0.30)
          
     ## Ratchet Option [2.3]:
        xmpOptions("\nNext: Ratchet Option > ")
        RatchetOption(TypeFlag = "c", S = 60, alpha = 1.1, time1 = c(1.00, 0.75), 
          Time2 = c(0.75, 0.50), r = 0.08, b = 0.04, sigma = 0.30)
          
     ## Time Switch Option [2.4]:
        xmpOptions("\nNext: Time Switch Option > ")
        TimeSwitchOption(TypeFlag = "c", S = 100, X = 110, Time = 1, 
         r = 0.06, b = 0.06, sigma = 0.26, A = 5, m = 0, dt = 1/365)
         
     ## Simple Chooser Option [2.5.1]:
        xmpOptions("\nNext: Simple Chooser Option > ")
        SimpleChooserOption(S = 50, X = 50, time1 = 1/4, Time2 = 1/2, 
          r = 0.08, b = 0.08, sigma = 0.25)  
             
     ## Complex Chooser Option [2.5.2]:
        xmpOptions("\nNext: Complex Chooser Option > ")
        ComplexChooserOption(S = 50, Xc = 55, Xp = 48, Time = 0.25, 
          Timec = 0.50, Timep = 0.5833, r = 0.10, b = 0.1-0.05, 
          sigma = 0.35, doprint = TRUE)
          
     ## Option On Option [2.6]:
        xmpOptions("\nNext: Option On Option > ")
        OptionOnOption(TypeFlag = "pc", S = 500, X1 = 520, X2 = 50, 
          time1 = 1/2, Time2 = 1/4, r = 0.08, b = 0.08-0.03, sigma = 0.35)
         
     ## Holder Extendible Option [2.7.1]:
        xmpOptions("\nNext: Holder Extendible Option > ")
        HolderExtendibleOption(TypeFlag = "c", S = 100, X1 = 100, 
          X2 = 105, time1 = 0.50, Time2 = 0.75, r = 0.08, b = 0.08, 
          sigma = 0.25, A = 1)
          
     ## Writer Extendible Option [2.7.2]:
        xmpOptions("\nNext: Writer Extendible Option > ")
        WriterExtendibleOption(TypeFlag = "c", S = 80, X1 = 90, X2 = 82,
          time1 = 0.50, Time2 = 0.75, r = 0.10, b = 0.10, sigma = 0.30)
      

