ECMT3150: Assignment 2 (Semester 2, 2024) Lecturer: Simon Kwok Due: 5pm, 18 October 2024 (Friday) NOTE: Please do not write your …nal answers in your R-script. You should summarise the outputs (e.g., plots) and include your discussion and …nal answers in the written response …le. Both your written response …le and R-script (i.e., the .R source …le, not the screenshot) need to be submitted. [Total: 30 marks (+ bonus)] Carol is undergoing a series of training at the pricing team in Goldman Sachs. She is studying a simple …nancial market consisting of a risk-free money account and a stock called BOB. Here is the single-period model under the risk-neutral probability measure Q:

Time length of the period is .

In the risk-free market account, a dollar at time 0 will grow into a = er at time , where r is the continuously compounded risk-free interest rate.

At time 0, the share price is S0. At time , the share price rises to S = S0u with probability q, and drops to S = S0d with probability 1 q. 1. [2 marks] Write down the risk-neutral probability distribution of S, the share price at time . Express the probability mass function in terms of u; d and q. 2. [3 marks] Show that q = ad ud . [Hint: the discounted share price is a martingale under Q.] 3. [3 marks] Find V ar(S), the variance of the share price at time ? Express your answer in terms of a, u and d. 4. [3 marks] Let u = e p

and d = 1 u = e p . Show that V ar(S)

S202 for small . [Hint: ex

1 + x if x is close to zero. The …nal result is obtained by dropping terms involving higher power of ] Carol wants to construct a binomial tree model for the price of BOB traded in an n- period market, where n is a positive integer. Here is the binomial tree model under Q (for i = 1; : : : ; n):

Time length of a period is . 1

In the risk-free market account, a dollar at time (i 1) will grow into a = er at time i, where r is the continuously compounded risk-free interest rate, which remains constant over time.

At time (i1) , the share price starts at S(i1). At time i, the share price rises to S(i1)u with probability q, or drops to S(i1)d with probability 1q. The probability q is as given in question 2, and u and d are as given in question 4 (i.e. u = e p

and d = 1 u = e p ). Assume that the price changes are independent across all n periods. 5. [3 marks] Let j denote the number of times by which the share price goes up over n periods. What is the probability distribution of j? For a given j, show that the share price at the end of period n is given by Sn = S0u jdnj: 6. [2 marks] Consider a European call option written on a share of BOB at time 0 with strike price X and time-to-maturity

= n. Show that its price is given by Cbin0 = E Q[ernmax(Sn X; 0)]: (1) Suppose we are at time 0, and the current share price of BOB is S0 = 50. Suppose r = 0:02 and

= 0:3. Write an R code that simulates 5000 sample paths of share price using the above binomial tree model with the following speci…cations: n = 63,

= 1=252.1 While simulating the random numbers, set the random seed to be the last 5 digits of your SID.2 [Hint: you may use rbinom(5000,n,p)to generate 5000 random integers from a binomial distribution with parameters n and p.] 7. [3 marks] Using your code, compute the time-0 price of an at-the-money European call option written on a share of BOB at time 0 with strike price X = S0 = 50 and expiring in 63 days (i.e.,

= 63). Correct your answer to 3 decimal places. 8. [3 marks] Compute analytically the time-0 price of the same call option using the Black-Scholes formula instead. Correct your answer to 3 decimal places. Compare it with your answer in question 7. Carol has recently moved to the product design team. She is currently designing an exotic option written on a share of BOB at time 0. This option will give the following payo¤ as a function of the share price S at time

g(S ) = 8<: X1 S for S < X1; 0 for X1

S

X2; S X2 for S > X2; where X1 < X2. Carol named this exotic option as “‡y-with-BOB,”after noting that the graph of the payo¤ function looks like the wings of an aeroplane. 1The time step

= 1252 , measured in years, is equivalent in length to a day out of 252 trading days in a year. The total length of n time steps, n = 312 , thus amounts to three months measured in years. 2This is to ensure that your answer will be di¤erent from that of other students. 2 9. [3 marks] Using your code, compute the time-0 price of a ‡y-with-BOB option with strike prices X1 = 45, X2 = 55 and expiring in 63 days (i.e.,

= 63). Correct your answer to 3 decimal places. 10. [3 marks] Compute analytically the time-0 price of a ‡y-with-BOB option using the Black-Scholes formula instead. Correct your answer to 3 decimal places. Compare it with your answer in question 9. 11. [2 marks] What type of investors will be interested in ‡y-with-BOB? 12. [Optional question for those who are up to the challenge; bonus marks will be given for correct solutions] Prove mathematically that Cbin0 as de…ned in question 6 converges to the Black-Scholes call price as ! 0 and n!1 while

= n remaining constant. 3 51作业君版权所有