Correct answers only in all the questions!

3.(i) Describe the rights of the holder of:

(a) a European call option.

(b) a European put option.

[2]

(ii) Draw on separate diagrams, defining all notation used:

(a) the payoff from buying a call option.

(b) the payoff from selling a call option.

[3]

A derivatives trader has presented an option strategy to their manager for review. The

trader believes that a lot of publicity surrounding a pharmaceutical company stock isunfounded and the stock price will remain stable over the coming months. They have

suggested a strategy to profit from this which combines four option contracts with the

same expiration date but three different strike prices:

? buying one call option for 8 at the lower strike price, 80 - A

? selling two call options for 10 (5 each) at the middle strike price, 100 - B

? buying one call option for 4 at the higher strike price, 120 - C

(iii) Draw the payoff for A, B and C together with the payoff for the overall strategy

suggested above. [3]

(iv) Comment on the strategy presented by the derivatives trader. [3]

4

The value of an investment asset follows the equation A(t) = exp(B,), where B, follows standard Brownian motion. (i) State the five defining properties that apply to B, as a standard Brownian motion. [5] An actuarial student invests $1,000 in the asset at time 0. (ii) Calculate the expected value of this investment at time 5. [2] (iii) Calculate the probability that the value of the student's investment is less than $10,000 at time 5. [2]An insurance company has a portfolio of insurance policies. Claims arise according to a Poisson process, and claim amounts have a probability distribution with parameter 0. (i) State one assumption the insurance company is likely to make when modelling aggregate claim amounts. [1] (ii) Explain what the Maximum Likelihood Estimate (MLE) of 0 represents. [2] (iii) State an alternative to using the MLE. [1] (iv) Suggest two complications that may arise for the insurance company when it uses past claims data to determine the MLE of 0. [2]Tarik and Liam are playing a zero-sum two-person game. From a deck of three cards numbered 1, 2 and 3 Tarik selects a card, making sure Liam does not know which one it is. Liam then proceeds to guess which card Tarik has picked. If Liam is wrong, he continues to guess until he has guessed correctly, at which point the game ends. After each guess, if Liam's guess is lower than the number on Tarik's card, Tarik says "Low", but if Liam's guess is higher than the number on Tarik's card, Tarik says "High". At the end of each game, Liam pays Tarik $1 for each guess he made. You should assume that Liam will never make a guess that contradicts the information provided by Tarik - for example, if Liam guesses "2" first, and Tarik says "Low", Liam would then always guess "3", rather than "1". Consider strategy A, where Liam will guess "1" first, and then guesses "2" if 1 is not correct. (i) Set out the four other strategies in addition to A (labelled B to E) which Liam could adopt. [4] (ii) Construct the payoff matrix to Tarik. [3] (iii) Explain whether or not there is a saddle point. [2]