Compare monopoly and perfectly competitive markets with respect to price output and allocative efficiency.
Distinguish between total product and marginal product.
Apply game theory to analyse strategic behaviour of firms in oligopoly.
Discuss derived demand.
(1) Explain the constraints on f(X(()) in order for Ito's lemma to apply. [2] (ii) Discuss the relevance of Ito's lemma in the valuation of derivatives. [3] Let X(() and Y (() be two stochastic processes adapted to the same Brownian motion X(1) = o(1)dw(1)+u(1)at, dF (1) = p(D)dW (1) + v(1)dt. (iii) Prove, using Ito's lemma, that: d(X(OF(0)) = X(1)dY (1) + Y(OX(() +o(f)p(t)at. [Hint: Consider the expression (X(()+ Y(1))-] [4] An asset has a price at time : of S(1), which satisfies the following stochastic differential equation: as(() =-58()di + 4dw ((). where W(() is standard Brownian motion. The solution for S(() is of the form: where A, B. C and D are integer constants. (iv) Calculate A, Cand D. [Hint: Let Y(?) =e"and find a suitable X(/) to enable the use of the result of part (mi).] [7] (v) Give an expression for the value of B. [1]An investment manager has a portfolio of options on a single underlying asset. Explain whether a position in this underlying asset could be used to make the portfolio gamma-neutral [2] The investment manager ensures that the portfolio is delta-hedged at the start of each business day. It would require the investment manager to purchase vanilla options to gamma hedge the portfolio. (ii) Explain why the gamma of the delta-hedged portfolio must be negative. [2] (iii) Derive the following approximation for a vanilla option: change in option value = A * change in underlying price + -I x (change in underlying price)' where A is the delta of the option and I is the gamma of the option. [3] The portfolio, on a given day, is delta-neutral at the start of the business day but has a gamma of -3, 706.2 (iv) Identify the potential variation in the value of the portfolio on this day, using your answer to part (iii) to justify this. [3] The delta and gamma of a particular traded option on the underlying asset are 0.70 and 1.74 respectively on the given day. (v) Determine the transactions in the traded option and or underlying asset which are required to make the portfolio both delta-neutral and gamma-neutral. [3] (vi) Propose alterations to the original portfolio to make it less gamma negative. [1] (vii) Recommend another action that the company should take to mitigate the risk of a large negative gamma position occurring in future. [1]
