Exotic Option Pricing
Arvind Saini
March 30, 2016
We will stimulate the stock prices that follow geometric brownian motion for the future and then use them to calculate the premium prices of Exotic Options.
Exotic Options
Pricing of Exotic Option using numerical methods. Monte carlo stimulation method
Asian Options using arithmetic as well as geometric averages
cat("\014")
library(sde)## Warning: package 'sde' was built under R version 3.2.4## Loading required package: MASS
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## sde 2.0.14
## Companion package to the book
## 'Simulation and Inference for Stochastic Differential Equations With R Examples'
## Iacus, Springer NY, (2008)
## To check the errata corrige of the book, type vignette("sde.errata")library(quantmod)## Loading required package: xts
## Loading required package: TTR
## Version 0.4-0 included new data defaults. See ?getSymbols.today='2016-03-29'
r=0.03
getSymbols('SPY', src = 'yahoo', from = '2005-01-01',to=today)## As of 0.4-0, 'getSymbols' uses env=parent.frame() and
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## from.m, : downloaded length 208809 != reported length 200## [1] "SPY"sigma=sd(dailyReturn(SPY))
s0=coredata(Cl(SPY[today]))
s0=s0[1]
T = 3
strike=210
nt=100000
n=200
#############Generate nt trajectories
dt=T/n
t=seq(0,T,by=dt)
X=matrix(rep(0,length(t)*nt), nrow=nt)
Y=matrix(rep(0,length(t)*nt), nrow=nt)
for (i in 1:nt)
{
X[i,]= GBM(x=s0,r=r,sigma=sigma,T=T,N=n)
}
##Plot
#bounds for simulated prices
plot(t,X[1,],t='l',col=1,
ylab="Price P(t)",xlab="Time t", main="Simulated stochastic future paths")
for(i in 2:nt)
{
lines(t,X[i,],col=i)
}
Aavg <- array(0,dim=c(0,nt))
Aoption.call <- array(0,dim=c(0,nt))
Aoption.put <- array(0,dim=c(0,nt))
for (i in 1:nt)
{
Aavg[i]=mean(X[i,])
Aoption.call[i]=exp(-r*T)*(max((Aavg[i]-strike),0))
Aoption.put[i]=exp(-r*T)*(max((-(Aavg[i]-strike)),0))
}
(arithmetic.asian.call=mean(Aoption.call))## [1] 4.274536(arithmetic.asian.put=mean(Aoption.put))## [1] 0.0430393Gavg <- array(0,dim=c(0,nt))
Goption.call <- array(0,dim=c(0,nt))
Goption.put <- array(0,dim=c(0,nt))
for (i in 1:nt)
{
Gavg[i]=exp(mean(log(X[i,])))
Goption.call[i]=exp(-r*T)*(max((exp(Gavg[i])-strike),0))
Goption.put[i]=exp(-r*T)*(max((-(exp(Gavg[i])-strike)),0))
}
(geometric.asian.call=mean(Goption.call))## [1] 5.890475e+94(geometric.asian.put=mean(Goption.put))## [1] 0path <- array(0,dim=c(0,n))
for (i in 0:n+1)
{
path[i]=mean(X[,i])
}
plot(t,path,t='l',col=3,xlab="Time(t)",ylab="Price($)",main="Average path of stock")
Analytical pricing of Asian Options
library(RQuantLib)## Warning: package 'RQuantLib' was built under R version 3.2.4library(fExoticOptions)## Warning: package 'fExoticOptions' was built under R version 3.2.4## Loading required package: timeDate
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## Analysing Markets and calculating Basic Statistics
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
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## Loading required package: fOptions## Warning: package 'fOptions' was built under R version 3.2.4##
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## Pricing and Evaluating Basic Options
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.orgTurnbullWakemanAsianApproxOption(TypeFlag = "c", S = s0, SA = s0,
X = strike, Time = T, time = T, tau = 0 , r = r,b = r, sigma = sigma)##
## Title:
## Turnbull Wakeman Asian Approximated Option
##
## Call:
## TurnbullWakemanAsianApproxOption(TypeFlag = "c", S = s0, SA = s0,
## X = strike, Time = T, time = T, tau = 0, r = r, b = r, sigma = sigma)
##
## Parameters:
## Value:
## TypeFlag c
## S 205.119995
## SA 205.119995
## X 210
## Time 3
## time 3
## tau 0
## r 0.03
## b 0.03
## sigma 0.0125693071624312
##
## Option Price:
## 4.278104
##
## Description:
## Sun Apr 17 16:01:06 2016TurnbullWakemanAsianApproxOption(TypeFlag = "p", S = s0, SA = s0,
X = strike, Time = T, time = T, tau = 0 , r = r,b = r, sigma = sigma)##
## Title:
## Turnbull Wakeman Asian Approximated Option
##
## Call:
## TurnbullWakemanAsianApproxOption(TypeFlag = "p", S = s0, SA = s0,
## X = strike, Time = T, time = T, tau = 0, r = r, b = r, sigma = sigma)
##
## Parameters:
## Value:
## TypeFlag p
## S 205.119995
## SA 205.119995
## X 210
## Time 3
## time 3
## tau 0
## r 0.03
## b 0.03
## sigma 0.0125693071624312
##
## Option Price:
## 0.04326591
##
## Description:
## Sun Apr 17 16:01:06 2016AsianOption("geometric", "call", underlying=s0, strike=strike, div=0.0,
riskFree=r, maturity=T, vol=sigma)## Concise summary of valuation for AsianOption
## value delta gamma vega theta rho divRho
## 4.2048 0.9145 0.0341 16.8684 -2.7229 268.7540 -281.3685AsianOption("geometric", "put", underlying=s0, strike=strike, div=0.0,
riskFree=r, maturity=T, vol=sigma)## Concise summary of valuation for AsianOption
## value delta gamma vega theta rho divRho
## 0.0439 -0.0415 0.0341 18.1007 0.0910 -12.8929 12.7612Lookback Options using monte carlo
Lmin <- array(0,dim=c(0,nt))
Lmax <- array(0,dim=c(0,nt))
Loption.call <- array(0,dim=c(0,nt))
Loption.put <- array(0,dim=c(0,nt))
for (i in 1:nt)
{
Lmin[i]=min(X[i,])
Lmax[i]=max(X[i,])
Loption.call[i]=exp(-r*T)*(max((X[i,n+1]-Lmin[i]),0))
Loption.put[i]=exp(-r*T)*(max(Lmax[i]-X[i,n+1],0))
}
(lookback.float.call=mean(Loption.call))## [1] 17.99624(lookback.float.put=mean(Loption.put))## [1] 0.3751179Barrier Options using Monte carlo
B=220
payoff <- array(0,dim=c(0,nt))
payoff1 <- array(0,dim=c(0,nt))
for (i in 1:nt)
{
if(Lmax[i]>B)
{
payoff[i]=max(X[i,n+1]-strike,0)
payoff1[i]=0
}
else
{
payoff[i]=0
payoff1[i]=max(X[i,n+1]-strike,0)
}
}
mean.payoff=mean(payoff)
mean.payoff1=mean(payoff1)
(barrier.option.upin = exp(-r*T)*mean.payoff)## [1] 12.18314(barrier.option.downin = exp(-r*T)*mean.payoff1)## [1] 1.011311Barrier Options using analytical approach
BarrierOption(barrType="upin", type="call", underlying=s0,
strike=strike, dividendYield=0, riskFreeRate=r,
maturity=T, volatility=sigma, barrier=B)## Concise summary of valuation for BarrierOption
## value delta gamma vega theta rho divRho
## 12.261 NA NA NA NA NA NA