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ECMT3150: Assignment 2 (Semester 2, 2024)
Due: 5pm, 18 October 2024 (Friday)
NOTE: Please do not write your final answers in your R-script. You should summarise the outputs (e.g., plots) and include your discussion and final answers in the written response file. Both your written response file and R-script (i.e., the .R source Öle, 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 financial 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△ = S0 dwith 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 = u-d/a-d . [Hint: the discounted share price is a martingale under
Q.]
3. [3 marks] Find Var(S△ ), the variance of the share price at time △? Express your answer in terms of a, u and d.
4. [3 marks] Let u = eσ √△ and d = 
= e-σ√△ . Show that Var(S△ ) ≈ S0(2)σ 2 △ for small
△ . [Hint: ex ≈ 1 + x if x is close to zero. The final 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 △ .
● 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 overtime.
● At time (i — 1)△ , the share price starts at S(i-1)△ . At time i△, the share price rises to S(i-1)△u with probability q, or drops to S(i-1)△dwith 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 = eσ √△ and d = u/1 = e-σ√△ ). 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△ = S0ujdn-j:
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 T = n△ . Show that its price is given by
C0(bin) = EQ [e-rn△ max(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 σ = 0:3. Write an R code that simulates 5000 sample paths of share price using the above binomial tree model with the following specifications: n = 63, △ = 1=252. While simulating the random numbers, set the random seed to be the last 5 digits of your SID. [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., T = 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 payoff as a function of the share price ST at time T
where X1 < X2 . Carol named this exotic option as ''fly-with-BOB,'' after noting that the graph of the payoff function looks like the wings of an aeroplane.
9. [3 marks] Using your code, compute the time-0 price of a fly-with-BOB option with strike prices X1 = 45, X2 = 55 and expiring in 63 days (i.e., τ = 63△). Correct your answer to 3 decimal places.
10. [3 marks] Compute analytically the time-0 price of a fly-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 fly-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 defined in question 6 converges to
the Black-Scholes call price as △ → 0 and n → ∞ while τ = n△ remaining constant.