BinaryOptions            package:fOptions            R Documentation

_V_a_l_u_a_t_i_o_n _o_f _B_i_n_a_r_y _O_p_t_i_o_n_s

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

     This is a collection of functions to valuate binary options. 
     Binary options, also known as digital options, have discontinuous 
     payoffs. They can be used as building blocks to develop options 
     with more complicated payoffs. For example, a regular European 
     call option is equivalent to a long position in an
     asset-or-nothing  call and a short position in a cash-or-nothing
     call, where the both  options have the same strike price and the
     cash payoff of the  cash-or-nothing option equals the strike
     price. Unlike standard  European style options, the payout for
     binary options does not  depend on how much it is in-the-money but
     rather whether or not  it is on the money. The options payoff is
     fixed at the options  inception and is based on the price of the
     underlying asset on  the expiration date. Binary options may also
     incorporate barriers,  as is the case with binary-barrier options. 

     The functions are:

       1  'GapOption'                    Gap Option,
       2  'CashOrNothingOption'          Cash Or Nothing Option,
       3  'TwoAssetCashOrNothingOption'  Two Asset Cash Or Nothing Option,
       4  'AssetOrNothingOption'         Asset Or Nothing Option,
       5  'SuperShareOption'             Super Share Option,
       6  'BinaryBarrierOption'          Binary Barrier Option.

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

     GapOption(TypeFlag, S, X1, X2, Time, r, b, sigma)
     CashOrNothingOption(TypeFlag, S, X, K, Time, r, b, sigma) 
     TwoAssetCashOrNothingOption(TypeFlag, S1, S2, X1, X2, K, Time, 
             r, b1, b2, sigma1, sigma2, rho)
     AssetOrNothingOption(TypeFlag, S, X, Time, r, b, sigma)
     SuperShareOption(S, XL, XH, Time, r, b, sigma)
     BinaryBarrierOption(TypeFlag, S, X, H, K, Time, r, b, sigma, eta, phi)

_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. 

  b1, b2: [TwoAssetCashOrNothing*] - the annualized cost-of-carry rate
          for the first and second asset, a numeric value. 

eta, phi: [BinaryBarrier*] - a set of parameters to price 28 different
          types of Binary  Barrier options:
           01: 'eta=+1, phi=NA, [S>H]' down-and-in
          cash-at-hit-or-nothing, 
           02: 'eta=-1, phi=NA, [S<H]' up-and-in
          cash-at-hit-or-nothing, 
           03: 'eta=+1, phi=NA, [S>H]' down-and-in
          asset-at-hit-or-nothing, 
           04: 'eta=-1, phi=NA, [S<H]' up-and-in
          asset-at-hit-or-nothing, 
           05: 'eta=+1, phi=-1, [S>H]' down-and-in
          cash-at-expiry-or-nothing, 
           06: 'eta=-1, phi=+1, [S<H]' up-and-in
          cash-at-expiry-or-nothing, 
           07: 'eta=+1, phi=-1, [S>H]' down-and-in
          asset-at-expiry-or-nothing, 
           08: 'eta=-1, phi=+1, [S<H]' up-and-in
          asset-at-expiry-or-nothing, 
           09: 'eta=+1, phi=+1, [S>H]' down-and-out cash-or-nothing, 
           10: 'eta=-1, phi=-1, [S<H]' up-and-out cash-or-nothing, 
           11: 'eta=+1, phi=+1, [S>H]' down-and-out asset-or-nothing, 
           12: 'eta=-1, phi=-1, [S<H]' up-and-out asset-or-nothing, 
           13: 'eta=+1, phi=+1, [S>H]' down-and-in cash-or-nothing
          call, 
           14: 'eta=-1, phi=+1, [S<H]' up-and-in cash-or-nothing call, 
           15: 'eta=+1, phi=+1, [S>H]' down-and-in asset-or-nothing
          call, 
           16: 'eta=-1, phi=+1, [S<H]' up-and-in asset-or-nothing call, 
           17: 'eta=+1, phi=-1, [S>H]' down-and-in cash-or-nothing put, 
           18: 'eta=-1, phi=-1, [S<H]' up-and-in cash-or-nothing put, 
           19: 'eta=+1, phi=-1, [S>H]' down-and-in asset-or-nothing
          put, 
           20: 'eta=-1, phi=-1, [S<H]' up-and-in asset-or-nothing put, 
           21: 'eta=+1, phi=+1, [S>H]' down-and-out cash-or-nothing
          call, 
           22: 'eta=-1, phi=+1, [S<H]' up-and-out cash-or-nothing call, 
           23: 'eta=+1, phi=+1, [S>H]' down-and-out asset-or-nothing
          call, 
           24: 'eta=-1, phi=-1, [S<H]' up-and-out asset-or-nothing
          call, 
           25: 'eta=+1, phi=-1, [S>H]' down-and-out cash-or-nothing
          put, 
           26: 'eta=-1, phi=-1, [S<H]' up-and-out cash-or-nothing put, 
           27: 'eta=+1, phi=-1, [S>H]' down-and-out asset-or-nothing
          put, 
           28: 'eta=-1, phi=-1, [S<H]' up-and-out asset-or-nothing put. 

       H: [BinaryBarrier*] - the barrier value, a numeric value. 

       K: [CashOrNothing*] -  the cash amount at expiry if the option
          is in the money,  a numerical value.
           [TwoAssetCashOrNothing*] - for the cash-or-nothing call the
          cash amount at expiry if asset 'S1' is above the strike 'X1'
          and asset  'S2' is above strike 'X2' at expiration,  
           for the cash-or-nothing put the cash amount at expiry if
          asset 'S1' is below the strike 'X1' and asset  'S2' is below
          strike 'X2' at expiration,  
           for the cash-or-nothing up-down the cash amount at expiry if
          asset 'S1' is above the strike 'X1' and asset  'S2' is below
          strike 'X2' at expiration,  
           for the cash-or-nothing down-up the cash amount at expiry if
          asset 'S1' is below the strike 'X1' and asset  'S2' is above
          strike 'X2' at expiration.
           [BinaryBarrier*] - the prespecified cash amount, a numeric
          value. 

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

     rho: [TwoAssetCashOrNothing*] - the correlation of the volatility
          between the first and second asset, a numeric value. 

       S: the asset price, a numeric value. 

  S1, S2: [TwoAssetCashOrNothing*] - the price of the first and second
          asset, a numeric value. 

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

sigma1, sigma2: [TwoAssetCashOrNothing*] - the annualized volatility of
          the first and second underlying  security, numeric values. 

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

TypeFlag: a character string either '"c"' for a call option or  a '"p"'
          for a put option. 
           [TwoAssetCashOrNothing*] - a character string either '"c"'
          for a call option, or  a '"p"' for a put option, or a '"ud"'
          for an  up-down option, or a '"du"' for a down-up option.
           [BinaryBarrier*] - an integer between 1 and 28, selecting
          one of the 28 types, for a definition lookup the arguments
          'eta' and  'phi'. 

       X: the exercise price, a numeric value. 

  X1, X2: [GapOption][TwoAssetCashOrNothing*]  - the first and the
          second exercise price, a numeric value. 

  XL, XH: [SuperShare*]  - the lower and upper boundary strike, a
          numeric value. 

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

     *Gap Options:* 

      The payoff on a gap option depends on the usual factors of a
     plain option,  but is also affected by a "gap"  amount of exercise
     prices, which may be  positive or negative. Note, that a gap call
     (put) option is equivalent to  being long (short) an
     asset-or-nothing call (put) and short (long) a  cash-or-nothing
     call (put). The option price is calculated analytically  according
     to Reiner and Rubinstein (1991). 
      [Haug's Book, Chapter 2.11.1] 

     *Cash-or-Nothing Options:* 

      For this option a predetermined amount is paid at expiration if
     the  asset is above for a call or below for a put some strike
     level. The  amount independent of the path taken. These options
     require no payment  of an exercise price. The exercise price
     determines whether or not the  option returns a payoff. The value
     of a cash-or-nothing call (put)  option is the present value of
     the fixed cash payoff multiplied by  the probability that the
     terminal price will be greater than (less than)  the exercise
     price. The option price is calculated analytically  according to
     Reiner and Rubinstein (1991). 
      [Haug's Book, Chapter 2.11.2] 

     *Two-Asset-Cash-Or-Nothing Options:* 

      These options are building blocks for constructing more complex
     exotic  options. There are four types of two-asset cash-or-nothing
     options, the  first two situationsa are: A
     two-asset-cash-or-nothing call pays out a  fixed cash amount if
     the price of the first asset is above (below) the  strike price of
     the first asset and the price of the second asset is also  above
     (below) the strike price of the second asset at expiration. The 
     other two situations arise under the following conditions: A
     two-asset  cash-or-nothing down-up pays out a fixed cash amount if
     the price of the  first asset is is below (above) the strike price
     of the first asset and  the price of the second asset is above
     (below) the strike price of the  second asset at expiration. The
     option price is calculated analytically  according to Heynen and
     Kat (1996). 
      [Haug's Book, Chapter 2.11.3] 

     *Asset-Or-Nothing Options:* 

      In this option a predetermined asset value is paid if the asset
     is, at  expiration, above for a call or below for a put some
     strike level,  independent of the path taken. For a call (put) the
     terminal price is  greater than (less than) the exercise price,
     the call (put) expires  worthless. The exercise price is never
     paid. Instead, the value of the  asset relative to the exercise
     price determines whether or not the  option returns a payoff. The
     value of an asset-or-nothing call (put)  option is the present
     value of the asset multiplied by the probability  that the
     terminal price will be greater than (less than) the exercise 
     price. The option price is calculated analytically according to
     Cox  and Rubinstein (1985). 
      [Haug's Book, Chapter 2.11.4] 

     *Supershare Options:* 

      These options represents a contingent claim on a fraction of the 
     underlying portfolio. The contingency is that the value of the
     portfolio  must lie between a lower and an upper bound at
     expiration. If the value  lies within these boundaries, the
     supershare is worth a proportion of the  assets underlying the
     portfolio, else the supershare expires worthless.  A supershare
     has a payoff that is basically like a spread of two 
     asset-or-nothing calls, in which the owner of a supershare
     purchases an  asset-or-nothing call with an strike price of the
     lower strike and sells  an asset-or-nothing call with an strike
     price of the upper strike. The  option price is calculated
     analytically according to Hakansson (1976). 
      [Haug's Book, Chapter 2.11.5] 

     *Binary Barrier Options:* 

      These options combine characteristics of both binary and barrier 
     options. They are path dependent with a discontinuous payoff.
     Similar  to barrier options, the payoff depends on whether or not
     the asset  price crosses a predetermined barrier. There are 28
     different types of  binary barrier options, which can be divided
     into two main categories:  Cash-or-nothing and Asset-or-nothing
     barrier options. Cash-or-nothing  barrier options pay out a
     predetermined cash amount or nothing, depending  on whether the
     asset price has hit the barrier. Asset-or-nothing barrier  options
     pay out the value of the asset or nothing, depending on whether 
     the asset price has crossed the barrier. The barrier monitoring
     frequency  can be adjusted to account for discrete monitoring
     using an approximation  developed by Broadie, Glasserman, and Kou
     (1995). Binary-barrier options  can be priced analytically using a
     model introduced by Reiner and  Rubinstein (1991). 
      [Haug's Book, Chapter 2.11.6]

_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 2.11 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:

     Cox J.C., Rubinstein M. (1985); _Innovations in Option Markets_,
     Prentice-Hall, New Jersey.

     Hakkansson N.H. (1976); _The Purchasing Power Fund: A New Kind of
     Financial Intermediary_, Financial Analysts Journal 32, 49-59.

     Haug E.G. (1997);  _The complete Guide to Option Pricing
     Formulas_,  Chapter 2.11, McGraw-Hill, New York.

     Heinen R.C., Kat H.M. (1996); _Brick by Brick_, Risk Magazine 9,
     6.

     Reiner E., Rubinstein M. (1991); _Unscrambling the Binary Code_;
     Risk Magazine 4, 9.

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

     ## Examples from Chapter 2.11 in E.G. Haug's Option Guide (1997)

     ## Gap Option [2.11.1]:
        xmpOptions("\nStart: Gap Option > ")
        GapOption(TypeFlag  =  "c", S  =  50, X1  =  50, X2  =  57, Time  =  0.5, 
              r  =  0.09, b  =  0.09, sigma  =  0.20)

     ## Cash Or Nothing Option [2.11.2]:
        xmpOptions("\nNext: Cash or Nothing Option > ")
        CashOrNothingOption(TypeFlag  =  "p", S  =  100, X  =  80, K  =  10, 
              Time  =  9/12, r  =  0.06, b  =  0, sigma  =  0.35) 

     ## Two Asset Cash Or Nothing Option [2.11.3]:
        xmpOptions("\nNext: Two Asset Cash Or Nothing Option > ")
        # Type 1 - call:
        TwoAssetCashOrNothingOption(TypeFlag  =  "c", S1  =  100, S2  =  100, 
              X1  =  110, X2  =  90, K  =  10, Time  =  0.5, r  =  0.10, b1  =  0.05, 
              b2  =  0.06, sigma1  =  0.20, sigma2  =  0.25, rho  =  0.5)
        # Type 2 - put:
        TwoAssetCashOrNothingOption(TypeFlag  =  "p", S1  =  100, S2  =  100, 
              X1  =  110, X2  =  90, K  =  10, Time  =  0.5, r  =  0.10, b1  =  0.05, 
              b2  =  0.06, sigma1  =  0.20, sigma2  =  0.25, rho  =  -0.5)
        # Type 3 - down-up:
        TwoAssetCashOrNothingOption(TypeFlag  =  "ud", S1  =  100, S2  =  100, 
              X1  =  110, X2  =  90, K  =  10, Time  =  1, r  =  0.10, b1  =  0.05, 
              b2  =  0.06, sigma1  =  0.20, sigma2  =  0.25, rho  =  0)
        # Type 4 - up-down:
        TwoAssetCashOrNothingOption(TypeFlag  =  "du", S1  =  100, S2  =  100, 
              X1  =  110, X2  =  90, K  =  10, Time  =  1, r  =  0.10, b1  =  0.05, 
              b2  =  0.06, sigma1  =  0.20, sigma2  =  0.25, rho  =  0)

     ## Asset Or Nothing Option [2.11.4]: 
        xmpOptions("\nNext: Asset Or Nothing Option > ")
        AssetOrNothingOption(TypeFlag  =  "p", S  =  70, X  =  65, Time  =  0.5, 
          r  =  0.07, b  =  0.07 - 0.05, sigma  =  0.27)

     ## Super Share Option [2.11.5]:  
        xmpOptions("\nNext: Super Share Option > ")
        SuperShareOption(S  =  100, XL  =  90, XH  =  110, Time  =  0.25, r  =  0.10, 
          b  =  0, sigma  =  0.20)

     ## Binary Barrier Option [2.11.6]: 
        xmpOptions("\nNext: Binary Barrier Option > ")
        BinaryBarrierOption(TypeFlag = "6", S = 95, X=102, H = 100, 
          K = 15, Time = 0.5, r = 0.1, b = 0.1, sigma = 0.20)
        BinaryBarrierOption(TypeFlag = "12", S = 95, X = 98, H = 100, 
          K = 15, Time = 0.5, r = 0.1, b = 0.1, sigma = 0.20)
          

