| Title: | Pricing of Vanilla and Exotic Option Contracts |
|---|---|
| Description: | Option pricing (financial derivatives) techniques mainly following textbook 'Options, Futures and Other Derivatives', 9ed by John C.Hull, 2014. Prentice Hall. Implementations are via binomial tree option model (BOPM), Black-Scholes model, Monte Carlo simulations, etc. This package is a result of Quantitative Financial Risk Management course (STAT 449 and STAT 649) at Rice University, Houston, TX, USA, taught by Oleg Melnikov, statistics PhD student, as of Spring 2015. |
| Authors: | Oleg Melnikov [aut, cre], Max Lee [ctb], Robert Abramov [ctb], Richard Huang [ctb], Liu Tong [ctb], Jake Kornblau [ctb], Xinnan Lu [ctb], Kiryl Novikau [ctb], Tongyue Luo [ctb], Le You [ctb], Jin Chen [ctb], Chengwei Ge [ctb], Jiayao Huang [ctb], Kim Raath [ctb] |
| Maintainer: | Oleg Melnikov <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0.1 |
| Built: | 2025-03-13 02:59:58 UTC |
| Source: | https://github.com/cran/QFRM |
OptPos class.Coerce an argument to OptPos class.
as.OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)as.OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)
o |
A |
Pos |
Specify position direction in your portfolio. |
Prem |
Option premium, i.e. cost of an option purchased or to be purchased. |
An object of class OptPos.
Oleg Melnikov
as.OptPos(Opt())as.OptPos(Opt())
Price Asian option using BS model
AsianBS(o = OptPx(Opt(Style = "Asian")))AsianBS(o = OptPx(Opt(Style = "Asian")))
o |
An object of class |
This pricing algorithm assumes average price is calculated continuously.
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.
Xinnan Lu, Department of Statistics, Rice University, Spring 2015
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.
(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(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
Calculates the price of an Asian option using Monte Carlo simulations to
determine expected payout.
Assumptions:
The option follows a General Brownian Motion (BM),
where .
The value of (the expected price increase) is o$r, the risk free rate of return (RoR).
The averaging period is the life of the option.
AsianMC(o = OptPx(o = Opt(Style = "Asian"), NSteps = 5), NPaths = 5)AsianMC(o = OptPx(o = Opt(Style = "Asian"), NSteps = 5), NPaths = 5)
o |
The |
NPaths |
The number of simulation paths to use in calculating the price, |
The option o with the price in the field PxMC based on MC simulations.
Jake Kornblau, Department of Statistics and Department of Computer Science, Rice University, 2016
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014.
Prentice Hall. ISBN 978-0-13-345631-8,
http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
http://www.math.umn.edu/~spirn/5076/Lecture16.pdf
(o = AsianMC())$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))$PxMC(o = AsianMC())$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))$PxMC
Calculates the price of an Average Strike option using Monte Carlo simulations
by determining the determine expected payout. Assumes that the input option follows a General
Brownian Motion where
Note that the value of (the expected price increase) is assumped to be
o$r, the risk free rate of return. Additionally, the averaging period is
assumed to be the life of the option.
AverageStrikeMC(o = OptPx(o = Opt(Style = "AverageStrike")), NPaths = 5)AverageStrikeMC(o = OptPx(o = Opt(Style = "AverageStrike")), NPaths = 5)
o |
The AverageStrike |
NPaths |
the number of simulations to use in calculating the price, |
The original option object o with the price in the field PxMC based on the MC simulations.
Jake Kornblau, Department of Statistics and Department of Computer Science, Rice University, Spring 2015
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
(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 =~ $5.6(o = AverageStrikeMC())$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 =~ $5.6
This function calculates the price of a Barrier option. This price is based on the assumptions that the probability distribution is lognormal and that the asset price is observed continuously.
BarrierBS(o = OptPx(Opt(Style = "Barrier")), dir = c("Up", "Down"), knock = c("In", "Out"), H = 40)BarrierBS(o = OptPx(Opt(Style = "Barrier")), dir = c("Up", "Down"), knock = c("In", "Out"), H = 40)
o |
The |
dir |
The direction of the option to price. Either Up or Down. |
knock |
Whether the option goes In or Out when the barrier is reached. |
H |
The barrier level |
To price the barrier option, we need to know whether the option is Up or Down | In or Out | Call or Put. Beyond that we also need the S0, K, r, q, vol, H, and ttm arguments from the object classes defined in the package.
The price of the barrier option o, which is based on the BSM-adjusted algorithm (see references).
Kiryl Novikau, Department of Statistics, Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. pp.606-607
(o = BarrierBS())$PxBS # Option with default arguments is valued at $9.71 #Down-and-In-Call o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=10) o = OptPx(o, r = .05, q = 0, vol = .25) o = BarrierBS(o, dir = "Down", knock = 'In', H = 40) #Down-and-Out Call o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=10) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Down", knock = 'Out', H = 40) #Up-and-In Call o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Call", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Up", knock = 'In', H = 60) #Up-and-Out Call o = Opt(Style='Barrier', S0 = 50, K = 50, ttm = 1, Right="Call", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Up", knock = 'Out', H = 60) #Down-and-In Put o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Down", knock = 'In', H = 40) #Down-and-Out Put o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Down", knock = 'Out', H = 40) #Up-and-In Put o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Up", knock = 'In', H = 60) #Up-and-Out Put o = Opt(Style='Barrier', S0=50, K=50, ttm=1, Right="Put", ContrSize=1) o = OptPx(o, r = .05, q = .02, vol = .25) o = BarrierBS(o, dir = "Up", knock = 'Out', H = 60)(o = BarrierBS())$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)
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.
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)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)
o |
An object of class |
dir |
A direction for the barrier, either |
knock |
The option is either a knock-in option or knock-out option. Default= |
H |
The barrier level. |
A list of class BarrierLT consisting of the input object OptPx
and the appended new parameters and option price.
Tong Liu, Department of Statistics, Rice University, Spring 2015
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.
# 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))$PxLT) c=(10:200) plot(visual~c,type="l",xlab="NSteps",ylab="Price",main="Price converence with NSteps") # Down and Knock-out Call Option with H=40 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir="Down",knock="Out",H=40) # Down and Knock-in Call Option with H=40 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir="Down",knock="In",H=40) # Up and Knock-out Call Option with H=60 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir='Up',knock="Out") # Down and Knock-out Put Option with H=40 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir="Down",knock="Out",H=40) # Down and Knock-in Put Option with H=40 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir="Down",knock="In",H=40) # Up and Knock-out Put Option with H=60 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir='Up',knock="Out") # Up and Knock-in Put Option with H=60 BarrierLT(OptPx(o=Opt(Style="Barrier",Right="Put")))# default Up and Knock-in Call Option with H=60, approximately 7.09 (o = BarrierLT())$PxLT #Visualization of price changes as Nsteps change. o = Opt(Style="Barrier") visual=sapply(10:200,function(n) BarrierLT(OptPx(o,NSteps=n))$PxLT) c=(10:200) plot(visual~c,type="l",xlab="NSteps",ylab="Price",main="Price converence with NSteps") # Down and Knock-out Call Option with H=40 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir="Down",knock="Out",H=40) # Down and Knock-in Call Option with H=40 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir="Down",knock="In",H=40) # Up and Knock-out Call Option with H=60 o = OptPx(o=Opt(Style="Barrier")) BarrierLT(o,dir='Up',knock="Out") # Down and Knock-out Put Option with H=40 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir="Down",knock="Out",H=40) # Down and Knock-in Put Option with H=40 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir="Down",knock="In",H=40) # Up and Knock-out Put Option with H=60 o = OptPx(o=Opt(Style="Barrier",Right="Put")) BarrierLT(o,dir='Up',knock="Out") # Up and Knock-in Put Option with H=60 BarrierLT(OptPx(o=Opt(Style="Barrier",Right="Put")))
Calculates the price of a Barrier Option using 10000 Monte Carlo simulations. The helper function BarrierCal() aims to calculate expected payout for each stock prices.
Important Assumptions:
The option follows a General Brownian Motion (GBM)
where .
The value of (the expected percent price increase) is assumed to be o$r-o$q.
BarrierMC(o = OptPx(o = Opt(Style = "Barrier")), knock = c("In", "Out"), B = 60, NPaths = 5)BarrierMC(o = OptPx(o = Opt(Style = "Barrier")), knock = c("In", "Out"), B = 60, NPaths = 5)
o |
The |
knock |
Defines the Barrier option to be " |
B |
The Barrier price level |
NPaths |
The number of simulation paths to use in calculating the price |
The option o with the price in the field PxMC based on MC simulations and the Barrier option
properties set by the users themselves
Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. Also, http://stackoverflow.com/questions/25946852/r-monte-carlo-simulation-price-path-converging-volatility-issue
(o = BarrierMC())$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 =~ $1(o = BarrierMC())$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 =~ $1
Compute option price via binomial option pricing model (recombining symmetric binomial tree)
Binary_BOPM(o = OptPx(Opt(Style = "Binary")), Type = c("cash-or-nothing", "asset-or-nothing"), Q = 1000, IncBT = FALSE)Binary_BOPM(o = OptPx(Opt(Style = "Binary")), Type = c("cash-or-nothing", "asset-or-nothing"), Q = 1000, IncBT = FALSE)
o |
|
Type |
Binary option type: |
Q |
A fixed amount of payoff |
IncBT |
TRUE/FALSE, indicates whether to include the full binomial tree in the returned object |
original OptPx object with Px.BOPM property and (optional) binomial tree
IncBT = FALSE: option price value (type double, class numeric)
IncBT = TRUE: binomial tree as a list (of length (o$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)
(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))$PxBT(o = Binary_BOPM())$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
S3 object pricing model for a binary option.
Two types of binary options are priced: 'cash-or-nothing' and 'asset-or-nothing'.
BinaryBS(o = OptPx(Opt(Style = "Binary")), Q = 1, Type = c("cash-or-nothing", "asset-or-nothing"))BinaryBS(o = OptPx(Opt(Style = "Binary")), Q = 1, Type = c("cash-or-nothing", "asset-or-nothing"))
o |
An object of class |
Q |
A fixed amount of payoff |
Type |
Binary option type: 'Cash or Nothing' or 'Asset or Nothing'.
Partial names are allowed, eg. |
A list of class Binary.BS consisting of the input object OptPx and the appended new parameters and option price.
Xinnan Lu, Department of Statistics, Rice University, Spring 2015
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
(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')(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.
BinaryMC(o = OptPx(Opt(Style = "Binary")), Q = 25, Type = c("cash-or-nothing", "asset-or-nothing"), NPaths = 5)BinaryMC(o = OptPx(Opt(Style = "Binary")), Q = 25, Type = c("cash-or-nothing", "asset-or-nothing"), NPaths = 5)
o |
An |
Q |
A fixed numeric amount of payoff |
Type |
Binary option type: |
NPaths |
The number of simulation paths to use in calculating the price
Partial names are allowed, eg. |
Two types of binary options are priced: 'cash-or-nothing' and 'asset-or-nothing'.
The original input object o with added parameters and option price PxMC
Tongyue Luo, Rice University, Spring 2015.
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.
(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"))$PxMC(o = BinaryMC())$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
Compute option price via binomial option pricing model (recombining symmetric binomial tree). If no tree requested for European option, vectorized algorithm is used.
BOPM(o = OptPx(), IncBT = TRUE)BOPM(o = OptPx(), IncBT = TRUE)
o |
An |
IncBT |
Values |
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$NSteps+1) of numeric matrices (2 x i)
Each matrix is a set of possible i outcomes at time step i columns: (underlying prices, option prices)
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315
#See Fig.13.11, Hull/9e/p291. #Create an option and price it o = Opt(Style='Eu', Right='C', S0 = 808, ttm = .5, K = 800) o = BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=2), IncBT=TRUE) o$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)$PxBT
#DerivaGem diplays up to 10 steps: o = Opt(Style='Am', Right='C', 810, .5, 800) BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=20), IncBT=TRUE)
#DerivaGem computes up to 500 steps: o = Opt(Style='American', Right='Put', 810, 0.5, 800) BOPM( OptPx(o, r=0.05, q=0.02, vol=0.2, NSteps=1000), IncBT=FALSE)
BOPM_Eu for European option via vectorized approach.
A helper function to price European options via a vectorized (fast, memory efficient) approach.
BOPM_Eu(o = OptPx())BOPM_Eu(o = OptPx())
o |
An |
A list of class OptPx with an element PxBT, which is an option price value (type double, class numeric)
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
Code adopted Gilli & Schumann's R implementation to Opt* objects
Gili, M. and Schumann, E. (2009) Implementing Binomial Trees, COMISEF Working Papers Series
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1341181 for original paper, BOPM for American option pricing.
#Fig.13.11, Hull/9e/p291: o = Opt(Style='European', Right='Call', S0=810, ttm=.5, K=800) (o <- BOPM_Eu( OptPx(o, r=.05, q=.02, vol=.2, NSteps=2)))$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))$PxBT#Fig.13.11, Hull/9e/p291: o = Opt(Style='European', Right='Call', S0=810, ttm=.5, K=800) (o <- BOPM_Eu( OptPx(o, r=.05, q=.02, vol=.2, NSteps=2)))$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))$PxBT
a wrapper function for BS_Simple; uses OptPx object as input.
BS(o = OptPx())BS(o = OptPx())
o |
An |
An original OptPx object with BS list as components of Black-Scholes formular.
See BS_Simple.
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315
#See Hull, p.338, Ex.15.6. #Create an option and price it o = Opt(Style='Eu', Right='Call', S0 = 42, ttm = .5, K = 40) o = BS( OptPx(o, r=.1, vol=.2, NSteps=NA)) o$PxBS #print call option price computed by Black-Scholes pricing model o$BS$Px$Put #print put option price computed by Black-Scholes pricing model#See Hull, p.338, Ex.15.6. #Create an option and price it o = Opt(Style='Eu', Right='Call', S0 = 42, ttm = .5, K = 40) o = BS( OptPx(o, r=.1, vol=.2, NSteps=NA)) o$PxBS #print call option price computed by Black-Scholes pricing model o$BS$Px$Put #print put option price computed by Black-Scholes pricing model
Black-Scholes (aka Black-Scholes-Merton, BS, BSM) formula for simple parameters
BS_Simple(S0 = 42, K = 40, r = 0.1, q = 0, ttm = 0.5, vol = 0.2)BS_Simple(S0 = 42, K = 40, r = 0.1, q = 0, ttm = 0.5, vol = 0.2)
S0 |
The spot price of the underlying security |
K |
The srike price of the underlying (same currency as S0) |
r |
The annualized risk free interest rate, as annual percent / 100 (i.e. fractional form. 0.1 is 10 percent per annum) |
q |
The annualized dividiend yield, same units as |
ttm |
The time to maturity, fraction of a year (annualized) |
vol |
The volatility, in units of standard deviation. |
Uses BS formula to calculate call/put option values and elements of BS model
a list of BS formula elements and BS price,
such as d1 for , d2 for , Nd1 for ,
Nd2 for , NCallPxBS for BSM call price, PutPxBS for BSM put price
Robert Abramov, Department of Statistics, Rice University, Spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod. http://amzn.com/0133456315 http://www.theresearchkitchen.com/archives/106
#See Hull p.339, Ex.15.6. (o <- BS_Simple(S0=42,K=40,r=.1,q=0,ttm=.5,vol=.2))$Px$Call #returns 4.759422 o$Px$Put # returns 0.8085994 as the price of the put BS_Simple(100,90,0.05,0,2,0.30) BS_Simple(50,60,0.1,.2,3,0.25) BS_Simple(90,90,0.15,0,.5,0.20) BS_Simple(15,15,.01,0.0,0.5,.5)#See Hull p.339, Ex.15.6. (o <- BS_Simple(S0=42,K=40,r=.1,q=0,ttm=.5,vol=.2))$Px$Call #returns 4.759422 o$Px$Put # returns 0.8085994 as the price of the put BS_Simple(100,90,0.05,0,2,0.30) BS_Simple(50,60,0.1,.2,3,0.25) BS_Simple(90,90,0.15,0,.5,0.20) BS_Simple(15,15,.01,0.0,0.5,.5)
Compute an exotic option that allow the holder decide the option
will be a call or put option at some predetermined future date.
In a simple case, both put and call option are plain vanilla option.
The value of the simple chooser option is .
The plain vanilla option is calculated based on the BS model.
ChooserBS(o = OptPx(Opt(Style = "Chooser")), t1 = 9/12, t2 = 3/12)ChooserBS(o = OptPx(Opt(Style = "Chooser")), t1 = 9/12, t2 = 3/12)
o |
An object of class |
t1 |
The time to maturity of the call option, measured in years. |
t2 |
The time to maturity of the put option, measured in years. |
A list of class SimpleChooserBS consisting of the original OptPx object
and the option pricing parameters t1, t2,
as well as the computed price PxBS.
Le You, Department of Statistics, Rice University, spring 2015
Hull, John C.,Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
Huang Espen G., Option Pricing Formulas, 2ed. http://down.cenet.org.cn/upfile/10/20083212958160.pdf
Wee, Lim Tiong, MFE5010 Exotic Options,Notes for Lecture 4 Chooser option. http://www.stat.nus.edu.sg/~stalimtw/MFE5010/PDF/L4chooser.pdf
Humphreys, Natalia A., ACTS 4302 Principles of Actuarial Models: Financial Economics. Lesson 14: All-or-nothing, Gap, Exchange and Chooser Options.
(o = ChooserBS())$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))$PxBS(o = ChooserBS())$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))$PxBS
Calculates the price of a Chooser option using a recombining binomial tree model. Has pricing capabilities for both simple European Chooser options as well as American Chooser Options, where exercise can occur any time as a call or put options.
ChooserLT(o = OptPx(Opt("Chooser", ttm = 1)), t1 = 0.5, t2 = 0.5, IncBT = FALSE)ChooserLT(o = OptPx(Opt("Chooser", ttm = 1)), t1 = 0.5, t2 = 0.5, IncBT = FALSE)
o |
The |
t1 |
The time to maturity of the call option, measured in years. |
t2 |
The time to maturity of the put option, measured in years. |
IncBT |
|
The American chooser option is interpreted as exercise of option being available at any point in time during the life of the option.
An original OptPx object with PxLT field as the price of the option and user-supplied ttc,
IncBT parameters attached.
Richard Huang, Department of Statistics, Rice University, spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall.
ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
Thomas S.Y. Ho et al., The Oxford Guide to Financial Modeling : Applications for Capital Markets. . .
(o = ChooserLT())$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(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)$PxLT
Price chooser option using Monte Carlo (MC) simulation.
ChooserMC(o = OptPx(Opt(Style = "Chooser")), isEu = TRUE, T1 = 1, NPaths = 5, plot = FALSE)ChooserMC(o = OptPx(Opt(Style = "Chooser")), isEu = TRUE, T1 = 1, NPaths = 5, plot = FALSE)
o |
An object of class |
isEu |
Values |
T1 |
The time when the choice is made whether the option is a call or put |
NPaths |
The number of Monte Carol simulation paths |
plot |
Values |
A chooser option (sometimes referred to as an as you like it option) has the feature that, after a specified period of time, the holder can choose whether the option is a call or a put. In this algorithm, we can price chooser options when the underlying options are both European or are both American. When the underlying is an American option, the option holder can exercise before and after T1.
A list of class ChooserMC consisting of original OptPx object,
option pricing parameters isEu, NPaths, and T1,
as well as the computed price PxMC for the chooser option.
Xinnan Lu, Department of Statistics, Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html. p.603.
(o = ChooserMC())$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)(o = ChooserMC())$PxMC o = OptPx(Opt(Right='Call',Style="Chooser")) ChooserMC(o,isEu=TRUE,NPaths=5, plot=TRUE) o = OptPx(Opt(Right='Put',Style="Chooser")) ChooserMC(o,isEu=TRUE,NPaths=5, plot=TRUE) o = Opt(Right='C',S0=100,K=110,ttm=4,Style="Chooser") o = OptPx(o,vol=0.2,r=0.05,q=0.04) ChooserMC(o,isEu=TRUE,T1=2,NPaths=5) o = Opt(Right='P',S0=110,K=100,ttm=4,Style="Chooser") o = OptPx(o,vol=0.2,r=0.05,q=0.04) ChooserMC(o,isEu=TRUE,T1=2,NPaths=5) o = Opt(Right='C',S0=50,K=50,ttm=0.5,Style="Ch") o = OptPx(o,vol=0.25,r=0.08,q=0.1) ChooserMC(o,isEu=FALSE,T1=0.25,NPaths=5)
Compound option valuation with Black-Scholes (BS) model
CompoundBS(o = OptPx(Opt(Style = "Compound")), K1 = 10, T1 = 0.5, Type = c("cc", "cp", "pp", "pc"))CompoundBS(o = OptPx(Opt(Style = "Compound")), K1 = 10, T1 = 0.5, Type = c("cc", "cp", "pp", "pc"))
o |
= |
K1 |
The first Strike Price (of the option on the option) |
T1 |
The time of first expiry (of the option on the option) |
Type |
Possible choices are
|
A list of object 'OptCompound' containing the option parameters binomial tree parameters and compound option parameters
Robert Abramov
(o <- CompoundBS())$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)(o <- CompoundBS())$PxBS #price compound option with default parameters o = OptPx(Opt(Style='Compound'), r=0.05, q=0.0, vol=0.25) CompoundBS(o,K1=10,T1=0.5) o = Opt(Style='Compound', S0=50, K=52, ttm=1) CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=6,T1=1.5) o = Opt(Style='Compound', S0=90, K=100, ttm=1.5) CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=15,T1=1) o = Opt(Style='Compound', S0=15, K=15, ttm=0.25) CompoundBS(o=OptPx(o, r=.05, q=0, vol=.25),K1=3,T1=1.5)
CompoundLT prices a compound option using the binomial tree (BT) method.
The inputs it takes are two OptPx objects.
It pulls the S from the o2 input which should be the option with the greater time to maturity.
CompoundLT(o1 = OptPx(Opt(Style = "Compound")), o2 = OptPx(Opt(Style = "Compound")))CompoundLT(o1 = OptPx(Opt(Style = "Compound")), o2 = OptPx(Opt(Style = "Compound")))
o1 |
The |
o2 |
The |
User-supplied o1 option with fields o2 and PxLT,
as the second option and calculated price, respectively.
Kiryl Novikau, Department of Statistics, Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
(o = CompoundLT())$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(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))$PxLT
A binomial tree pricer of a Deferred Payment option. An American option that has payment at expiry no matter when exercise, causing differences in present value (PV) of a payoff.
DeferredPaymentLT(o = OptPx(Opt(Style = "DeferredPayment")))DeferredPaymentLT(o = OptPx(Opt(Style = "DeferredPayment")))
o |
An object of class |
An object of class OptPx with price included
Max Lee, Department of Statistics, Rice University, Spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html
(o = DeferredPaymentLT())$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(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 via Black-Scholes (BS) model
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"))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"))
o |
An object of class |
I1 |
A spot price of the underlying security 1 (usually I1) |
I2 |
A spot price of the underlying security 2 (usually I2) |
sigma1 |
a vector of implied volatilities for the associated security 1 |
sigma2 |
a vector of implied volatilities for the associated security 2 |
g1 |
is the payout rate of the first stock |
rho |
is the correlation between asset 1 and asset 2 |
Type |
ForeignEquity option type: 'Foreign' or 'Domestic' |
Two types of ForeignEquity options are priced: 'Foreign' and 'Domestic'.
See "Exotic Options", 2nd, Peter G. Zhang for more details.
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.
Chengwei Ge, Department of Statistics, Rice University, 2015
Zhang, Peter G. Exotic Options, 2nd, 1998.
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')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')
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
ForwardStartBS(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1)ForwardStartBS(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1)
o |
an |
tts |
Time to start of the option (in years) |
A standard European option starts at a future time tts.
The original user-supplied OptPX object
with price field PxBS and any other provided user-supplied parameters.
Tongyue Luo, Department of Statistics, Rice University, Spring 2015
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
(o = ForwardStartBS())$PxBS o = OptPx(Opt(Style='ForwardStart', Right='Put')) (o = ForwardStartBS(o))$PxBS(o = ForwardStartBS())$PxBS o = OptPx(Opt(Style='ForwardStart', Right='Put')) (o = ForwardStartBS(o))$PxBS
S3 object pricing model for a forward start European option using Monte Carlo simulation
ForwardStartMC(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1, NPaths = 5)ForwardStartMC(o = OptPx(Opt(Style = "ForwardStart")), tts = 0.1, NPaths = 5)
o |
An object of class |
tts |
Time to start of the option, in years. |
NPaths |
The number of MC simulation paths. |
A standard European option starts at a future time tts.
A list of class ForwardStartMC consisting of the input object
OptPx and the appended new parameters and option price.
Tongyue Luo, Rice University, Spring 2015.
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8,
http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
http://investexcel.net/forward-start-options/
(o = ForwardStartMC())$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(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')))$PxMC
S3 object constructor for price of gap option using BS model
GapBS(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm = 1, ContrSize = 1, SName = "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q = 0, vol = 0.2), K2 = 350000)GapBS(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm = 1, ContrSize = 1, SName = "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q = 0, vol = 0.2), K2 = 350000)
o |
An object of class |
K2 |
Strike price that determine if the option pays off. |
An original OptPx object with PxBS field as the price of the option
and user-supplied K2 parameter
Tong Liu, Department of Statistics, Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8. http://www.mathworks.com/help/fininst/gapbybls.html
#See J.C.Hull, OFOD'2014, 9-ed, Example 26.1, p.601 (o <- GapBS())$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))$PxBS#See J.C.Hull, OFOD'2014, 9-ed, Example 26.1, p.601 (o <- GapBS())$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))$PxBS
A binomial tree pricer of Gap options that takes the average results for given step sizes in NSteps. Large step sizes should be used for optimal accuracy but may take a minute or so.
GapLT(o = OptPx(Opt(Style = "Gap")), K2 = 60, on = c(100, 200))GapLT(o = OptPx(Opt(Style = "Gap")), K2 = 60, on = c(100, 200))
o |
An object of class |
K2 |
A numeric strike price above used in calculating if option is in the money or not, known as trigger. |
on |
A vector of number of steps to be used in binomial tree averaging, vector of positive intergers. |
An onject of class OptPx including price
Max Lee, Department of Statistics, Rice University, Spring 2015
Hull, John C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall.
ISBN 978-0-13-345631-8. http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
Humphreys, Natalia. University of Dallas.
(o = GapLT())$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)))$PxLT o = Opt(Style="Gap", Right='Call',S0 = 65, ttm = 1,K = 70) o = OptPx(o,r = .05, q=.02,vol =.1)(o = GapLT())$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)))$PxLT o = Opt(Style="Gap", Right='Call',S0 = 65, ttm = 1,K = 70) o = OptPx(o,r = .05, q=.02,vol =.1)
GapMC prices a gap option using the MC method.
The call payoff is when , where is the trigger strike.
The payoff is increased by , which can be positive or negative.
The put payoff is when .
Default values are from policyholder-insurance example 26.1, p.601, from referenced OFOD, 9ed, text.
GapMC(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm = 1, ContrSize = 1, SName = "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q = 0, vol = 0.2), K2 = 350000, NPaths = 5)GapMC(o = OptPx(Opt(Style = "Gap", Right = "Put", S0 = 5e+05, K = 4e+05, ttm = 1, ContrSize = 1, SName = "Insurance coverage example #26.1, p.601, OFOD, J.C.Hull, 9ed."), r = 0.05, q = 0, vol = 0.2), K2 = 350000, NPaths = 5)
o |
The |
K2 |
The trigger strike price. |
NPaths |
The number of paths (trials) to simulate. |
An OptPx object. The price is stored under o$PxMC.
Kiryl Novikau, Department of Statistics, Rice University, Spring 2015
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
(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))$PxMC(o = GapMC())$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
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.
HolderExtendibleBS(o = OptPx(Opt(Style = "HolderExtendible")), k = 105, t1 = 0.5, t2 = 0.75, A = 1)HolderExtendibleBS(o = OptPx(Opt(Style = "HolderExtendible")), k = 105, t1 = 0.5, t2 = 0.75, A = 1)
o |
An object of class |
k |
The exercise price of the option at t2, a numeric value. |
t1 |
The time to maturity of the call option, measured in years. |
t2 |
The time to maturity of the put option, measured in years. |
A |
The corresponding asset price has exceeded the exercise price X. |
The original OptPx object
and the option pricing parameters t1, t2,k,A, and computed price PxBS.
Le You, Department of Statistics, Rice University, Spring 2015
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.
(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))$PxBS(o = HolderExtendibleBS())$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
Opt?Tests the argument for the specific class type.
is.Opt(o)is.Opt(o)
o |
Any object |
TRUE if and only if an argument is of Opt class.
Oleg Melnikov
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.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}
OptPos?Tests the argument for the specific class type.
is.OptPos(o)is.OptPos(o)
o |
Any object |
TRUE if and only if an argument is of OptPos class.
Oleg Melnikov
is.OptPos(OptPos())is.OptPos(OptPos())
OptPx?Tests the argument for the specific class type.
is.OptPx(o)is.OptPx(o)
o |
Any object |
TRUE if and only if an argument is of OptPx class.
Oleg Melnikov
is.OptPx(OptPx(Opt(S0=20), r=0.12))is.OptPx(OptPx(Opt(S0=20), r=0.12))
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)
where .
The value of (the expected price increase) is assumed to be o$r, the risk free rate of return.
LadderMC(o = OptPx(o = Opt(Style = "Ladder"), NSteps = 5), NPaths = 5, L = c(60, 80, 100))LadderMC(o = OptPx(o = Opt(Style = "Ladder"), NSteps = 5), NPaths = 5, L = c(60, 80, 100))
o |
The |
NPaths |
The number of simulation paths to use in calculating the price |
L |
A series of ladder strike price. |
The option o with the price in the field PxMC based on MC simulations
and the ladder strike price L set by the users themselves
Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015
(o = LadderMC())$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)))$PxMC # Price = ~10.25(o = LadderMC())$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)))$PxMC # Price = ~10.25
Calculates the price of a lookback option using a BSM-adjusted algorithm; Carries the assumption that the asset price is observed continuously.
LookbackBS(o = OptPx(Opt(Style = "Lookback")), Smax = 50, Smin = 50, Type = c("Floating", "Fixed"))LookbackBS(o = OptPx(Opt(Style = "Lookback")), Smax = 50, Smin = 50, Type = c("Floating", "Fixed"))
o |
An object of class |
Smax |
The maximum asset price observed to date. |
Smin |
The minimum asset price observed to date. |
Type |
Specifies the Lookback option as either Floating or Fixed- default argument is Floating. |
To price the lookback option, we require the Smax/Smin, S0, r, q, vol, and ttm arguments from the object classes defined in the package. An example of a complete OptLookback option object can be found in the examples.
An original OptPx object with PxBS field as the price of the option
and user-supplied Smin, Smax, and Type lookback parameters attached.
Richard Huang, Department of Statistics, Rice University, Spring 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall. ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
(o = LookbackBS())$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")(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")
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.
LookbackMC(o = OptPx(Opt(Style = "Lookback"), r = 0.05, q = 0, vol = 0.3), NPaths = 5, div = 1000, Type = c("Floating", "Fixed"))LookbackMC(o = OptPx(Opt(Style = "Lookback"), r = 0.05, q = 0, vol = 0.3), NPaths = 5, div = 1000, Type = c("Floating", "Fixed"))
o |
The |
NPaths |
How many time of the simulation are applied. Coustomer defined. |
div |
number to decide length of each interval |
Type |
Specifies the Lookback option as either Floating or Fixed- default argument is Floating. |
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.
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).
Tong Liu, Department of Statistics, Rice University, Spring 2015
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
(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)(o = LookbackMC())$PxMC #Use default arguments, Output: approximately 16.31. # Floating & Put o=OptPx(Opt(S0=50,K=50,ttm=0.25,Right='Put',Style="Lookback"),r=0.1,vol=.4) LookbackMC(o,NPaths=5,div=1000) #Output: 7.79 from Hull 9e Example 26.2 Pg 608. # Floating & Call o=OptPx(Opt(S0=50,K=50,ttm=0.25,Right='Call',Style="Lookback"),r=0.1,vol=.4) LookbackMC(o,NPaths=5,div=1000) #Output: 8.04 from Hull 9e Example 26.2 Pg 608 # Fixed & Put o=OptPx(Opt(S0=50,K=60,ttm=1,Right='Put',Style="Lookback"),r=0.05,q=0.02,vol=.25) LookbackMC(o,Type="Fixed",NPaths=5,div=1000) # Fixed & Call o=OptPx(Opt(S0=50,K=55,ttm=1,Right='Call',Style="Lookback"),r=0.1,vol=.25) LookbackMC(o,Type="Fixed",NPaths=5,div=1000)
Opt object constructorAn S3 object constructor for an option contract (financial derivative)
Opt(Style = c("European", "American", "Asian", "Binary", "AverageStrike", "Barrier", "Chooser", "Compound", "DeferredPayment", "ForeignEquity", "ForwardStart", "Gap", "HolderExtendible", "Ladder", "Lookback", "MOPM", "Perpetual", "Quotient", "Rainbow", "Shout", "SimpleChooser", "VarianceSwap"), Right = c("Call", "Put", "Other"), S0 = 50, ttm = 2, K = 52, Curr = "$", ContrSize = 100, SName = "A stock share", SSymbol = "")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 = "")
Style |
An option style: |
Right |
An option right: |
S0 |
A spot price of the underlying security (usually, today's stock price, |
ttm |
A time to maturity, in units of time matching r units; usually years |
K |
A strike price |
Curr |
An optional currency units for monetary values of the underlying security and an option |
ContrSize |
A contract size, i.e. number of option shares per contract |
SName |
A (optional) descriptful name of the underlying. Eg. Microsoft Corp |
SSymbol |
An (optional) official ticker of the underlying. Eg. MSFT |
A list of class Opt
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
Opt() #Creates an S3 object for an option contract Opt(Right='Put') #See J. C. Hull, OFOD'2014, 9-ed, Fig.13.10, p.289Opt() #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 constructorS3 object constructor for lattice-pricing specs of an option contract. Inherits Opt object.
OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)OptPos(o = Opt(), Pos = c("Long", "Short"), Prem = 0)
o |
An object of class |
Pos |
A position direction (to the holder) with values |
Prem |
A option premim (i.e. market cost or price), a non-negative amount to be paid for the option contract being modeled. |
A list of class OptPx
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
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.289OptPos() # 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 constructorAn S3 object constructor for lattice-pricing specifications for an option contract. Opt object is inhereted.
OptPx(o = Opt(), r = 0.05, q = 0, rf = 0, vol = 0.3, NSteps = 3)OptPx(o = Opt(), r = 0.05, q = 0, rf = 0, vol = 0.3, NSteps = 3)
o |
An object of class |
r |
A risk free rate (annualized) |
q |
A dividend yield (as annualized rate), Hull/p291 |
rf |
A foreign risk free rate (annualized), Hull/p.292 |
vol |
A volaility (as Sd.Dev, sigma) |
NSteps |
A number of time steps in BOPM calculation |
A list of class OptPx with parameters supplied to Opt and OptPx constructors
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
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)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)
Bivariate Standard Normal CDF Calculator For Given Values of x, y, and rho
pbnorm(x = 0, y = 0, rho = 0)pbnorm(x = 0, y = 0, rho = 0)
x |
The |
y |
The |
rho |
The correlation between variables |
This runs a bivariate standard normal pdf then calculates the cdf from that based on the input parameters
Density under the bivariate standard normal distribution
Robert Abramov, Department of Statistics, Rice University, 2015
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
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)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)
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.
PerpetualBS(o = OptPx(Opt(Style = "Perpetual"), q = 0.1))PerpetualBS(o = OptPx(Opt(Style = "Perpetual"), q = 0.1))
o |
AN object of class |
A list of class Perpetual.BS consisting of the input object OptPx
Kim Raath, Department of Statistics, Rice University, Spring 2015.
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
#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#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
Profit(o = OptPos(), S = o$S0)Profit(o = OptPos(), S = o$S0)
o |
An object of class |
S |
A (optional) vector or value of stock price(s) (double) at which to compute profits |
A numeric matrix of size [length(S), 2]. Columns: stock prices, corresponding option profits
Oleg Melnikov, Department of Statistics, Rice University, Spring 2015
Profit(o=Opt()) graphics::plot( print( Profit(OptPos(Prem=2.5), S=40:60)), type='l'); grid()Profit(o=Opt()) graphics::plot( print( Profit(OptPos(Prem=2.5), S=40:60)), type='l'); grid()
Quotient Option via Black-Scholes (BS) model
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)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)
o |
An object of class |
I1 |
A spot price of the underlying security 1 (usually I1) |
I2 |
A spot price of the underlying security 2 (usually I2) |
g1 |
Payout rate of the first stock |
g2 |
Payout rate of the 2nd stock |
sigma1 |
a vector of implied volatilities for the associated security 1 |
sigma2 |
a vector of implied volatilities for the associated security 2 |
rho |
is the correlation between asset 1 and asset 2 |
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.
Chengwei Ge, Department of Statistics, Rice University, Spring 2015
Zhang Peter G., Exotic Options, 2nd, 1998. http://amzn.com/9810235216.
(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)(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)
Calculates the price of a Quotient option using Monte-Carlo simulations.
QuotientMC(o = OptPx(Opt(Style = "Quotient")), S0_2 = 100, NPaths = 5)QuotientMC(o = OptPx(Opt(Style = "Quotient")), S0_2 = 100, NPaths = 5)
o |
The |
S0_2 |
The spot price of the second underlying asset. |
NPaths |
Number of monte-carlo simulations to run. Larger number of trials lower variability at the expense of computation time. |
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.
An original OptPx object with Px.MC field as the price of the option and user-supplied S0_2, NPaths parameters attached.
Richard Huang, Department of Statistics, Rice University, Spring 2015
http://www.investment-and-finance.net/derivatives/q/quotient-option.html
(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)(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 via Black-Scholes (BS) model
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"))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"))
o |
An object of class |
S1 |
A spot price of the underlying security 1 (usually S1) |
S2 |
A spot price of the underlying security 2 (usually S2) |
D1 |
A percent yield per annum from the underlying security 1 |
D2 |
A percent yield per annum from the underlying security 2 |
sigma1 |
a vector of implied volatilities for the associated security 1 |
sigma2 |
a vector of implied volatilities for the associated security 2 |
rho |
is the correlation between asset 1 and asset 2 |
Type |
Rainbow option type: 'Max' or 'Min'. |
Two types of Rainbow options are priced: 'Max' and 'Min'.
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.
Chengwei Ge,Department of Statistics, Rice University, Spring 2015
Zhang Peter G., Exotic Options, 2nd ed, 1998.
(o = RainbowBS())$PxBS o = OptPx(Opt(Style = 'Rainbow', Right = "Put"), r = 0.08) RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min') o = OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Put"), r = 0.08) RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min') o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Put"), r = 0.08) RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Max') o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Call"), r = 0.08) RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Min') o=OptPx(Opt(Style = 'Rainbow', K = 102, ttm = 1, Right = "Call"), r = 0.08) RainbowBS(o, S1=100, S2=95, D1=0,D2=0,sigma1=0.15,sigma2=0.2, rho=0.75,Type='Max')(o = RainbowBS())$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
ShoutFD(o = OptPx(Opt(Style = "Shout")), N = 100, M = 20, Smin = 0, Smax = 100)ShoutFD(o = OptPx(Opt(Style = "Shout")), N = 100, M = 20, Smin = 0, Smax = 100)
o |
An object of class |
N |
The number of equally spaced intervals. Default is 100. |
M |
The number of equally spaced stock price. Default is 20. |
Smin |
similar to Smax |
Smax |
A stock price sufficiently high that, when it is reached, the put option has virtually no value. The level of Smax should be chosen in such a way that one of these equally spaced stock prices is the current stock price. |
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 .
A list of class OptPx, including
option pricing parameters N, M, Smin, and Smax,
as well as the computed option price PxFD.
Xinnan Lu, Department of Statistics, Rice University, 2015
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.
(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))$PxFD(o = ShoutFD(OptPx(Opt(Right="C", Style="Shout"))))$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
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.
ShoutLT(o = OptPx(Opt(Style = "Shout")), IncBT = TRUE)ShoutLT(o = OptPx(Opt(Style = "Shout")), IncBT = TRUE)
o |
An object of class |
IncBT |
TRUE/FALSE indicating whether to include binomial tree (list object) with output |
A list of class ShoutLT consisting of the original OptPx object,
binomial tree stepBT and the computed price PxBS.
Le You, Department of Statistics, Rice University, Spring 2015
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
(o = ShoutLT( OptPx(Opt(Style='Shout'))))$PxLT o = Opt(Style='Shout', Right='Call', S0=60, ttm=.25, K=60) ShoutLT( OptPx(o,r=.1, q=.02, vol=.45, NSteps=10)) o = Opt(Style='Shout', Right='Call', S0=60, ttm=.25, K=60)(o = ShoutLT( OptPx(Opt(Style='Shout'))))$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)
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.
ShoutLTVectorized(o = OptPx(o = Opt(Style = "Shout")))ShoutLTVectorized(o = OptPx(o = Opt(Style = "Shout")))
o |
An object of class |
A list of class ShoutLT consisting of the original OptPx object,
binomial tree step BT and the computed price PxBS.
Le You, Department of Statistics, Rice University, Spring 2015
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
(o = ShoutLTVectorized( OptPx(Opt(Style='Shout'))))$PxLT o = Opt(Style='Shout') (o = ShoutLTVectorized( OptPx(o, r=.1, q=.02, vol=.45, NSteps=10)))$PxLT(o = ShoutLTVectorized( OptPx(Opt(Style='Shout'))))$PxLT o = Opt(Style='Shout') (o = ShoutLTVectorized( OptPx(o, r=.1, q=.02, vol=.45, NSteps=10)))$PxLT
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, where .
Note that the value of (the expected price increase) is assumped to be
o$r, the risk free rate of return.
ShoutMC(o = OptPx(o = Opt(Style = "Shout")), NPaths = 10)ShoutMC(o = OptPx(o = Opt(Style = "Shout")), NPaths = 10)
o |
The |
NPaths |
The number of simulation paths to use in calculating the price; must be >= 10 |
The option object o with the price in the field PxMC based on the MC simulations.
Jake Kornblau, Department of Statistics, Rice University, 2015
Hull, J.C., Options, Futures and Other Derivatives, 9ed, 2014. Prentice Hall.
ISBN 978-0-13-345631-8, http://www-2.rotman.utoronto.ca/~hull/ofod/index.html.
Also: http://www.math.umn.edu/~spirn/5076/Lecture16.pdf
(o = ShoutMC())$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(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
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)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)
o |
An object of class |
K |
A vector of non-negative strike prices |
Vol |
a vector of non-negative, less than zero implied volatilities for the associated strikes |
notional |
A numeric positive amount to be invested |
varrate |
A numeric positive varaince rate to be swapped |
An object of class OptPx with value included
Max Lee, Department of Statistics, Rice University, Spring 2015
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.
(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(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
Calculates the price of a VarianceSwap Option using 500 Monte Carlo simulations.
Important Assumptions:
The option o followes a General Brownian Motion
where .
The value of (the expected price increase) is assumed to be o$r-o$q.
VarianceSwapMC(o = OptPx(o = Opt(Style = "VarianceSwap")), var = 0.2, NPaths = 5)VarianceSwapMC(o = OptPx(o = Opt(Style = "VarianceSwap")), var = 0.2, NPaths = 5)
o |
The |
var |
The variance strike level |
NPaths |
The number of simulation paths to use in calculating the price, |
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
Huang Jiayao, Risk Management and Business Intelligence at Hong Kong University of Science and Technology, Exchange student at Rice University, Spring 2015
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
(o = VarianceSwapMC())$PxMC #Price = ~0.0245 (o = VarianceSwapMC(NPaths = 5))$PxMC # Price = ~0.0245 (o = VarianceSwapMC(var=0.4))$PxMC # Price = ~-0.1565(o = VarianceSwapMC())$PxMC #Price = ~0.0245 (o = VarianceSwapMC(NPaths = 5))$PxMC # Price = ~0.0245 (o = VarianceSwapMC(var=0.4))$PxMC # Price = ~-0.1565