NOTE: Please do not write your Önal answers in your R-script. You should summarise theoutputs (e.g., plots) and include your discussion and Önal answers in the written responseÖle. Both your written response Öle and R-script (i.e., the .R source Öle, not the screenshot)need to be submitted.
1. Consider a single-period market consisting of a risk-free money account and a riskystock ANZ. The time length of the period is . Let S0 denote the price of a share ofANZ at time 0. At the end of the period (time ), its price either goes up to S = S0uor down to S = S0d. Let p be the probability that the share price goes up under therisk-neutral probability measure Q. The risk-free interest rate is r. Let a = er.(a) Show that p = a d u d . [Hint: the discounted share price is a martingale under Q.](b) Find V ar(S), the variance of the share price at time ?
Express your answerin terms of a, u and d.(c) Let u = ep and d = 1u = e p. Show that V ar(S) S202 for small .[Hint: ex 1 + x if x is close to zero. The Önal result is obtained by droppingterms involving higher power of ]Now consider an n-period market, where n is a positive integer. In period i (i = 1; : : : ; n), the ANZ share price starts at S(i 1), and it either goes up to S(i 1)u withQ-probability p, or goes down to S(i 1)d with Q-probability 1 p. The probability pis given in part (a), u = ep, and d = 1u = e p.
Assume that the price changes areindependent over all n periods. This is the binomial tree model for stock price.(d) Let j denote the number of times by which ANZ goes up over n periods. Whatis the probability distribution of j? For a given j, show that the ANZ share priceat the end of period n is given bySn = S0ujdn j :(e) Consider a European call option written on a share of ANZ at time 0 with strikeprice X and time-to-maturity = n. Show that its price is given byCbin0 = EQ[ern max(Sn X; 0)]: (1)1Suppose r = 0:01 and = 0:4.
Write an R code that simulates 1000 sample paths ofANZ share price using the above binomial tree model with the following speciÖcations:n = 21, = 1=252.1 While simulating the random numbers, set the random seed tobe the last 4 digits of your SID.2[Hint: you may use rbinom(1000,n,p)to generate1000 random integers from a binomial distribution with parameters n and p.](f) Using your code, compute the time-0 price of an at-the-money European calloption with S0 = X = 25 and expiring in 21 days.(g) Compute the time-0 price of the same call option using the Black-Scholes formulainstead.
Compare it with your answer in part (f).(h) [Optional question for those who are up to the challenge; bonus marks will begiven for correct solutions] Prove mathematically that Cbin0 converges to the BlackScholes call price as ! 0 and n ! 1 while = n remaining constant.2. Suppose St is a geometric Brownian motion with instantaneous drift and volatility. The law of motion is given bydSt = Stdt + Stdwt; (2)where wt is the standard Brownian motion.
(a) Derive the law of motion for Ut = Stet. Identify this stochastic process.
(b) Write an R code that simulates 1000 sample paths of Ut by discretising the law ofmotion in part (a) with discretisation step = 1252 and number of steps n = 252.Set U0 = 1.3 Assume = 0:4.
While simulating the random numbers, set therandom seed to be the last 4 digits of your SID. Produce a plot showing the Örst30 sample paths.
(c) Suppose Ut represents the time-t value of a portfolio in which you have an initialinvestment of U0 = 1 at time 1. Assume = 0:4. Compute the 5% Valueat-Risk of your portfolio over a 1-year horizon. [Hint: use the R commandquantile(uloss,0.95), where uloss is the vector of 1000 realised values of theloss U[1] U[253] obtained from the 1000 sample paths.]
(d) Derive the law of motion for Vt = ln Ut . Identify this stochastic process.(e) Based on the result in part (d), derive the distribution of the log-return ln( Ut+ Ut )over the period [t; t + ].(f) With the help of your result in part (e), compute manually the 5% Value-at-Riskof your portfolio over a 1-year horizon. Why is your answer di§erent from that inpart (c)?1The time step = 1252 , measured in years, is equivalent in length to a day out of 252 trading days in ayear. The total length of n time steps, n = 112 , therefore amounts to a month measured in years.2This is to ensure that your answer will be di§erent from that of other students.3Store fUjg252j=0 as an array in R: U[j + 1] for j = 0; 1; : : : ; 252, where U[1] = 1 is the starting point.
Requirements: 2questions | .doc file
