Asian option valuation via Black-Scholes (BS) model

Description

Price Asian option using BS model

Usage

AsianBS(o = OptPx(Opt(Style = "Asian")))

Arguments

Details

This pricing algorithm assumes average price is calculated continuously.

Value

A list of class AsianBS consisting of the original OptPx object and the option pricing parameters M1, M2, F0, and sigma as well as the computed option price PxBS.

Author(s)

Xinnan Lu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html pp.609-611.

Examples

(o = AsianBS())$PxBS #Price = ~4.973973,  using default values

 o = Opt(Style='Asian',S0=100,K=90,ttm=3)
 (o = AsianBS(OptPx(o,r=0.03,q=0,vol=0.3)))$PxBS

 o = Opt(Style='Asian',Right='P',S0=100,K=110,ttm=0.5)
 (o = AsianBS(OptPx(o,r=0.03,q=0.01,vol=0.3)))$PxBS

 #See J.C.Hull, OFOD'2014, 9-ed, ex.26.3, pp.610. The price is 5.62.
 o = Opt(Style='Asian',Right='Call',S0=50,K=50,ttm=1)
 (o = AsianBS(OptPx(o,r=0.1,q=0,vol=0.4)))$PxBS

Asian option valuation with Monte Carlo (MC) simulation.

Description

Calculates the price of an Asian option using Monte Carlo simulations to determine expected payout.
Assumptions:
The option follows a General Brownian Motion (BM),
ds = mu * S * dt + sqrt(vol) * S * dW where dW ~ N(0,1).
The value of mu (the expected price increase) is o$r, the risk free rate of return (RoR).
The averaging period is the life of the option.

Usage

AsianMC(o = OptPx(o = Opt(Style = "Asian"), NSteps = 5), NPaths = 5)

Arguments

Value

The option o with the price in the field PxMC based on MC simulations.

Author(s)

Jake Kornblau, Department of Statistics and Department of Computer Science, Rice University, 2016

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8,
http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
http://www.math.umn.edu/~spirn/5076/Lecture16.pdf

Examples

(o = AsianMC())$PxMC #Price = ~5.00,  using default values

  o = OptPx(Opt(Style='Asian'), NSteps = 5)
  (o = AsianMC(o, NPaths=5))$PxMC #Price = ~$5

  (o = AsianMC(NPaths = 5))$PxMC # Price = ~$5

  o = Opt(Style='Asian', Right='Put',S0=10, K=15)
  o = OptPx(o, r=.05, vol=.1, NSteps = 5)
  (o = AsianMC(o, NPaths = 5))$PxMC # Price = ~$4

  #See J.C.Hull, OFOD'2014, 9-ed, ex.26.3, pp.610.
 o = Opt(Style='Asian',S0=50,K=50,ttm=1)
 o = OptPx(o,r=0.1,q=0,vol=0.4,NSteps=5)
 (o = AsianBS(o))$PxBS   #Price is 5.62.
 (o = AsianMC(o))$PxMC

Average Strike option valuation via Monte Carlo (MC) simulation

Description

Calculates the price of an Average Strike option using Monte Carlo simulations by determining the determine expected payout. Assumes that the input option follows a General Brownian Motion ds = mu * S * dt + sqrt(vol) * S * dz where dz ~ N(0,1) Note that the value of mu (the expected price increase) is assumped to be o$r, the risk free rate of return. Additionally, the averaging period is assumed to be the life of the option.

Usage

AverageStrikeMC(o = OptPx(o = Opt(Style = "AverageStrike")), NPaths = 5)

Arguments

Value

The original option object o with the price in the field PxMC based on the MC simulations.

Author(s)

Jake Kornblau, Department of Statistics and Department of Computer Science, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html Also, http://www.math.umn.edu/~spirn/5076/Lecture16.pdf

Examples

(o = AverageStrikeMC())$PxMC   #Price =~ $3.6

  o = OptPx(o = Opt(Style='AverageStrike'), NSteps = 5)
  (o = AverageStrikeMC(o))$PxMC # Price =~ $6

  (o = AverageStrikeMC(NPaths = 20))$PxMC  #Price =~ $3.4

  o = OptPx(o = Opt(Style='AverageStrike'), NSteps = 5)
  (o = AverageStrikeMC(o, NPaths = 20))$PxMC  #Price =~ $5.6

Binomial option pricing model

Description

Compute option price via binomial option pricing model (recombining symmetric binomial tree). If no tree requested for European option, vectorized algorithm is used.

Usage

BOPM(o = OptPx(), IncBT = TRUE)

Arguments

Value

An original OptPx object with PxBT field as the binomial-tree-based price of an option and (an optional) the fullly-generated binomial tree in BT field.

• IncBT = FALSE: option price value (type double, class numeric)

• IncBT = TRUE: binomial tree as a list (of length (o$NSteps+1) of numeric matrices (2 x i)

Each matrix is a set of possible i outcomes at time step i columns: (underlying prices, option prices)

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315

#See Fig.13.11, Hull/9e/p291. #Create an option and price it o = Opt(Style='Eu', Right='C', S0 = 808, ttm = .5, K = 800) o = BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=2), IncBT=TRUE) o$PxBT #print added calculated price to PxBT field

#Fig.13.11, Hull/9e/p291: o = Opt(Style='Eu', Right='C', S0=810, ttm=.5, K=800) BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=2), IncBT=TRUE)$PxBT

#DerivaGem diplays up to 10 steps: o = Opt(Style='Am', Right='C', 810, .5, 800) BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=20), IncBT=TRUE)

#DerivaGem computes up to 500 steps: o = Opt(Style='American', Right='Put', 810, 0.5, 800) BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=1000), IncBT=FALSE)

See Also

BOPM_Eu for European option via vectorized approach.

European option valuation (vectorized computation).

Description

A helper function to price European options via a vectorized (fast, memory efficient) approach.

Usage

BOPM_Eu(o = OptPx())

Arguments

Value

A list of class OptPx with an element PxBT, which is an option price value (type double, class numeric)

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015 Code adopted Gilli & Schumann's R implementation to Opt* objects

References

Gili, M. and Schumann, E. (2009) Implementing Binomial Trees, COMISEF Working Papers Series

See Also

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1341181 for original paper, BOPM for American option pricing.

Examples

#Fig.13.11, Hull/9e/p291:
o = Opt(Style='European', Right='Call', S0=810, ttm=.5, K=800)
(o <- BOPM_Eu( OptPx(o, r=.05, q=.02, vol=.2, NSteps=2)))$PxBT

o = Opt('Eu', 'C', 0.61, .5, 0.6, SName='USD/AUD')
o = OptPx(o, r=.05, q=.02, vol=.12, NSteps=2)
(o <- BOPM_Eu(o))$PxBT

Black-Scholes (BS) pricing model

Description

a wrapper function for BS_Simple; uses OptPx object as input.

Usage

BS(o = OptPx())

Arguments

Value

An original OptPx object with BS list as components of Black-Scholes formular. See BS_Simple.

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315

Examples

#See Hull, p.338, Ex.15.6. #Create an option and price it
o = Opt(Style='Eu', Right='Call', S0 = 42, ttm = .5, K = 40)
o = BS( OptPx(o, r=.1, vol=.2, NSteps=NA))
o$PxBS #print call option price computed by Black-Scholes pricing model
o$BS$Px$Put #print put option price computed by Black-Scholes pricing model

Black-Scholes formula

Description

Black-Scholes (aka Black-Scholes-Merton, BS, BSM) formula for simple parameters

Usage

BS_Simple(S0 = 42, K = 40, r = 0.1, q = 0, ttm = 0.5, vol = 0.2)

Arguments

Details

Uses BS formula to calculate call/put option values and elements of BS model

Value

a list of BS formula elements and BS price, such as d1 for d_1, d2 for d_2, Nd1 for N(d_1), Nd2 for N(d_2), NCallPxBS for BSM call price, PutPxBS for BSM put price

Author(s)

Robert Abramov, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315 http://www.theresearchkitchen.com/archives/106

Examples

#See Hull p.339, Ex.15.6.
(o <- BS_Simple(S0=42,K=40,r=.1,q=0,ttm=.5,vol=.2))$Px$Call #returns 4.759422
o$Px$Put # returns 0.8085994 as the price of the put

BS_Simple(100,90,0.05,0,2,0.30)
BS_Simple(50,60,0.1,.2,3,0.25)
BS_Simple(90,90,0.15,0,.5,0.20)
BS_Simple(15,15,.01,0.0,0.5,.5)

Barrier option pricing via Black-Scholes (BS) model

Description

This function calculates the price of a Barrier option. This price is based on the assumptions that the probability distribution is lognormal and that the asset price is observed continuously.

Usage

BarrierBS(o = OptPx(Opt(Style = "Barrier")), dir = c("Up", "Down"),
  knock = c("In", "Out"), H = 40)

Arguments

Details

To price the barrier option, we need to know whether the option is Up or Down | In or Out | Call or Put. Beyond that we also need the S0, K, r, q, vol, H, and ttm arguments from the object classes defined in the package.

Value

The price of the barrier option o, which is based on the BSM-adjusted algorithm (see references).

Author(s)

Kiryl Novikau, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. pp.606-607

Examples

(o = BarrierBS())$PxBS # Option with default arguments is valued at $9.71

 #Down-and-In-Call
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=10)
 o = OptPx(o,  r = .05, q = 0, vol = .25)
 o = BarrierBS(o, dir = "Down", knock = 'In', H = 40)

 #Down-and-Out Call
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=10)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Down", knock = 'Out', H = 40)

 #Up-and-In Call
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Up", knock = 'In', H = 60)

 #Up-and-Out Call
 o = Opt(Style='Barrier', S0 = 50, K = 50, ttm = 1, Right="Call", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Up", knock = 'Out', H = 60)

 #Down-and-In Put
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Down", knock = 'In', H = 40)

 #Down-and-Out Put
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Down", knock = 'Out', H = 40)

 #Up-and-In Put
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Up", knock = 'In', H = 60)

 #Up-and-Out Put
 o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1)
 o = OptPx(o, r = .05, q = .02, vol = .25)
 o = BarrierBS(o, dir = "Up", knock = 'Out', H = 60)

Barrrier option valuation via lattice tree (LT)

Description

Use Binomial Tree to price barrier options with relatively large NSteps (NSteps > 100) steps. The price may be not as percise as BSM function cause the convergence speed for Binomial Tree is kind of slow.

Usage

BarrierLT(o = OptPx(Opt(Style = "Barrier"), vol = 0.25, r = 0.05, q = 0.02,
  NSteps = 5), dir = c("Up", "Down"), knock = c("In", "Out"), H = 60)

Arguments

Value

A list of class BarrierLT consisting of the input object OptPx and the appended new parameters and option price.

Author(s)

Tong Liu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
p.467-468. Trinomial Trees, p.604-606: Barrier Options.

Examples

# default Up and Knock-in Call Option with H=60, approximately 7.09
(o = BarrierLT())$PxLT

#Visualization of price changes as Nsteps change.
o = Opt(Style="Barrier")
visual=sapply(10:200,function(n) BarrierLT(OptPx(o,NSteps=n))$PxLT)

c=(10:200)
plot(visual~c,type="l",xlab="NSteps",ylab="Price",main="Price converence with NSteps")

# Down and Knock-out Call Option with H=40
o = OptPx(o=Opt(Style="Barrier"))
BarrierLT(o,dir="Down",knock="Out",H=40)

# Down and Knock-in Call Option with H=40
o = OptPx(o=Opt(Style="Barrier"))
BarrierLT(o,dir="Down",knock="In",H=40)

# Up and Knock-out Call Option with H=60
o = OptPx(o=Opt(Style="Barrier"))
BarrierLT(o,dir='Up',knock="Out")

# Down and Knock-out Put Option with H=40
o = OptPx(o=Opt(Style="Barrier",Right="Put"))
BarrierLT(o,dir="Down",knock="Out",H=40)

# Down and Knock-in Put Option with H=40
o = OptPx(o=Opt(Style="Barrier",Right="Put"))
BarrierLT(o,dir="Down",knock="In",H=40)

# Up and Knock-out Put Option with H=60
o = OptPx(o=Opt(Style="Barrier",Right="Put"))
BarrierLT(o,dir='Up',knock="Out")

# Up and Knock-in Put Option with H=60
BarrierLT(OptPx(o=Opt(Style="Barrier",Right="Put")))

Barrier option valuation via Monte Carlo (MC) simulation.

Description

Calculates the price of a Barrier Option using 10000 Monte Carlo simulations. The helper function BarrierCal() aims to calculate expected payout for each stock prices.

Important Assumptions: The option follows a General Brownian Motion (GBM) ds = mu * S * dt + sqrt(vol) * S * dW where dW ~ N(0,1). The value of mu (the expected percent price increase) is assumed to be o$r-o$q.

Usage

BarrierMC(o = OptPx(o = Opt(Style = "Barrier")), knock = c("In", "Out"),
  B = 60, NPaths = 5)

Arguments

Value

The option o with the price in the field PxMC based on MC simulations and the Barrier option properties set by the users themselves

Author(s)

Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. Also, http://stackoverflow.com/questions/25946852/r-monte-carlo-simulation-price-path-converging-volatility-issue

Examples

(o = BarrierMC())$PxMC #Price =~ $11

 o = OptPx(o=Opt(Style='Barrier'),NSteps = 10)
 (o = BarrierMC(o))$PxMC #Price =~ $14.1

 (o = BarrierMC(NPaths = 5))$PxMC # Price =~ $11

 (o = BarrierMC(B=65))$PxMC # Price =~ $10

 (o = BarrierMC(knock="Out"))$PxMC #Price =~ $1

Binary option valuation with Black-Scholes (BS) model

Description

S3 object pricing model for a binary option. Two types of binary options are priced: 'cash-or-nothing' and 'asset-or-nothing'.

Usage

BinaryBS(o = OptPx(Opt(Style = "Binary")), Q = 1,
  Type = c("cash-or-nothing", "asset-or-nothing"))

Arguments

Value

A list of class Binary.BS consisting of the input object OptPx and the appended new parameters and option price.

Author(s)

Xinnan Lu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. pp.606-607

Examples

(o = BinaryBS())$PxBS

#This example should produce price 4.33 (see Derivagem, DG201.xls)
o = Opt(Style="Binary", Right='Call', S0=50, ttm=5/12, K=52)
o = OptPx(o, r=.1, vol=.40, NSteps=NA)
(o = BinaryBS(o, Q = 10, Type='cash-or-nothing'))$PxBS

BinaryBS(OptPx(Opt(Style="Binary"), q=.01), Type='asset-or-nothing')
BinaryBS(OptPx(Opt(Style="Binary", S0=100, K=80),q=.01))
o = Opt(Style="Binary", Right="Put", S0=50, K=60)
BinaryBS(OptPx(o,q=.04), Type='asset-or-nothing')

Binary option valuation via Monte-Carlo (via) simulation.

Description

Binary option valuation via Monte-Carlo (via) simulation.

Usage

BinaryMC(o = OptPx(Opt(Style = "Binary")), Q = 25,
  Type = c("cash-or-nothing", "asset-or-nothing"), NPaths = 5)

Arguments

Details

Two types of binary options are priced: 'cash-or-nothing' and 'asset-or-nothing'.

Value

The original input object o with added parameters and option price PxMC

Author(s)

Tongyue Luo, Rice University, Spring 2015.

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. pp.606-607.

Examples

(o = BinaryMC())$PxMC

o = OptPx(Opt(Style="Binary"))
(o = BinaryMC(o, Type="cash"))$PxMC

o = OptPx(Opt(Style="Binary"),q=0.01)
(o = BinaryMC(o, Type="asset"))$PxMC

o = OptPx(Opt(Style="Binary", S0=100, K=80),q=0.01)
(o = BinaryMC(o, Type="cash"))$PxMC

o = OptPx(Opt(Style="Binary", Right="Put", S0=50, K=60),q=0.04)
(o = BinaryMC(o, Type="asset"))$PxMC

Binary option valuation vialattice tree (LT) implementation

Description

Compute option price via binomial option pricing model (recombining symmetric binomial tree)

Usage

Binary_BOPM(o = OptPx(Opt(Style = "Binary")), Type = c("cash-or-nothing",
  "asset-or-nothing"), Q = 1000, IncBT = FALSE)

Arguments

Value

original OptPx object with Px.BOPM property and (optional) binomial tree IncBT = FALSE: option price value (type double, class numeric) IncBT = TRUE: binomial tree as a list (of length (o$n+1) of numeric matrices (2 x i). Each matrix is a set of possible i outcomes at time step i columns: (underlying prices, option prices)

Examples

(o = Binary_BOPM())$PxBT

o = OptPx(o=Opt(Style='Binary'))
(o = Binary_BOPM(o, Type='cash', Q=100, IncBT=TRUE))$PxBT

o = OptPx(Opt(Style='Binary'), r=0.05, q=0.02, rf=0.0, vol=0.30, NSteps=5)
(o = Binary_BOPM(o, Type='cash', Q=1000, IncBT=FALSE))$PxBT

o = OptPx(o=Opt(Style='Binary'), r=0.15, q=0.01, rf=0.05, vol=0.35, NSteps=5)
(o = Binary_BOPM(o,Type='asset',Q=150, IncBT=FALSE))$PxBT

o = OptPx(o=Opt(Style='Binary'), r=0.025, q=0.001, rf=0.0, vol=0.10, NSteps=5)
(o = Binary_BOPM(o, Type='cash', Q=20, IncBT=FALSE))$PxBT

Chooser option valuation via Black-Scholes (BS) model

Description

Compute an exotic option that allow the holder decide the option will be a call or put option at some predetermined future date. In a simple case, both put and call option are plain vanilla option. The value of the simple chooser option is \max{C(S,K,t_1),P(S,K,t_2)}. The plain vanilla option is calculated based on the BS model.

Usage

ChooserBS(o = OptPx(Opt(Style = "Chooser")), t1 = 9/12, t2 = 3/12)

Arguments

Value

A list of class SimpleChooserBS consisting of the original OptPx object and the option pricing parameters t1, t2, as well as the computed price PxBS.

Author(s)

Le You, Department of Statistics, Rice University, spring 2015

References

• Hull, John C.,Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html

• Huang Espen G., Option Pricing Formulas, 2ed. http://down.cenet.org.cn/upfile/10/20083212958160.pdf

• Wee, Lim Tiong, MFE5010 Exotic Options,Notes for Lecture 4 Chooser option. http://www.stat.nus.edu.sg/~stalimtw/MFE5010/PDF/L4chooser.pdf

• Humphreys, Natalia A., ACTS 4302 Principles of Actuarial Models: Financial Economics. Lesson 14: All-or-nothing, Gap, Exchange and Chooser Options.

Examples

(o = ChooserBS())$PxBS

o = Opt(Style='Chooser',Right='Other',S0=50, K=50)
(o = ChooserBS(OptPx(o, r=0.06, q=0.02, vol=0.2),9/12, 3/12))$PxBS

o = Opt(Style='Chooser',Right='Other',S0=50, K=50)
(o = ChooserBS (OptPx(o,r=0.08, q=0, vol=0.25),1/2, 1/4))$PxBS

o = Opt(Style='Chooser',Right='Other',S0=100, K=50)
(o = ChooserBS(OptPx(o,r=0.08, q=0.05, vol=0.3),1/2, 1/4))$PxBS

Chooser option valuation via Lattice Tree (LT) Model

Description

Calculates the price of a Chooser option using a recombining binomial tree model. Has pricing capabilities for both simple European Chooser options as well as American Chooser Options, where exercise can occur any time as a call or put options.

Usage

ChooserLT(o = OptPx(Opt("Chooser", ttm = 1)), t1 = 0.5, t2 = 0.5,
  IncBT = FALSE)

Arguments

Details

The American chooser option is interpreted as exercise of option being available at any point in time during the life of the option.

Value

An original OptPx object with PxLT field as the price of the option and user-supplied ttc, IncBT parameters attached.

Author(s)

Richard Huang, Department of Statistics, Rice University, spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
Thomas S.Y. Ho et al., The Oxford Guide to Financial Modeling : Applications for Capital Markets. . .

Examples

(o = ChooserLT())$PxLT    #Default Chooser option price. (See Ho pg 234 in references)

o = Opt('Eu', S0=100, ttm=1, K=100)
o = OptPx(o, r=0.10, q=0, vol=0.1, NSteps=5)
(o = ChooserLT(o, t1 = .5, t2 =.5, IncBT=TRUE))$PxLT

#American Chooser, higher price than European equivalent
o = Opt('Am', S0=100, ttm=1, K=100)
o = OptPx(o, r=0.10, q=0, vol=0.1, NSteps=5)
ChooserLT(o,t1=.5, t2=.5,IncBT=FALSE)$PxLT

o = Opt('Eu', S0=50, ttm=1, K=50)
o = OptPx(o, r=0.05, q=0.02, vol=0.25, NSteps=5)
ChooserLT(o, t1 = .75, t2 = .75, IncBT=FALSE)$PxLT

o = Opt('Eu', S0=50, ttm=1, K=50)
o = OptPx(o, r=0.05, q=0.5, vol=0.25, NSteps=5)
ChooserLT(o, t1 = .75, t2 = .75, IncBT=FALSE)$PxLT

Chooser option valuation via Monte Carlo (MC) simulations

Description

Price chooser option using Monte Carlo (MC) simulation.

Usage

ChooserMC(o = OptPx(Opt(Style = "Chooser")), isEu = TRUE, T1 = 1,
  NPaths = 5, plot = FALSE)

Arguments

Details

A chooser option (sometimes referred to as an as you like it option) has the feature that, after a specified period of time, the holder can choose whether the option is a call or a put. In this algorithm, we can price chooser options when the underlying options are both European or are both American. When the underlying is an American option, the option holder can exercise before and after T1.

Value

A list of class ChooserMC consisting of original OptPx object, option pricing parameters isEu, NPaths, and T1, as well as the computed price PxMC for the chooser option.

Author(s)

Xinnan Lu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. p.603.

Examples

(o = ChooserMC())$PxMC

o = OptPx(Opt(Right='Call',Style="Chooser"))
 ChooserMC(o,isEu=TRUE,NPaths=5, plot=TRUE)

 o = OptPx(Opt(Right='Put',Style="Chooser"))
 ChooserMC(o,isEu=TRUE,NPaths=5, plot=TRUE)

 o = Opt(Right='C',S0=100,K=110,ttm=4,Style="Chooser")
 o = OptPx(o,vol=0.2,r=0.05,q=0.04)
 ChooserMC(o,isEu=TRUE,T1=2,NPaths=5)

 o = Opt(Right='P',S0=110,K=100,ttm=4,Style="Chooser")
 o = OptPx(o,vol=0.2,r=0.05,q=0.04)
 ChooserMC(o,isEu=TRUE,T1=2,NPaths=5)

 o = Opt(Right='C',S0=50,K=50,ttm=0.5,Style="Ch")
 o = OptPx(o,vol=0.25,r=0.08,q=0.1)
 ChooserMC(o,isEu=FALSE,T1=0.25,NPaths=5)

Compound option valuation with Black-Scholes (BS) model

Description

Compound option valuation with Black-Scholes (BS) model

Usage

CompoundBS(o = OptPx(Opt(Style = "Compound")), K1 = 10, T1 = 0.5,
  Type = c("cc", "cp", "pp", "pc"))

Arguments

Value

A list of object 'OptCompound' containing the option parameters binomial tree parameters and compound option parameters

Author(s)

Robert Abramov

Examples

(o <- CompoundBS())$PxBS #price compound option with default parameters

o = OptPx(Opt(Style='Compound'), r=0.05, q=0.0, vol=0.25)
CompoundBS(o,K1=10,T1=0.5)

o = Opt(Style='Compound', S0=50, K=52, ttm=1)
CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=6,T1=1.5)

o = Opt(Style='Compound', S0=90, K=100, ttm=1.5)
CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=15,T1=1)

o = Opt(Style='Compound', S0=15, K=15, ttm=0.25)
CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=3,T1=1.5)

Compound option valuation via lattice tree (LT) model

Description

CompoundLT prices a compound option using the binomial tree (BT) method. The inputs it takes are two OptPx objects. It pulls the S from the o2 input which should be the option with the greater time to maturity.

Usage

CompoundLT(o1 = OptPx(Opt(Style = "Compound")), o2 = OptPx(Opt(Style =
  "Compound")))

Arguments

Value

User-supplied o1 option with fields o2 and PxLT, as the second option and calculated price, respectively.

Author(s)

Kiryl Novikau, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.

Examples

(o = CompoundLT())$PxLT # Uses default arguments

#Put option on a Call:
o = Opt(Style="Compound", S0=50, ttm=.5, Right="P", K = 50)
o1 = OptPx(o, r = .1, vol = .4, NSteps = 5)
o = Opt(Style="Compound", S0=50, ttm=.75, Right="C", K = 60)
o2 = OptPx(o, r = .1, vol = .4, NSteps = 5)
(o = CompoundLT(o1, o2))$PxLT

#Call option on a Call:
o = Opt(Style = "Compound", S0 = 50, ttm= .5, Right = "Call", K = 50)
o1 = OptPx(o, r = .1, vol = .4, NSteps = 5)
o = Opt(Style = "Compound", S0 = 50, ttm= .75, Right = "Call", K = 5)
o2 = OptPx(o, r = .1, vol = .4, NSteps = 5)
(o = CompoundLT(o1, o2))$PxLT

#Put option on a Put:
o = Opt(Style = "Compound", S0 = 50, ttm= .5, Right = "Put", K = 40)
o1 = OptPx(o, r = .1, vol = .4, NSteps = 5)
o = Opt(Style = "Compound", S0 = 50, ttm= .75, Right = "Put", K = 50)
o2 = OptPx(o, r = .1, vol = .4, NSteps = 5)
(o = CompoundLT(o1, o2))$PxMC

#Call option on a Put:
o = Opt(Style = "Compound", S0 = 50, ttm= .5, Right = "Call", K = 30)
o1 = OptPx(o, r = .1, vol = .4, NSteps = 5)
o = Opt(Style = "Compound", S0 = 50, ttm= .75, Right = "Put", K = 80)
o2 = OptPx(o, r = .1, vol = .4, NSteps = 5)
(o = CompoundLT(o1, o2))$PxLT

DeferredPaymentLT

Description

A binomial tree pricer of a Deferred Payment option. An American option that has payment at expiry no matter when exercise, causing differences in present value (PV) of a payoff.

Usage

DeferredPaymentLT(o = OptPx(Opt(Style = "DeferredPayment")))

Arguments

Value

An object of class OptPx with price included

Author(s)

Max Lee, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html

Examples

(o = DeferredPaymentLT())$PxLT

o = Opt(Style='DeferredPayment', Right="Call", S0=110,ttm=.5,K=110)
(o = DeferredPaymentLT(OptPx(o,r=.05,q=.04,vol=.2,NSteps=5)))$PxLT

o = Opt(Style='DeferredPayment', Right="Put", S0 = 50, ttm=2,K=47)
(o = DeferredPaymentLT(OptPx(o,r=.05,q=.04,vol=.25,NSteps=3)))$PxLT

ForeignEquity option valuation via Black-Scholes (BS) model

Description

ForeignEquity Option via Black-Scholes (BS) model

Usage

ForeignEquityBS(o = OptPx(Opt(Style = "ForeignEquity")), I1 = 1540,
  I2 = 1/90, sigma1 = 0.14, sigma2 = 0.18, g1 = 0.02, rho = -0.3,
  Type = c("Foreign", "Domestic"))

Arguments

Details

Two types of ForeignEquity options are priced: 'Foreign' and 'Domestic'. See "Exotic Options", 2nd, Peter G. Zhang for more details.

Value

A list of class ForeignEquityBS consisting of the original OptPx object and the option pricing parameters I1,I2, Type, isForeign, and isDomestic as well as the computed price PxBS.

Author(s)

Chengwei Ge, Department of Statistics, Rice University, 2015

References

Zhang, Peter G. Exotic Options, 2nd, 1998.

Examples

o = OptPx(Opt(Style = 'ForeignEquity', Right = "Put"), r= 0.03)
ForeignEquityBS(o, I1=1540, I2=1/90, g1=.02, sigma1=.14,sigma2=0.18, rho=.03,Type='Foreign')

o = OptPx(Opt(Style = 'ForeignEquity',  Right = "Put", ttm=9/12, K=1600), r=.03)
ForeignEquityBS(o, I1=1540, I2=1/90, g1=.02, sigma1=.14,sigma2=0.18, rho=0.03,Type='Foreign')

o = OptPx(Opt(Style = 'ForeignEquity', Right = "C", ttm=9/12, K=1600), r=.03)
ForeignEquityBS(o, I1=1540, I2=1/90, g1=.02, sigma1=.14,sigma2=0.18, rho=0.03,Type='Foreign')

o = OptPx(Opt(Style = 'ForeignEquity', Right = "C", ttm=9/12, K=1600), r=.03)
ForeignEquityBS(o, I1=1540, I2=1/90, g1=.02, sigma1=.14,sigma2=0.18, rho=0.03,Type='Domestic')

o = OptPx(Opt(Style = 'ForeignEquity', Right = "P", ttm=9/12, K=1600), r=.03)
ForeignEquityBS(o, I1=1540, I2=1/90, g1=.02, sigma1=.14,sigma2=0.18, rho=0.03,Type='Domestic')

ForwardStart option valuation via Black-Scholes (BS) model

Description

Compute the price of Forward Start options using BSM. A forward start option is a standard European option whose strike price is set equal to current asset price at some prespecified future date. Employee incentive options are basically forward start option

Usage

ForwardStartBS(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1)

Arguments

Details

A standard European option starts at a future time tts.

Value

The original user-supplied OptPX object with price field PxBS and any other provided user-supplied parameters.

Author(s)

Tongyue Luo, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8.http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. p.602

Examples

(o = ForwardStartBS())$PxBS

o = OptPx(Opt(Style='ForwardStart', Right='Put'))
(o = ForwardStartBS(o))$PxBS

Forward Start option valuation via Monte-Carlo (MC) simulation

Description

S3 object pricing model for a forward start European option using Monte Carlo simulation

Usage

ForwardStartMC(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1,
  NPaths = 5)

Arguments

Details

A standard European option starts at a future time tts.

Value

A list of class ForwardStartMC consisting of the input object OptPx and the appended new parameters and option price.

Author(s)

Tongyue Luo, Rice University, Spring 2015.

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
http://investexcel.net/forward-start-options/

Examples

(o = ForwardStartMC())$PxMC

o = OptPx(Opt(Style='ForwardStart'), q = 0.03, r = 0.1, vol = 0.15)
(o = ForwardStartMC(o, tts=0.25))$PxMC

ForwardStartMC(o = OptPx(Opt(Style='ForwardStart', Right='Put')))$PxMC

Gap option valuation via Black-Scholes (BS) model

Description

S3 object constructor for price of gap option using BS model

Usage

GapBS(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm =
  1, ContrSize = 1, SName =
  "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q
  = 0, vol = 0.2), K2 = 350000)

Arguments

Value

An original OptPx object with PxBS field as the price of the option and user-supplied K2 parameter

Author(s)

Tong Liu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www.mathworks.com/help/fininst/gapbybls.html

Examples

#See J.C.Hull, OFOD'2014, 9-ed, Example 26.1, p.601
(o <- GapBS())$PxBS

GapBS(o=OptPx(Opt(Style='Gap',Right='Put',K=57)))

#See http://www.mathworks.com/help/fininst/gapbybls.html
o = Opt(Style='Gap',Right='Put',K=57,ttm=0.5,S0=52)
o = GapBS(OptPx(o,vol=0.2,r=0.09),K2=50)

o = Opt(Style='Gap',Right='Put',K=57,ttm=0.5,S0=50)
(o <- GapBS(OptPx(o,vol=0.2,r=0.09),K2=50))$PxBS

Gap option valuation via lattice tree (LT) model

Description

A binomial tree pricer of Gap options that takes the average results for given step sizes in NSteps. Large step sizes should be used for optimal accuracy but may take a minute or so.

Usage

GapLT(o = OptPx(Opt(Style = "Gap")), K2 = 60, on = c(100, 200))

Arguments

Value

An onject of class OptPx including price

Author(s)

Max Lee, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
Humphreys, Natalia. University of Dallas.

Examples

(o = GapLT())$PxLT

o = Opt(Style="Gap",Right='Put',S0 = 500000, ttm = 1,K = 400000)
o = OptPx(o,r = .05, q=0, vol =.2)
(o = GapLT(o,K2 = 350000,on=c(498,499,500,501,502)))$PxLT

o = Opt(Style="Gap", Right='Call',S0 = 65, ttm = 1,K = 70)
o = OptPx(o,r = .05, q=.02,vol =.1)

Gap option valuation via Monte Carlo (MC) simulation

Description

GapMC prices a gap option using the MC method. The call payoff is S_T-K when S_T>K2, where K_2 is the trigger strike. The payoff is increased by K_2-K, which can be positive or negative. The put payoff is K-S_T when S_T<K_2. Default values are from policyholder-insurance example 26.1, p.601, from referenced OFOD, 9ed, text.

Usage

GapMC(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm =
  1, ContrSize = 1, SName =
  "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q
  = 0, vol = 0.2), K2 = 350000, NPaths = 5)

Arguments

Value

An OptPx object. The price is stored under o$PxMC.

Author(s)

Kiryl Novikau, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. p.601

Examples

(o = GapMC())$PxMC  #example 26.1, p.601

o = Opt(Style='Gap', Right='Call', S0=50, K=40, ttm=1)
o = OptPx(o, vol=.2, r=.05, q = .02)
(o = GapMC(o, K2 = 45, NPaths = 5))$PxMC

o = Opt(Style='Gap', Right='Call', S0 = 50, K = 60, ttm = 1)
o = OptPx(o, vol=.25,r=.15, q = .02)
(o = GapMC(o, K2 = 55, NPaths = 5))$PxMC

o = Opt(Style='Gap', Right = 'Put', S0 = 50, K = 57, ttm = .5)
o = OptPx(o, vol = .2, r = .09, q = .2)
(o = GapMC(o, K2 = 50, NPaths = 5))$PxMC

o = Opt(Style='Gap', Right='Call', S0=500000, K=400000, ttm=1)
o = OptPx(o, vol=.2,r=.05, q = 0)
(o = GapMC(o, K2 = 350000, NPaths = 5))$PxMC

Holder Extendible option valuation via Black-Scholes (BS) model

Description

Computes the price of exotic option (via BS model) which gives the holder the right to extend the option's maturity at an additional premium.

Usage

HolderExtendibleBS(o = OptPx(Opt(Style = "HolderExtendible")), k = 105,
  t1 = 0.5, t2 = 0.75, A = 1)

Arguments

Value

The original OptPx object and the option pricing parameters t1, t2,k,A, and computed price PxBS.

Author(s)

Le You, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
Haug, Espen G.,Option Pricing Formulas, 2ed.

Examples

(o = HolderExtendibleBS())$PxBS

o = Opt(Style='HolderExtendible',Right='Call', S0=100, ttm=0.5, K=100)
o = OptPx(o,r=0.08,q=0,vol=0.25)
(o = HolderExtendibleBS(o,k=105,t1=0.5,t2=0.75,A=1))$PxBS

o = Opt("HolderExtendible","Put", S0=100, ttm=0.5, K=100)
o = OptPx(o,r=0.08,q=0,vol=0.25)
(o = HolderExtendibleBS(o,k=90,t1=0.5,t2=0.75,A=1))$PxBS

Ladder option valuation via Monte Carlo (MC) simulation.

Description

Calculates the price of a Ladder Option using 5000 Monte Carlo simulations. The helper function LadderCal() aims to calculate expected payout for each stock prices.

Important Assumptions: The option o follows a General Brownian Motion (BM) ds = mu * S * dt + sqrt(vol) * S * dW where dW ~ N(0,1). The value of mu (the expected price increase) is assumed to be o$r, the risk free rate of return.

Usage

LadderMC(o = OptPx(o = Opt(Style = "Ladder"), NSteps = 5), NPaths = 5,
  L = c(60, 80, 100))

Arguments

Value

The option o with the price in the field PxMC based on MC simulations and the ladder strike price L set by the users themselves

Author(s)

Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015

References

http://stackoverflow.com/questions/25946852/r-monte-carlo-simulation-price-path-converging-volatility-issue

Examples

(o = LadderMC())$PxMC #Price = ~12.30

 o = OptPx(o=Opt(Style='Ladder'), NSteps = 5)
 (o = LadderMC(o))$PxMC        #Price = ~11.50

 o = OptPx(Opt(Style='Ladder', Right='Put'))
 (o = LadderMC(o, NPaths = 5))$PxMC   # Price = ~12.36

 (o = LadderMC(L=c(55,65,75)))$PxMC   # Price = ~10.25

Lookback option valuation with Black-Scholes (BS) model

Description

Calculates the price of a lookback option using a BSM-adjusted algorithm; Carries the assumption that the asset price is observed continuously.

Usage

LookbackBS(o = OptPx(Opt(Style = "Lookback")), Smax = 50, Smin = 50,
  Type = c("Floating", "Fixed"))

Arguments

Details

To price the lookback option, we require the Smax/Smin, S0, r, q, vol, and ttm arguments from the object classes defined in the package. An example of a complete OptLookback option object can be found in the examples.

Value

An original OptPx object with PxBS field as the price of the option and user-supplied Smin, Smax, and Type lookback parameters attached.

Author(s)

Richard Huang, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.

Examples

(o = LookbackBS())$PxBS
  LookbackBS(OptPx(Opt(Style = 'Lookback'))) #Uses default arguments

  # See Hull 9e Example 26.2, p.608; gives price of 7.79
  o = Opt(Style = 'Lookback', S0 = 50, ttm= .25, Right = "Put")
  o = OptPx(o,r = .1, vol = .4)
  o = LookbackBS(o, Type = "Floating")

  # See Hull 9e Example 26.2, p.608; gives price of 8.04
  o = Opt(Style = 'Lookback', S0 = 50, ttm= .25, Right = "Call")
  o = OptPx(o, r = .1, vol = .4)
  o = LookbackBS(o, Type = "Floating")

  # Price = 17.7129
  o = Opt(Style = 'Lookback', S0 = 50, ttm= 1, Right = "Put", K = 60)
  o = OptPx(o,r = .05, q = .02, vol = .25)
  o = LookbackBS(o, Type = "Fixed")

  # Price = 8.237
  o = Opt(Style = 'Lookback', S0 = 50, ttm= 1, Right = "Call", K = 55)
  o = OptPx(o,r = .1, q = .02, vol = .25)
  o = LookbackBS(o, Type = "Fixed")

Lookback option valuation via Monte Carlo (MC) simulation

Description

Calculates the price of a lookback option using a Monte Carlo (MC) Simulation. Carries the assumption that the asset price is observed continuously. Assumes that the the option o followes ds = mu * S * dt + sqrt(vol) * S * dz where dz is a Wiener Process. Assume that without dividends, mu are default to be r.

Usage

LookbackMC(o = OptPx(Opt(Style = "Lookback"), r = 0.05, q = 0, vol = 0.3),
  NPaths = 5, div = 1000, Type = c("Floating", "Fixed"))

Arguments

Details

To price the lookback option, we require the S0, K, and ttm arguments from object Opt and r, q, vol from object OptPx defined in the package. The results of simulation would unstable without setting seeds.

Value

A list of class LookbackMC consisting of the input object OptPx and the price of the lookback option based on Monte Carlo Simulation (see references).

Author(s)

Tong Liu, Department of Statistics, Rice University, Spring 2015

References

Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod

Examples

(o = LookbackMC())$PxMC   #Use default arguments, Output: approximately 16.31.

 # Floating & Put
 o=OptPx(Opt(S0=50,K=50,ttm=0.25,Right='Put',Style="Lookback"),r=0.1,vol=.4)
 LookbackMC(o,NPaths=5,div=1000) #Output: 7.79 from Hull 9e Example 26.2 Pg 608.

 # Floating & Call
 o=OptPx(Opt(S0=50,K=50,ttm=0.25,Right='Call',Style="Lookback"),r=0.1,vol=.4)
 LookbackMC(o,NPaths=5,div=1000) #Output: 8.04 from Hull 9e Example 26.2 Pg 608

 # Fixed & Put
 o=OptPx(Opt(S0=50,K=60,ttm=1,Right='Put',Style="Lookback"),r=0.05,q=0.02,vol=.25)
 LookbackMC(o,Type="Fixed",NPaths=5,div=1000)

 # Fixed & Call
 o=OptPx(Opt(S0=50,K=55,ttm=1,Right='Call',Style="Lookback"),r=0.1,vol=.25)
 LookbackMC(o,Type="Fixed",NPaths=5,div=1000)

Opt object constructor

Description

An S3 object constructor for an option contract (financial derivative)

Usage

Opt(Style = c("European", "American", "Asian", "Binary", "AverageStrike",
  "Barrier", "Chooser", "Compound", "DeferredPayment", "ForeignEquity",
  "ForwardStart", "Gap", "HolderExtendible", "Ladder", "Lookback", "MOPM",
  "Perpetual", "Quotient", "Rainbow", "Shout", "SimpleChooser", "VarianceSwap"),
  Right = c("Call", "Put", "Other"), S0 = 50, ttm = 2, K = 52,
  Curr = "$", ContrSize = 100, SName = "A stock share", SSymbol = "")

Arguments

Value

A list of class Opt

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

Examples

Opt()  #Creates an S3 object for an option contract
Opt(Right='Put')   #See J. C. Hull, OFOD'2014, 9-ed, Fig.13.10, p.289

OptPos object constructor

Description

S3 object constructor for lattice-pricing specs of an option contract. Inherits Opt object.

Usage

OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)

Arguments

Value

A list of class OptPx

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

Examples

OptPos()  # Creates an S3 object for an option contract
OptPos(Opt(Right='Put'))  #See J.C.Hull, OFOD'2014, 9-ed, Fig.13.10, p.289

OptPx object constructor

Description

An S3 object constructor for lattice-pricing specifications for an option contract. Opt object is inhereted.

Usage

OptPx(o = Opt(), r = 0.05, q = 0, rf = 0, vol = 0.3, NSteps = 3)

Arguments

Value

A list of class OptPx with parameters supplied to Opt and OptPx constructors

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

Examples

OptPx()  #Creates an S3 object for an option contract

#See J.C.Hull, OFOD'2014, 9-ed, Fig.13.10, p.289
OptPx(Opt(Right='Put'))

o = OptPx(Opt(Right='Call', S0=42, ttm=.5, K=40), r=.1, vol=.2)

Perpetual option valuation via Black-Scholes (BS) model

Description

An exotic option is an option which has features making it more complex than commonly traded options. A perpetual option is non-standard financial option with no fixed maturity and no exercise limit. While the life of a standard option can vary from a few days to several years, a perpetual option (XPO) can be exercised at any time. Perpetual options are considered an American option. European options can be exercised only on the option's maturity date.

Usage

PerpetualBS(o = OptPx(Opt(Style = "Perpetual"), q = 0.1))

Arguments

Value

A list of class Perpetual.BS consisting of the input object OptPx

Author(s)

Kim Raath, Department of Statistics, Rice University, Spring 2015.

References

Chi-Guhn Lee, The Black-Scholes Formula, Courses, Notes, Note2, Sec 1.5 and 1.6 http://www.mie.utoronto.ca/courses/mie566f/materials/note2.pdf

Examples

#Perpetual American Call and Put
#Verify pricing with \url{http://www.coggit.com/freetools}
(o <- PerpetualBS())$PxBS # Approximately valued at $8.54

#This example should produce approximately $33.66
o = Opt(Style="Perpetual", Right='Put', S0=50, K=55)
o = OptPx(o, r = .03, q = 0.1, vol = .4)
(o = PerpetualBS(o))$PxBS

#This example should produce approximately $10.87
o = Opt(Style="Perpetual", Right='Call', S0=50, K=55)
o = OptPx(o, r = .03, q = 0.1, vol = .4)
(o <- PerpetualBS(o))$PxBS

Computes payout/profit values

Description

Computes payout/profit values

Usage

Profit(o = OptPos(), S = o$S0)

Arguments

Value

A numeric matrix of size [length(S), 2]. Columns: stock prices, corresponding option profits

Author(s)

Oleg Melnikov, Department of Statistics, Rice University, Spring 2015

Examples

Profit(o=Opt())
graphics::plot( print( Profit(OptPos(Prem=2.5), S=40:60)), type='l'); grid()

Quotient option valuation via Black-Scholes (BS) model

Description

Quotient Option via Black-Scholes (BS) model

Usage

QuotientBS(o = OptPx(Opt(Style = "Quotient")), I1 = 100, I2 = 100,
  g1 = 0.04, g2 = 0.03, sigma1 = 0.18, sigma2 = 0.15, rho = 0.75)

Arguments

Value

A list of class QuotientBS consisting of the original OptPx object and the option pricing parameters I1,I2, Type, isForeign, and isDomestic as well as the computed price PxBS.

Author(s)

Chengwei Ge, Department of Statistics, Rice University, Spring 2015

References

Zhang Peter G., Exotic Options, 2nd, 1998. http://amzn.com/9810235216.

Examples

(o = QuotientBS())$PxBS

o = OptPx(Opt(Style = 'Quotient', Right = "Put"), r= 0.05)
(o = QuotientBS(o, I1=100, I2=100, g1=0.04, g2=0.03, sigma1=0.18,sigma2=0.15, rho=0.75))$PxBS

o = OptPx(Opt(Style = 'Quotient',  Right = "Put", ttm=1, K=1), r= 0.05)
QuotientBS(o, I1=100, I2=100, g1=0.04, g2=0.03, sigma1=0.18,sigma2=0.15, rho=0.75)

o = OptPx(Opt(Style = 'Quotient',  Right = "Call", ttm=1, K=1), r= 0.05)
QuotientBS(o, I1=100, I2=100, g1=0.04, g2=0.03, sigma1=0.18,sigma2=0.15, rho=0.75)

Quotient option valuation via Monte Carlo (MC) model

Description

Calculates the price of a Quotient option using Monte-Carlo simulations.

Usage

QuotientMC(o = OptPx(Opt(Style = "Quotient")), S0_2 = 100, NPaths = 5)

Arguments

Details

The Monte-Carlo simulations assume the underlying price undergoes Geometric Brownian Motion (GBM). Payoffs are discounted at risk-free rate to price the option. A thorough understanding of the object class construction is recommended. Please see OptPx, Opt for more information.

Value

An original OptPx object with Px.MC field as the price of the option and user-supplied S0_2, NPaths parameters attached.

Author(s)

Richard Huang, Department of Statistics, Rice University, Spring 2015

References

http://www.investment-and-finance.net/derivatives/q/quotient-option.html

Examples

(o = QuotientMC())$PxMC #Default Quotient option price.

o = OptPx(Opt(S0=100, ttm=1, K=1.3), r=0.10, q=0, vol=0.1)
(o = QuotientMC(o, S0_2 = 180, NPaths=5))$PxMC

QuotientMC(OptPx(Opt()), S0_2 = 180, NPaths=5)

QuotientMC(OptPx(), S0_2 = 201, NPaths = 5)

QuotientMC(OptPx(Opt(S0=500, ttm=1, K=2)), S0_2 = 1000, NPaths=5)

Rainbow option valuation via Black-Scholes (BS) model

Description

Rainbow Option via Black-Scholes (BS) model

Usage

RainbowBS(o = OptPx(Opt(Style = "Rainbow")), S1 = 100, S2 = 95, D1 = 0,
  D2 = 0, sigma1 = 0.15, sigma2 = 0.2, rho = 0.75, Type = c("Max",
  "Min"))

Arguments

Details

Two types of Rainbow options are priced: 'Max' and 'Min'.

Value

A list of class RainbowBS consisting of the original OptPx object and the option pricing parameters S1, Type, isMax, and isMin as well as the computed price PxBS.

Author(s)

Chengwei Ge,Department of Statistics, Rice University, Spring 2015

References

Zhang Peter G., Exotic Options, 2nd ed, 1998.

Examples

(o = RainbowBS())$PxBS

  o = OptPx(Opt(Style = 'Rainbow',  Right = "Put"), r = 0.08)
  RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min')

  o = OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Put"), r = 0.08)
  RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min')

  o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Put"), r = 0.08)
  RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Max')

  o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Call"), r = 0.08)
  RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min')

  o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Call"), r = 0.08)
  RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Max')

Shout option valuation via finite differences (FD) method

Description

Shout option valuation via finite differences (FD) method

Usage

ShoutFD(o = OptPx(Opt(Style = "Shout")), N = 100, M = 20, Smin = 0,
  Smax = 100)

Arguments

Details

A shout option is a European option where the holder can 'shout' to the writer at one time during its life. At the end of the life of the option, the option holder receives either the usual payoff from a European option or the intrinsic value at the time of the shout, whichever is greater. An explicit finite difference method (Page 482 in Hull's book) is used here to price the shout put option. Similar to pricing American options, the value of the option is consolidated at each node of the grid to see if shouting would be optimal. The corresponding shout call option is priced using the Put-Call-Parity in the finite difference method .

Value

A list of class OptPx, including option pricing parameters N, M, Smin, and Smax, as well as the computed option price PxFD.

Author(s)

Xinnan Lu, Department of Statistics, Rice University, 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html pp.609.

Examples

(o = ShoutFD(OptPx(Opt(Right="C", Style="Shout"))))$PxFD

 o = OptPx(Opt(Right="C", Style="Shout"))
 (o = ShoutFD(o, N=10))$PxFD # very differnt result for N=10

 (o = ShoutFD(OptPx(Opt(Right="P", Style="Shout"))))$PxFD

 o = Opt(Right='P', S0=100, K=110, ttm=0.5, Style='Shout')
 o = OptPx(o, vol=0.2, r=0.05, q=0.04)
 (o = ShoutFD(o,N=100,Smax=200))$PxFD

 o = Opt(Right="C", S0=110, K=100, ttm=0.5, Style="Shout")
 o = OptPx(o, vol=0.2, r=0.05, q=0.04)
 (o = ShoutFD(o,N=100,Smax=200))$PxFD

Shout option valuation via lattice tree (LT)

Description

A shout option is a European option where the holder can shout to the writer at one time during its life. At the end of the life of the option, the option holder receives either the usual payoff from a European option or the instrinsic value at the time of the shout, which ever is greater. max(0,S_T-S_tau)+(S_tau-K)

Usage

ShoutLT(o = OptPx(Opt(Style = "Shout")), IncBT = TRUE)

Arguments

Value

A list of class ShoutLT consisting of the original OptPx object, binomial tree stepBT and the computed price PxBS.

Author(s)

Le You, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315

Examples

(o = ShoutLT( OptPx(Opt(Style='Shout'))))$PxLT

o = Opt(Style='Shout', Right='Call', S0=60, ttm=.25, K=60)
ShoutLT( OptPx(o,r=.1, q=.02, vol=.45, NSteps=10))

o = Opt(Style='Shout', Right='Call', S0=60, ttm=.25, K=60)

Shout option valuation via lattice tree (LT)

Description

A shout option is a European option where the holder can shout to the writer at one time during its life. At the end of the life of the option, the option holder receives either the usual payoff from a European option or the instrinsic value at the time of the shout, which ever is greater. max(0,S_T-S_tau)+(S_tau-K)

Usage

ShoutLTVectorized(o = OptPx(o = Opt(Style = "Shout")))

Arguments

Value

A list of class ShoutLT consisting of the original OptPx object, binomial tree step BT and the computed price PxBS.

Author(s)

Le You, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315

Examples

(o = ShoutLTVectorized( OptPx(Opt(Style='Shout'))))$PxLT

o = Opt(Style='Shout')
(o = ShoutLTVectorized( OptPx(o, r=.1, q=.02, vol=.45, NSteps=10)))$PxLT

Shout option valuation via Monte Carlo (MC) simulations.

Description

Calculates the price of a shout option using Monte Carlo simulations to determine expected payout. Assumes that the option follows a General Brownian Motion (GBM) process, ds = mu * S * dt + sqrt(vol) * S * dW where dW ~ N(0,1). Note that the value of mu (the expected price increase) is assumped to be o$r, the risk free rate of return.

Usage

ShoutMC(o = OptPx(o = Opt(Style = "Shout")), NPaths = 10)

Arguments

Value

The option object o with the price in the field PxMC based on the MC simulations.

Author(s)

Jake Kornblau, Department of Statistics, Rice University, 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
Also: http://www.math.umn.edu/~spirn/5076/Lecture16.pdf

Examples

(o = ShoutMC())$PxMC # Approximately valued at $11

  o = Opt(Style='Shout')
  (o = ShoutMC(OptPx(o, NSteps = 5)))$PxMC # Approximately valued at $18.6

  o = Opt(Style='Shout',S0=110,K=100,ttm=.5)
  o = OptPx(o, r=.05, vol=.2, q=0, NSteps = 10)
  (o = ShoutMC(o, NPaths = 10))$PxMC

Variance Swap valuation via Black-Scholes (BS) model

Description

Variance Swap valuation via Black-Scholes (BS) model

Usage

VarianceSwapBS(o = OptPx(Opt(Style = "VarianceSwap", Right = "Other", ttm =
  0.25, S0 = 1020), r = 0.04, q = 0.01), K = seq(800, 1200, 50),
  Vol = seq(0.2, 0.24, 0.005), notional = 10^8, varrate = 0.045)

Arguments

Value

An object of class OptPx with value included

Author(s)

Max Lee, Department of Statistics, Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod.

Examples

(o = VarianceSwapBS())$PxBS

o = Opt(Style="VarianceSwap",Right="Other",ttm=.25,S0=1020)
o = OptPx(o,r=.04,q=.01)
Vol = Vol=c(.29,.28,.27,.26,.25,.24,.23,.22,.21)
(o = VarianceSwapBS(o,K=seq(800,1200,50),Vol=Vol,notional=10^8,varrate=.045))$PxBS

o = Opt(Style="VarianceSwap",Right="Other",ttm=.25,S0=1020)
o = OptPx(o,r=.04,q=.01)
Vol=c(.2,.205,.21,.215,.22,.225,.23,.235,.24)
(o =VarianceSwapBS(o,K=seq(800,1200,50),Vol=Vol,notional=10^8,varrate=.045))$PxBS

o = Opt(Style="VarianceSwap",Right="Other",ttm=.1,S0=100)
o = OptPx(o,r=.03,q=.02)
Vol=c(.2,.19,.18,.17,.16,.15,.14,.13,.12)
(o =VarianceSwapBS(o,K=seq(80,120,5),Vol=Vol,notional=10^4,varrate=.03))$PxBS

VarianceSwap option valuation via Monte Carlo (MC) simulation.

Description

Calculates the price of a VarianceSwap Option using 500 Monte Carlo simulations.
Important Assumptions: The option o followes a General Brownian Motion ds = mu * S * dt + sqrt(vol) * S * dW where dW ~ N(0,1). The value of mu (the expected price increase) is assumed to be o$r-o$q.

Usage

VarianceSwapMC(o = OptPx(o = Opt(Style = "VarianceSwap")), var = 0.2,
  NPaths = 5)

Arguments

Value

The option o with the price in the field PxMC based on MC simulations and the Variance Swap option properties set by the users themselves

Author(s)

Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015

References

Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod.
http://stackoverflow.com/questions/25946852/r-monte-carlo-simulation-price-path-converging-volatility-issue

Examples

(o = VarianceSwapMC())$PxMC #Price = ~0.0245

 (o = VarianceSwapMC(NPaths = 5))$PxMC # Price = ~0.0245

 (o = VarianceSwapMC(var=0.4))$PxMC # Price = ~-0.1565

Coerce an argument to OptPos class.

Description

Coerce an argument to OptPos class.

Usage

as.OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)

Arguments

Value

An object of class OptPos.

Author(s)

Oleg Melnikov

Examples

as.OptPos(Opt())

Is an object Opt?

Description

Tests the argument for the specific class type.

Usage

is.Opt(o)

Arguments

Value

TRUE if and only if an argument is of Opt class.

Author(s)

Oleg Melnikov

Examples

is.Opt(Opt())  #verifies that Opt() returns an object of class \code{Opt}
is.Opt(1:3)    #verifies that code{1:3} is not an object of class \code{Opt}

Is an object OptPos?

Description

Tests the argument for the specific class type.

Usage

is.OptPos(o)

Arguments

Value

TRUE if and only if an argument is of OptPos class.

Author(s)

Oleg Melnikov

Examples

is.OptPos(OptPos())

Is an object OptPx?

Description

Tests the argument for the specific class type.

Usage

is.OptPx(o)

Arguments

Value

TRUE if and only if an argument is of OptPx class.

Author(s)

Oleg Melnikov

Examples

is.OptPx(OptPx(Opt(S0=20), r=0.12))

Bivariate Standard Normal CDF

Description

Bivariate Standard Normal CDF Calculator For Given Values of x, y, and rho

Usage

pbnorm(x = 0, y = 0, rho = 0)

Arguments

Details

This runs a bivariate standard normal pdf then calculates the cdf from that based on the input parameters

Value

Density under the bivariate standard normal distribution

Author(s)

Robert Abramov, Department of Statistics, Rice University, 2015

References

Adapted from "Bivariate normal distribution with R", Edouard Tallent's blog from Sep 21, 2012
https://quantcorner.wordpress.com/2012/09/21/bivariate-normal-distribution-with-r

Examples

pbnorm(1, 1, .5)
#pbnorm(2, 2, 0)
#pbnorm(-1, -1, .35)
#pbnorm(0, 0, 0)

ttl = 'cdf of x, at y=0'
X = seq(-5,5,1)
graphics::plot(X, sapply(X, function(x) pbnorm(0,x,0)), type='l', main=ttl)