RI revision programme stats (hard)

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 3 Revision (Zeta) Topic(s): (C) Statistics (Solutions) ___________________________________________ Y6 H2 Math Term 3 Revision Lecture: (C) Statistics Page 1 of 14 Source of Question: NJC Prelim 9758/2023/02/Q11 1 (a) In a one-period binomial model, the stock price initially at 0S , either rises to 0uS with a probability p or drops to 0 1 Su with a probability ( )1 p− at the end of one period, where 1u> . Suppose that 0 10S = , 1.2u= and 0.55p= . Show that the expected value of the stock price S at the end of the period is 10.35 and find the variance of the stock price. [3] (b) The above model can be extended to a n-period binomial model. For each period, the stock price S either rises to uS with a probability p or goes drops to 1 Su with a probability 1 p− at the end of this period. (i) In the first quarter, the initial price of the stock at the start of the first period is 0S . Suppose the stock price rises 4 times and drops twice over 6 periods. Find the stock price in terms of 0S and u at the end of the 6 periods. [1] (ii) Show that the probability for part (b)(i) to happen is ( ) 2415 1pp − . [1] Initial price Price at the end of one period … price at the end of 2nd period initial price price at the end of 1st period … … 1st period 2nd period …

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________________ ___________________________________________ Y6 H2 Math Term 3 Revision Lecture: (C) Statistics Page 2 of 14 (iii) In the second quarter, the initial price of the stock at the start of the first period is 1S . Find the probability that the stock price is higher than 1S at the end of 6 periods in terms of p. [2] (c) A contract that gives investors the right to buy a stock at a specified price at the end of a specified period is termed “call option”. The specified price is called “strike price” and the amount needed to buy this contract is called “premium”. Each unit of a stock’s call option has a return value V at the end of the period. Its value is given by strike strike strike if , 0 if , SS S SV SS −>= ≤ where strikeS is the strike price and S is the stock price at the end of the period. Stock K, currently priced at $10, is modelled by the n- period binomial model in part (b) with 6n= , 1.2u= and 0.55p= . A call option of stock K has a strike price of $10. (i) Fill in the missing numbers in the following probability distribution table of V for this call option, giving the values to a suitable degree of accuracy. [2] The call option currently has a premium of $2.50 per unit. Peter plans to invest $10 000 on purchasing 4000 units of the call option and sell them at the end