Question 1. We will derive a two-state put option value in this problem. Data: S0 = 100; X = 110; 1 + r =1.10. or rate=10% The two possibilities for ST are 130 and 80.

  1. Show that the range of S is 50, whereas that of P is 30 across the two states. What is the hedge ratio of the put?

Note: Value of put option when the stock goes up is: 0 Value of put option when the stock goes down is: X-uS0

Hedge Ratio

(0-30)/50
## [1] -0.6
  1. & (C) Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?

Based on the hedge ratio, there are three shares and five put options

Payoff <- matrix(c(240,390,150,0,390,390),ncol=2,byrow=3)
colnames(Payoff) <- c("Stock Price Down","Stock Price Up")
rownames(Payoff) <- c("3 Shares","5 Puts","Total Pay-off")
Payoff <- as.table(Payoff)
Payoff
##               Stock Price Down Stock Price Up
## 3 Shares                   240            390
## 5 Puts                     150              0
## Total Pay-off              390            390
#Nonrandom Payoff of Portfolio
390
## [1] 390
#Present Discounted Value of Portfolio
(390)/(1+0.10)
## [1] 354.5455
  1. Given that the stock currently is selling at 100, solve for the value of the put.
(354.5455-(100*3))/5
## [1] 10.9091

Question 2. Use the Black-Scholes formula to find the value of a call option on the following stock:

Time to expiration 6 months Standard deviation 50% per year Exercise price $50 Stock price $50 Annual interest rate 3% Dividend 0

#Call Option
d1 <- (log(50/50) + (0.03 + 0.5^2/2)*0.5) / (0.5*sqrt(0.5))
d1
## [1] 0.2192031
d2 <- d1 - 0.5*sqrt(0.5)
d2
## [1] -0.1343503
N_d1 <- pnorm(d1)
N_d1
## [1] 0.5867541
N_d2 <- pnorm(d2)
N_d2
## [1] 0.4465628
BlackScholesHedgeRatio <- 50*pnorm(d1) - 50*exp(-.03*0.5)*pnorm(d2)
BlackScholesHedgeRatio
## [1] 7.341987
#Put Option
d1 <- (log(50/50) + (0.03 + 0.5^2/2)*0.5) / (0.5*sqrt(0.5))
d2 <- d1 - 0.5*sqrt(0.5)
  
value <-(50*exp(-0.03*0.5)*pnorm(-d2) - 50*pnorm(-d1))
value
## [1] 6.597584

Question 3. Find the Black-Scholes value of a put option on the stock in Problem 11 with the same exercise price and expiration as the call option. Formula for Put Call Parity: C+Xexp(-rt)-S=P

#Using Put-Call Parity 
7.341987+50*exp(-.03*0.5)-50
## [1] 6.597584

Question 5. Recalculate the value of the call option in Problem 11, successively substituting one of the changes below while keeping the other parameters as in Problem 11:

  1. Time to expiration = 3 months.
  2. Standard deviation = 25% per year.
  3. Exercise price = $55.
  4. Stock price = $55.
  5. Interest rate = 5%.

Consider each scenario independently. Confirm that the option value changes in accordance with the prediction of Table 21.1.

#Call Option (Expiration sate changes to 0.25 or 3 months)
d1 <- (log(50/50) + (0.03 + 0.5^2/2)*0.25) / (0.5*sqrt(0.25))
d1
## [1] 0.155
d2 <- d1 - 0.5*sqrt(0.25)
d2
## [1] -0.095
N_d1 <- pnorm(d1)
N_d1
## [1] 0.5615893
N_d2 <- pnorm(d2)
N_d2
## [1] 0.4621574
BlackScholesHedgeRatio1 <- 50*pnorm(d1) - 50*exp(-.03*0.25)*pnorm(d2)
BlackScholesHedgeRatio1
## [1] 5.144257
#Call Option (Standard Deviation reduced to 25%)
d1 <- (log(50/50) + (0.03 + 0.25^2/2)*0.5) / (0.25*sqrt(0.5))
d1
## [1] 0.1732412
d2 <- d1 - 0.25*sqrt(0.5)
d2
## [1] -0.003535534
N_d1 <- pnorm(d1)
N_d1
## [1] 0.5687691
N_d2 <- pnorm(d2)
N_d2
## [1] 0.4985895
BlackScholesHedgeRatio2 <- 50*pnorm(d1) - 50*exp(-.03*0.5)*pnorm(d2)
BlackScholesHedgeRatio2
## [1] 3.880128
#Call Option (Increase execize price to $55)
d1 <- (log(50/55) + (0.03 + 0.25^2/2)*0.5) / (0.25*sqrt(0.5))
d1
## [1] -0.3659146
d2 <- d1 - 0.25*sqrt(0.5)
d2
## [1] -0.5426913
N_d1 <- pnorm(d1)
N_d1
## [1] 0.3572144
N_d2 <- pnorm(d2)
N_d2
## [1] 0.2936712
BlackScholesHedgeRatio3 <- 50*pnorm(d1) - 55*exp(-.03*0.5)*pnorm(d2)
BlackScholesHedgeRatio3
## [1] 1.949276
#Call Option (Increase Stock Price to 55)
d1 <- (log(55/50) + (0.03 + 0.25^2/2)*0.5) / (0.25*sqrt(0.5))
d1
## [1] 0.712397
d2 <- d1 - 0.25*sqrt(0.5)
d2
## [1] 0.5356203
N_d1 <- pnorm(d1)
N_d1
## [1] 0.7618905
N_d2 <- pnorm(d2)
N_d2
## [1] 0.7038895
BlackScholesHedgeRatio4 <- 55*pnorm(d1) - 50*exp(-.03*0.5)*pnorm(d2)
BlackScholesHedgeRatio4
## [1] 7.233481
#Call Option (Increase Interest Rate to 5%)
d1 <- (log(50/50) + (0.05 + 0.5^2/2)*0.5) / (0.5*sqrt(0.5))
d1
## [1] 0.2474874
d2 <- d1 - 0.5*sqrt(0.5)
d2
## [1] -0.106066
N_d1 <- pnorm(d1)
N_d1
## [1] 0.5977345
N_d2 <- pnorm(d2)
N_d2
## [1] 0.457765
BlackScholesHedgeRatio5 <- 50*pnorm(d1) - 50*exp(-0.05*0.5)*pnorm(d2)
BlackScholesHedgeRatio5
## [1] 7.563587