MA4269 Assignment 1 (4 Questions) Important:due on Wednesday 20 September 2017. Submit to the boxes in the front ofLT33 before the beginning of the lecture,according to your tutorial group number. Please remember to write your name & matric number & tutorial group number. Question 1. Suppose the annual risk-free interest rate is 7% with continuous compounding.The 5-month forward price for the XYZ stock is $120. Consider the following premiums for the XYZ options with 5 months to expiration: Strike Call Put $110 $20 $12 Assume that the stock pays no dividends and it costs nothing to enter into the forward contract.Is there an arbitrage opportunity? If so, construct a portfolio that generates an arbitrage. Question 2.Suppose K 1 < K 2 < K 3. Suppose all options are European and have the same maturity and underlying asset, and K3 K 2 = K 2 K 1. Consider the following portfolios Portfolio A: short 2 calls with strike price K one call with strike price K 2, buy 1, buy one call with strike price 3. K Portfolio B: short 2 puts with strike price K one put with strike price K 2, buy 1, buy one put with strike price3. K Use the put-call parity to show that the values of the above portfolios are equal. Question 3. Suppose the time nowFor is 0. an interest rate of 5% per annum with continuous compounding, construct a portfolio of shares, position in cash (invested or borrowed),and European calls and European puts with different strike prices that has the following payoff at maturity T : 0 if ST 80 3ST 240 if 80 < S T 95 T = 45 if 95 < S T 100 2ST + 245 if 100 < TS 1 Question 4.This exercise is to brush up your skills in probability. My advice is that you should memorize part (a), which is very important. Suppose the random variable Y is normally distributed with mean and variance 2, i.e. Y N (, 2). Y Consider the random variable . (e eY is said to have a log-normal distribution.) Y Prove that the expectation and variance are of egiven by: 1 2 (a) E[eY ] = e+ 2 . 2 2 (b) Var[eY ] = e2+ (e 1). (c) E[(Y )4] = 34, E[(Y )3] = 0. Hint: We can write (x )2 22x = x 2 + 2 2x( + 2) 2 2 2 2 = x 2 + ( + ) 2x( + 2) + 2 ( + ) 2 2 4 2 = (x ( + )) 2. Recall: Z Y E[e ] = ex 1 22 e 2 ( 1 (x) 2 ) dx 2 Note that 2Y is normally distributed with mean 2 and variance . 4 2Y Var[eY ] = E[e E[eY ] 2 Question 5. Suppose that a stock price S geometric Brownian motion with t follows expected rate of return and volatility : dSt = St dt + S t dWt where = 0.08 and = 0.30. Suppose the current stock price S 0 = $100. (a) What is the probability that the stock price S year is less than $102? 1 in one (b) How does the probability P(S increases or decreases) 1 > 102) change (i.e. when you increase while other parameters remain unchanged? Justify your answer. 2 Question 6. Suppose f g t and t are two geometric Brownian motions (driven by the same Brownian motiont , W t 0) given by dft = (r + AB)ft dt + Bft dWt , dgt = (r + A2)gt dt + Ag t dWt where A, B and r are constants. Let Vt = terms of dt and dW t. (a) dVt (b) d (ln V t) 3 ft . Compute the following differential in gt