BasicAmericanOptions: Valuation of Basic American Options

In fOptions: Rmetrics - Pricing and Evaluating Basic Options

Description

A collection and description of functions to valuate basic American options. Approximative formulas for American calls are given for the Roll, Geske and Whaley Approximation, for the Barone-Adesi and Whaley Approximation, and for the Bjerksund and Stensland Approximation.

The functions are:

RollGeskeWhaleyOption	Roll, Geske and Whaley Approximation,
BAWAmericanApproxOption	Barone-Adesi and Whaley Approximation,
BSAmericanApproxOption	Bjerksund and Stensland Approximation.

Usage

RollGeskeWhaleyOption(S, X, time1, Time2, r, D, sigma,
    title = NULL, description = NULL) 
BAWAmericanApproxOption(TypeFlag, S, X, Time, r, b, sigma,
    title = NULL, description = NULL)
BSAmericanApproxOption(TypeFlag, S, X, Time, r, b, sigma,
    title = NULL, description = NULL) 

Arguments

b	the annualized cost-of-carry rate, a numeric value; e.g. 0.1 means 10% pa.
D	a single dividend with time to dividend payout t1.
description	a character string which allows for a brief description.
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.
time1, Time2	[RollGeskeWhaley*] - the first value measures time to dividend payout in years, e.g. 0.25 denotes a quarter, and the second value measures time to maturity measured in years, a numeric value; e.g. 0.5 means 6 months.
title	a character string which allows for a project title.
TypeFlag	a character string either "c" for a call option or a "p" for a put option.
X	the exercise price, a numeric value.

Details

Roll-Geske-Whaley Option:

The function RollGeskeWhaleyOption valuates American calls on a stock paying a single dividend with specified time to dividend payout according to the pricing formula derived by Roll, Geske and Whaley (1977).

Approximations for American Options:

The function BSAmericanApproxOption valuates American calls or puts on an underlying asset for a given cost-of-carry rate according to the quadratic approximation method due to Barone-Adesi and Whaley (1987). The function BSAmericanApproxOption valuates American calls or puts on stocks, futures, and currencies due to the approximation method of Bjerksund and Stensland (1993).

Value

RollGeskeWhaleyOption
BAWAmericanApproxOption
return the option price, a numeric value.

BSAmericanApproxOption
returns a list with the following two elements: Premium the option price, and TriggerPrice the trigger price.

Note

The functions implement the algorithms to valuate basic American options as described in Chapter 1.4 of Haug's Option Guide (1997).

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

References

Barone-Adesi G., Whaley R.E. (1987); Efficient Analytic Approximation of American Option Values, Journal of Finance 42, 301–320.

Bjerksund P., Stensland G. (1993); Closed Form Approximation of American Options, Scandinavian Journal of Management 9, 87–99.

Geske R. (1979); A Note on an Analytical Formula for Unprotected American Call Options on Stocks with known Dividends, Journal of Financial Economics 7, 63–81.

Haug E.G. (1997); The Complete Guide to Option Pricing Formulas, Chapter 1, McGraw-Hill, New York.

Roll R. (1977); An Analytic Valuation Formula for Unprotected American Call Options on Stocks with known Dividends, Journal of Financial Economics 5, 251–258.

Examples

## All the examples are from Haug's Option Guide (1997)

## CHAPTER 1.4: ANALYTICAL MODELS FOR AMERICAN OPTIONS
       
## Roll-Geske-Whaley American Calls on Dividend Paying 
   # Stocks [Haug 1.4.1]
   RollGeskeWhaleyOption(S = 80, X = 82, time1 = 1/4, 
     Time2 = 1/3, r = 0.06, D = 4, sigma = 0.30)
      
## Barone-Adesi and Whaley Approximation for American 
   # Options [Haug 1.4.2] vs. Black76 Option on Futures:
   BAWAmericanApproxOption(TypeFlag = "p", S = 100, 
     X = 100, Time = 0.5, r = 0.10, b = 0, sigma = 0.25)
   Black76Option(TypeFlag = "c", FT = 100, X = 100, 
     Time = 0.5, r = 0.10, sigma = 0.25)  
     
## Bjerksund and Stensland Approximation for American Options:
   BSAmericanApproxOption(TypeFlag = "c", S = 42, X = 40, 
     Time = 0.75, r = 0.04, b = 0.04-0.08, sigma = 0.35)

Example output

Loading required package: timeDate
Loading required package: timeSeries
Loading required package: fBasics


Rmetrics Package fBasics
Analysing Markets and calculating Basic Statistics
Copyright (C) 2005-2014 Rmetrics Association Zurich
Educational Software for Financial Engineering and Computational Science
Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
https://www.rmetrics.org --- Mail to: [e-mail]


Rmetrics Package fOptions
Pricing and Evaluating Basic Options
Copyright (C) 2005-2014 Rmetrics Association Zurich
Educational Software for Financial Engineering and Computational Science
Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
https://www.rmetrics.org --- Mail to: [e-mail]

Title:
 Roll Geske Whaley Option 

Call:
 RollGeskeWhaleyOption(S = 80, X = 82, time1 = 1/4, Time2 = 1/3, 
     r = 0.06, D = 4, sigma = 0.3)

Parameters:
       Value:           
 S     80               
 X     82               
 time1 0.25             
 Time2 0.333333333333333
 r     0.06             
 D     4                
 sigma 0.3              

Option Price:
 4.38603 

Description:
 Wed Aug 16 12:46:01 2017 


Title:
 BAW American Approximated Option 

Call:
 BAWAmericanApproxOption(TypeFlag = "p", S = 100, X = 100, Time = 0.5, 
     r = 0.1, b = 0, sigma = 0.25)

Parameters:
          Value:
 TypeFlag p     
 S        100   
 X        100   
 Time     0.5   
 r        0.1   
 b        0     
 sigma    0.25  

Option Price:
 6.801362 

Description:
 Wed Aug 16 12:46:01 2017 


Title:
 Black 76 Option Valuation 

Call:
 Black76Option(TypeFlag = "c", FT = 100, X = 100, Time = 0.5, 
     r = 0.1, sigma = 0.25)

Parameters:
          Value:
 TypeFlag c     
 FT       100   
 X        100   
 Time     0.5   
 r        0.1   
 sigma    0.25  

Option Price:
 6.699709 

Description:
 Wed Aug 16 12:46:01 2017 


Title:
 BS American Approximated Option 

Call:
 BSAmericanApproxOption(TypeFlag = "c", S = 42, X = 40, Time = 0.75, 
     r = 0.04, b = 0.04 - 0.08, sigma = 0.35)

Parameters:
             Value:          
 TypeFlag    c               
 S           42              
 X           40              
 Time        0.75            
 r           0.04            
 b           -0.04           
 sigma       0.35            
 TrigerPrice 57.5994499306841

Option Price:
 5.270405 

Description:
 Wed Aug 16 12:46:01 2017 

fOptions documentation built on May 2, 2019, 2:27 p.m.