ST330 Stochastic and Actuarial Methods in Finance
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Summer 2022 Timed R Coursework
ST330
Stochastic and Actuarial Methods in Finance
2021/22 syllabus only
Instructions to candidates
This paper contains two questions. Answer ALL TWO.
Question 1: 40 marks
Question 2: 60 marks
The marks in brackets relect marks for each part of a question.
Your work should show all steps and all results. R code should be included either as part of the .pdf ile you submit or as a separate .R or .Rmd ile. Any plots should be included in the .pdf ile. No iles other than .pdf, .R and .Rmd are acceptable.
Make sure your five-digit candidate number and course code are written or included at the top of every page of your answers.
Time allowed Reading Time: None
Writing Time: 2 hours
Additional time for assessment upload: one hour
1. Consider the CRR model with stock price S with value at time 0 given by S0 = 30. Over a single period, S can either move up to 40 or down to 20, with equal probability. The (continuously compounded) interest rate is 10% in each period.
(a) Use R to compute the risk neutral probability of an upward move. [5 marks]
(b) In the single period case, compute the value at time zero of a European put option with maturity T = 1 and strike price equal to the sum of 30 and the last digit of your candidate number. [10 marks]
(c) Consider now the six period case, where in each step the upward factor is 4/3 and downward factor is 2/3. You may use here the following function which generates the values of the stock at any time between t = 0 and t = 6.
Stocktree = matrix(0, nrow=7, ncol=7)
u = 4/3
d = 2/3
for (i in 1:7) {
for (j in 1:i) {
Stocktree[i,j] = 30 * u^(j-1) * d^((i-1)-(j-1))
}
}
Stocktree
i. Use the Stocktree command to ind the mean of S6 . [5 marks]
ii. Compute the value at any node of an American put option on S in this six-period model, with maturity T = 6 and strike price equal to the sum of 20 and the last digit of your candidate number. [20 marks]
2. This question concerns Brownian motion and the Black Scholes model. Let S0 = 10,
T = 2, µ = 2 and σ 2 = 4 i.e. one risky asset which has value at time T given by ST = S0e(µ−σ2 /2)T+σBT
where B denotes standard Brownian motion. Recall that BT follows a normal distribution with mean 0 and variance T.
(a) With the values µ,σ,T,S0 as deined use R to compute the mean m of X := log(ST) given by
m <- log(S_0)+(mu-sigma^2/2)T. [5 marks]
(b) Use the dnorm command to plot the density dens of log ST on the domain deined by
x <- seq(-5,9,by=0.01). [10 marks]
(c) Find the 5% quantile of log(ST). You may use the qnorm command for this. [5 marks]
(d) Highlight in red on the plot the area under the density between the 10% and 80% quantiles. This can be done by following the plot command by
i <- x >=A & x <= B
polygon(c( A,x[i],B), c(0,dens[i],0), col="red") where A and B need to be speciied [10 marks]
(e) Assume that for the risk free rate r we have r = 1 and consider a European call option on S with maturity T = 2 and strike price K equal to the sum of 10 and the last digit of your candidate number. You may use without proof any result derived in the lectures.
i. Use R to ind the value at time 0 of this European call option. [10 marks]
ii. Now let S0 be general and plot the value at time 0 of this European call option as a function of S0 . [10 marks]
iii. Plot a graph of the gamma of this European call option. [10 marks]
2023-05-12