fOptions (version 251.70)

MultipleExercisesOptions: Valuation of Mutiple Exercises Options

Description

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:

ll{ 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. }

Usage

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

Arguments

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

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) wil

b

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

description

a character string which allows for a brief description.

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

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

title

a character string which allows for a project title.

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 ca

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.

Value

• The option valuation programs return an object of class "fOPTION" with the following slots:
• @call
  the function call.
• @parameters
  a list with the input parameters.
• @price
  a numeric value with the value of the option.
• @title
  a character string with the name of the test.
• @description
  a character string with a brief description of the test.

Details

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]

References

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.

Examples

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

## ExecutiveStockOption [2.1]:
   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]:
   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]:
   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]:
   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]:
   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]:
   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]:
   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]:
   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]:
   WriterExtendibleOption(TypeFlag = "c", S = 80, X1 = 90, X2 = 82,
     time1 = 0.50, Time2 = 0.75, r = 0.10, b = 0.10, sigma = 0.30)