Question A4 in the paper attach below:

For a(i) is a long call european option, and I am not sure how to find the price? I am think that may be use black-scholes. Dont need to show the working process, just give me ideas. And I know a(ii), there is an combination, one is bull from call option and with another binary option. For the bull, how can I find the price? Is it also use black-scholes model? Thank you so much.

STAT3006 Examination Paper 2015/2016 Page 1 STAT3006: Stochastic Methods in Finance I 2015/2016 Answer ALL questions. Section A carries 40% of the total marks and Section B carries 60%. The relative weights attached to each question are as follows: A1 (9), A2 (7), A3 (10), A4 (14), B1 (6), B2 (9), B3 (10), B4 (8), B5 (7), B6 (20). The numbers in square brackets indicate the relative weight attached to each part question. Marks will not only be given for the final (numerical) answer but also for the accuracy and clarity of the answer. So make sure to write down workings, e.g. formulas, calculations, reasoning. Show your full working for all questions. Do not write formulas alone without any comment about what you are calculating. Except where otherwise stated, interest is compounded continuously and there are no transaction charges or buy-sell spreads. Assume that a positive risk-free interest rate always exists, and is the same for all maturities and is constant over time unless otherwise stated. All risk-free rates are expressed on an annualised basis. Assume also that all assets are tradable, that limitless short selling is always allowed, and that there is no counter-party credit default risk for all transactions. Time allowed: 2 12 hours. All data in this exam are fictional. T URN OVER STAT3006 Examination Paper 2015/2016 Page 2 Formula sheet 1. Ito's formula for a function G(x, t) can be written as: \u0012 \u0013 G G 1 2 G 2 dx + + b (x, t) dt, dG = x t 2 x2 for an Ito process following dx = a(x, t)dt + b(x, t)dB, with the usual notation. 2. The Black-Scholes formula for the price of a European call option under the standard assumptions, with strike price K and time to expiry T , is S0 N (d1 ) KerT N (d2 ), where N () denotes the cumulative distribution function of a standard Normal, and \u0010 \u0011 2 ln( SK0 ) + r + 2 T d1 = \u0010 T \u0011 2 ln( SK0 ) + r 2 T d2 = = d1 T T The Black-Scholes formula for the price of a European put option with strike price K and time to expiry T , is KerT N (d2 ) S0 N (d1 ), where the notation is the same as above. 3. The Black-Scholes-Merton partial differential equation is f 1 2f f + 2 S 2 2 + rS = rf t 2 S S with the usual notation. C ONTINUED STAT3006 Examination Paper 2015/2016 Page 3 Section A A1 A short forward contract that was negotiated some time ago will expire in six months and has a delivery price of 35. The current forward price for six-month forward contracts is 40. The six-month risk-free interest rate (with continuous compounding) is 5%. (a) What is the current value of the short forward contract? [3] (b) Three months later, the price of the stock is 50 and the risk-free interest rate is still 5%. What is the value of the short forward contract at this time? What is the price of a contract entered into at this time with the same expiration date as the first short forward contract? [6] A2 (a) Provide two reasons why geometric Brownian motion is a more suitable model for stock prices than generalised Brownian motion. [2] (b) Carefully define floating rate bonds. [2] (c) Construct an example of a self-financing portfolio using stocks and bonds. Explain your assumptions carefully. [3] A3 A stock price is currently 60. Over each of the next two four-month periods it is expected to go up by 10% or down by 5%. The risk-free rate is 7% per annum with continuous compounding. (a) Calculate the value today of an eight-month European put option with strike price 70. [5] (b) When, if ever, would it be worth exercising an eight-month American put option with strike price 70? [3] (c) What is the value today of an eight-month American call option with strike price 70? [2] A4 For each of the following derivatives with the given payoffs (in ) assume that the Black-Scholes assumptions hold, that all derivatives have maturity dates 1 year from now and that the underlying asset price process follows the stochastic differential equation dS = Sdt + SdB, with parameters = 0.3 and = 0.7. Assume also that the underlying asset is currently priced at 7 and that the risk-free rate is 0%. T URN OVER STAT3006 Examination Paper 2015/2016 Page 4 (a) Find the price of the derivative if it has: (i) A payoff of 0 if ST 5, and a payoff of ST 5 if ST 5. [4] (ii) A payoff of 0.4 if ST 5, a payoff of ST 5 if 5 ST 7, and a payoff of 1 if ST 7. [8] (b) What is the Gamma of the derivative in part (a) (i)? [2] C ONTINUED STAT3006 Examination Paper 2015/2016 Page 5 Section B B1 A stock price St follows geometric Brownian motion with drift = 0.15 and volatility = 0.35. Assume that S0 = 0.25 and that ST is the stock price at the end of six months. Find x such that P(ST 5. Derive the probability that S8 > 500. [7] T URN OVER STAT3006 Examination Paper 2015/2016 Page 6 B6 Assume that (Bt , t 0) is a standard Brownian motion. (a) Let Xt = tB1 . Is Xt a Brownian motion? Explain briefly your answer. [4] (b) Let Yt = (1 + t)B t tB1 . Compute the expectation and covariance of Yt . What kind t+1 of process is Yt ? Explain your reasoning. [7] \u0001 t/2 (c) Let Zt = e 2 eBt + eBt . Show that Zt is a martingale. [5] (d) Compute var(Bt7 ). [4] E ND OF PAPER