ECMT3150: Assignment 2 (Semester 1, 2022)
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ECMT3150: Assignment 2 (Semester 1, 2022)
[Total: 30 marks (+ bonus)] Bob is a budding investment banker in the pricing team. He proposes the following toy model for a single-period market that consists of a risk-free money account and the stock CBA. The time length of the period is A. Let S0 denote the price of a share of CBA at time 0. At the end of the period (time A), its price either goes up to SA = S0u or down to SA = S0 d. Let q denote the probability that the share price goes up under the risk-neutral probability measure Q. The risk-free interest rate is r. Let a = erA .
1. [2 marks] Write down the risk-neutral probability distribution of SA , the share price at time A. Express the probability mass function in terms of u;d and q .
2. [3 marks] Show that q = u(a)一(一)d(d). [Hint: the discounted share price is a martingale under Q.]
3. [3 marks] Find Var(SA ), the variance of the share price at time A? Express your answer in terms of a, u and d.
4. [3 marks] Let u = ea ′A and d = = e一a ′A. Show that Var(SA ) ≈ S0(2)a A2 for small A. [Hint: ex ≈ 1 + x if x is close to zero. The Önal result is obtained by dropping terms involving higher power of A]
Now Bob wants to build a binomial tree model for the share price of CBA stock traded in an n-period market, where n is a positive integer. The binomial tree model is given as follows. In period i (i = 1;:::;n), the CBA share price starts at S(i一1)A , and it either goes up to S(i一1)Au with Q-probability q, or goes down to S(i一1)A d with Q-probability 1 ━ q. The
probability q is as given in question 2, and u and d are as given in question 4 (i.e. u = ea ′A and d = = e一a ′A). Assume that the price changes are independent across all n periods.
5. [3 marks] Let j denote the number of times by which CBA goes up over n periods. What is the probability distribution of j? For a given j, show that the CBA share price at the end of period n is given by
SnA = S0uj dn一j :
6. [2 marks] Consider a European call option written on a share of CBA stock at time 0 with strike price X and time-to-maturity r = nA. Show that its price is given by
C0(bin) = EQ [e一rnA max(SnA ━ X;0)]: (1)
Suppose we are at time 0, and the current CBA share price is S0 = 100. Suppose r = 0:01 and a = 0:4. Write an R code that simulates 1000 sample paths of CBA share price using the above binomial tree model with the following speciÖcations: n = 21, A = 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(1000,n,p)to generate 1000 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 CBA stock at time 0 with strike price X = S0 = 100 and expiring in 21 days (i.e., r = 21A).
8. [3 marks] Compute analytically the time-0 price of the same call option using the Black-Scholes formula instead. Compare it with your answer in question 7.
Bob has recently moved to the product design team. He is currently designing an exotic option written on a share of CBA stock at time 0. This option will give the following payo§ as a function of the share price Sr at time r
( X1 ━ Sr for Sr < X1 ;
g(Sr ) = 1 0 for X1 ≤ Sr ≤ X2 ;
1 Sr ━ X2 for Sr > X2 ;
where X1 < X2 . Bob named this exotic option as ìáy-with-Bob,îafter noting that the graph of the payo§ function looks like the wings of an aeroplane.
9. [3 marks] Using your code, compute the time-0 price of a áy-with-Bob option with strike prices X1 = 90, X2 = 110 and expiring in 21 days (i.e., r = 21A).
10. [3 marks] Compute analytically the time-0 price of a áy-with-Bob option using the Black-Scholes formula instead. Compare it with your answer in question 9.
11. [2 marks] What type of investors will be interested in áy-with-Bob?
12. [Optional question for those who are up to the challenge; bonus marks will be given for correct solutions] Prove mathematically that C0(bin) as deÖned in question 6 converges to the Black-Scholes call price as A → 0 and n → ~ while r = nA remaining constant.
2022-05-03