fOptions (version 200.10058)

LookbackOptions: Valuation of Lookback Options

Description

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: ll{ 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.}

Usage

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)

Arguments

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 betwee

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 underl

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, [PTFixedStrikeLookback*] - the predetermined time time1 where t

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

X

the exercise price, a numeric value.

Value

• The option price, a numeric value.

Details

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]

References

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.

Examples

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

## Floating Strike Lookback Option [2.9.1]:
   xmpOptions("Start: 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("Next: 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("Next: 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("Next: 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("Next: 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)