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HW 4

The document contains 9 exercises involving the valuation and payoff profiles of various options strategies including: 1) Plotting the profit profiles of selling a call and buying a put. 2) Combining options in a bull spread, bear spread, and box spread. 3) Using a binomial tree to value an option and calculate the probability of being in-the-money. 4) Valuing options under the Black-Scholes model and examining the effect of changing parameters. 5) Considering the early exercise feature of American put options.

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140 views12 pages

HW 4

The document contains 9 exercises involving the valuation and payoff profiles of various options strategies including: 1) Plotting the profit profiles of selling a call and buying a put. 2) Combining options in a bull spread, bear spread, and box spread. 3) Using a binomial tree to value an option and calculate the probability of being in-the-money. 4) Valuing options under the Black-Scholes model and examining the effect of changing parameters. 5) Considering the early exercise feature of American put options.

Uploaded by

api-285777244
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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C183 HW4

YIK LUN, KEI


704-115-065
2017/5/30

Exercise 1

S1 <- seq(0,80,by=1)
C <- 4
E <- 50
x2 <- ifelse(S1 > E, E-S1+C, C)
plot(S1,x2,ylim=c(-30,30),type="l",
main="Profit from selling a call: C=4,E=50",ylab="Profit",xlab="S1")

Profit from selling a call: C=4,E=50


30
20
10
Profit

0
10
30

0 20 40 60 80

S1

Exercise 2

S1 <- seq(0,80,by=1)
P <- 3
E <- 40
x2 <- ifelse(S1 < E, E-S1-P,-P)
plot(S1,x2,ylim=c(-20,50),type="l",
main="Profit from buying a put: P=4,E=40",ylab="Profit",xlab="S1")

1
Profit from buying a put: P=4,E=40
10 20 30 40 50
Profit

0
20

0 20 40 60 80

S1

Exercise 3

S1 <- seq(0,80,by=1)
P <- 6
E <- 50
x2 <- ifelse(S1 < E, (E-S1-P)*2,-P*2)
plot(S1,x2,ylim=c(-30,90),type="l",
main="Profit from buying two puts: P=6,E=50",ylab="Profit",xlab="S1")

2
Profit from buying two puts: P=6,E=50
80
60
40
Profit

20
0
20

0 20 40 60 80

S1

S1 <- seq(0,80,by=1)
C <- 5
E <- 50
x2 <- ifelse(S1 > E, S1-E-C,-C)
plot(S1,x2,ylim=c(-30,40),type="l",
main="Profit from buying a call: C=5,E=50",ylab="Profit",xlab="S1")

3
Profit from buying a call: C=5,E=50
10 20 30 40
Profit

10 0
30

0 20 40 60 80

S1

S1 <- seq(0,80,by=1)
P <- 6
E <- 50
C <- 5
x1 <- ifelse(S1 > E, S1-E-C,-C)
x2 <- ifelse(S1 < E, (E-S1-P)*2,-P*2)
x3 <- x1+x2
plot(S1,x3,ylim=c(-20,85),type="l",
main="Profit from buying two puts and one call: C=5, P=6, E=50",ylab="Profit",xlab="S1")

4
Profit from buying two puts and one call: C=5, P=6, E=50
80
60
40
Profit

20
0
20

0 20 40 60 80

S1

Exercise 4

S1 <- seq(0,70,by=1)
sC <- 5
sE <- 45
bC <- 8
bE <- 40
x1 <- ifelse(S1 > bE, S1-bE-bC,-bC)
x2 <- ifelse(S1 > sE, -(S1-sE-sC)*2,sC*2)
x3 <- x1+x2
plot(S1,x3,ylim=c(-20,10),type="l",
main="Profit from buying one call and selling two calls",ylab="Profit",xlab="S1")

5
Profit from buying one call and selling two calls
10
5
0
Profit

5
10
20

0 10 20 30 40 50 60 70

S1

Exercise 5

ST buy call sell call buy put sell put Total


ST > E2 ST E1 E2 ST 0 0 E2 E1
E1 < ST < E2 ST E1 0 E2 ST 0 E2 E1
ST < E1 0 0 E2 ST ST E1 E2 E1

S1 <- seq(0,90,by=1)
sCE <- 60
bCE <- 50
sPE <- 50
bPE <- 60
x1 <- ifelse(S1 > bCE, S1-bCE,0)
x2 <- ifelse(S1 > sCE, -(S1-sCE),0)
x3 <- ifelse(S1 < sPE, -(sPE-S1),0)
x4 <- ifelse(S1 < bPE, bPE-S1,0)
x5 <- x1+x2+x3+x4
bull <- x1+x2
bear <- x3+x4

6
plot(S1,bull,ylim=c(-1,11),type="l",
main="bull spread",ylab="Payoff",xlab="S1")

bull spread
10
8
6
Payoff

4
2
0

0 20 40 60 80

S1
plot(S1,bear,ylim=c(-1,11),type="l",
main="bear spread",ylab="Payoff",xlab="S1")

bear spread
10
8
6
Payoff

4
2
0

0 20 40 60 80

S1

7
plot(S1,x5,ylim=c(5,15),type="l",
main="box spread",ylab="Payoff",xlab="S1")

box spread
14
12
Payoff

10
8
6

0 20 40 60 80

S1

Exercise 6

k should be round up to 6
k <- log(60/((1/1.2)^10 * 50 ))/log(1.2/(1/1.2));k

## [1] 5.5

u <- 1.2
d <- 1/1.2
S <- 50
E <- 60
Su10 <- 50 * u^10
Su9d1 <- 50 * u^9* d
Su8d2 <- 50 * u^8* d^2
Su7d3 <- 50 * u^7* d^3
Su6d4 <- 50 * u^6* d^4
Su5d5 <- 50 * u^5* d^5
Su4d6 <- 50 * u^4* d^6
Su3d7 <- 50 * u^3* d^7

8
Su2d8 <- 50 * u^2* d^8
Su1d9 <- 50 * u^1* d^9
Sd10 <- 50 * d^10
period10 <- c(Su10,Su9d1,Su8d2,Su7d3,Su6d4,Su5d5,Su4d6,Su3d7,Su2d8,Su1d9,Sd10)
period10

## [1] 309.586821 214.990848 149.299200 103.680000 72.000000 50.000000


## [7] 34.722222 24.112654 16.744899 11.628402 8.075279

c10 <- ifelse(period10 > E, period10-E , 0)


c10

## [1] 249.5868 154.9908 89.2992 43.6800 12.0000 0.0000 0.0000


## [8] 0.0000 0.0000 0.0000 0.0000

p <- (1+0.1-d)/(u-d)
pprime <- (p*u)/(1+0.1)
50*pbinom(5,10,pprime, lower.tail=FALSE) -
(60/(1+0.1)^10)*pbinom(5,10, p, lower.tail=FALSE)

## [1] 27.48628

p <- (1 + 0.1 - d) / (u-d)

pro <- c(p^10 * choose(10,0), p^9 * (1-p) * choose(10,1), p^8 * (1-p)^2 * choose(10,2),
p^7 * (1-p)^3 * choose(10,3), p^6 * (1-p)^4 * choose(10,4), p^5 * (1-p)^5* choose(10,5),
p^4 * (1-p)^6* choose(10,6),p^3 * (1-p)^7* choose(10,7),p^2 * (1-p)^8* choose(10,8),
p^1 * (1-p)^9* choose(10,9), (1-p)^10* choose(10,10))

sum(pro * c10) / (1+0.1)^10

## [1] 27.48628

Exercise 7

u <- 1+0.06
d <- 1-0.05
S <- 50
E <- 51
t<-1/4
rp <- 0.05

9
p <- (exp(rp*t)-d)/(u-d)

Su <- S * u
Sd <- S * d
Sud <- S * u * d
Suu <- S * u * u
Sdd <- S * d * d

bioptions <- matrix(rep("",15),nrow=5,ncol=3)

Cuu <- max(Suu-E,0)


Cud <- max(Sud-E,0)
Cdd <- max(Sdd-E,0)
Cu <- (Cuu*p + Cud*(1-p))/(exp(rp*t))
Cd <- (Cud*p + Cdd*(1-p))/(exp(rp*t))
C <- (Cu*p + Cd*(1-p))/(exp(rp*t))

bioptions[3,1] <- paste("S0=",S,", C=",round(C,4))


bioptions[2,2] <- paste("Su=",Su,", Cu=",round(Cu,3));
bioptions[4,2] <- paste("Sd=",Sd,", Cd=",round(Cd,3))
bioptions[1,3] <- paste("Suu=",Suu,", Cuu=",Cuu)
bioptions[3,3] <- paste("Sdu=",Sud,", Cud=",Cud)
bioptions[5,3] <- paste("Sdd=",Sdd,", Cdd=",Cdd)
bioptions

## [,1] [,2] [,3]


## [1,] "" "" "Suu= 56.18 , Cuu= 5.18"
## [2,] "" "Su= 53 , Cu= 2.91" ""
## [3,] "S0= 50 , C= 1.6351" "" "Sdu= 50.35 , Cud= 0"
## [4,] "" "Sd= 47.5 , Cd= 0" ""
## [5,] "" "" "Sdd= 45.125 , Cdd= 0"

Exercise 8

Puu <- max(E-Suu,0)


Pud <- max(E-Sud,0)
Pdd <- max(E-Sdd,0)
Pu <- (Puu*p + Pud*(1-p))/(exp(rp*t))
Pd <- (Pud*p + Pdd*(1-p))/(exp(rp*t))
P <- (Pu*p + Pd*(1-p))/(exp(rp*t))

bioptions[3,1] <- paste("S0=",S,", P=",round(P,3))


bioptions[2,2] <- paste("Su=",Su,", Pu=",round(Pu,3));
bioptions[4,2] <- paste("Sd=",Sd,", Pd=",round(Pd,3))
bioptions[1,3] <- paste("Suu=",Suu,", Puu=",Puu)
bioptions[3,3] <- paste("Sdu=",Sud,", Pud=",round(Pud,3))
bioptions[5,3] <- paste("Sdd=",Sdd,", Pdd=",Pdd)
bioptions

## [,1] [,2] [,3]


## [1,] "" "" "Suu= 56.18 , Puu= 0"
## [2,] "" "Su= 53 , Pu= 0.277" ""

10
## [3,] "S0= 50 , P= 1.376" "" "Sdu= 50.35 , Pud= 0.65"
## [4,] "" "Sd= 47.5 , Pd= 2.866" ""
## [5,] "" "" "Sdd= 45.125 , Pdd= 5.875"
C + E/(exp(rp*(1/2)));P+S # t =1/2

## [1] 51.37588
## [1] 51.37588
Cu + E/(exp(rp*(1/4)));Pu+Su # t =1/4

## [1] 53.27674
## [1] 53.27674
Cd + E/(exp(rp*(1/4)));Pd+Sd

## [1] 50.36647
## [1] 50.36647
Cuu + E/(exp(rp*0));Puu+Suu # t = 0

## [1] 56.18
## [1] 56.18
Cud + E/(exp(rp*0));Pud+Sud

## [1] 51
## [1] 51
Cdd + E/(exp(rp*0));Pdd+Sdd

## [1] 51
## [1] 51

Exercise 9
If the put option were American, we need to calculate the value of the put at each node and compare it to
the payoff from early exercise. The put value at any node will be the greater beteen these two values.
early0 <- max(E-S,0)
earlyu <- max(E-Su,0)
earlyd <- max(E-Sd,0)
earlyuu <- max(E-Suu,0)
earlyud <- max(E-Sud,0)
earlydd <- max(E-Sdd,0)
bioptions[3,1] <- paste("P=",round(P,3),", EE=",early0)
bioptions[2,2] <- paste("Pu=",round(Pu,3),", EE=",earlyu);
bioptions[4,2] <- paste("Pd=",round(Pd,3),", EE=",earlyd)
bioptions[1,3] <- paste("Puu=",Puu,", EE=",earlyuu)
bioptions[3,3] <- paste("Pud=",round(Pud,3),", EE=",round(earlyud,3))
bioptions[5,3] <- paste("Pdd=",Pdd,", EE=",earlydd)
bioptions

## [,1] [,2] [,3]

11
## [1,] "" "" "Puu= 0 , EE= 0"
## [2,] "" "Pu= 0.277 , EE= 0" ""
## [3,] "P= 1.376 , EE= 1" "" "Pud= 0.65 , EE= 0.65"
## [4,] "" "Pd= 2.866 , EE= 3.5" ""
## [5,] "" "" "Pdd= 5.875 , EE= 5.875"
P <- (Pu*p + earlyd*(1-p))/(exp(rp*t));P

## [1] 1.645603
We can see that when t=1 and stock price goes down, the price for put option is 2.866. However, if we can
early exercise the put option, the payoff is 3.5, which is greater than 2.866. Hence it is optimal to early
exercise the put option at that node. In addition, if we had early exercised the put option at t=1, then the
put price at t=0 should also be changed based on the put price at t=1. Eventually, the put price at t=0
should be $1.645603

12

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