金融数学代考 MT4551代写 Financial Mathematics代写

2023/05/15数学代写金融代写 Financial Mathematics代写 MT4551代写金融数学代写金融数学代考182

MT4551 Financial Mathematics

金融数学代考 1. (a) The following three risk-free bonds are available, (i) Costs 82p now and pays out 87p in 6 months. (ii) Costs 87p now and pays an interest of

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1. 金融数学代考

(a) The following three risk-free bonds are available,

(i) Costs 82p now and pays out 87p in 6 months.

(ii) Costs 87p now and pays an interest of 1.5p in 6 months and a final payment of 98p in 12 months.

(iii) Costs Xp now and pays 2p in 6 months and 102p in 12 months.

Using discrete interest, determine the 6 and 12 month interest rates (expressed in terms of annual values) and calculate the price X of bond (iii). [4]

(b) Consider a trading strategy consisting of the following options,

(i) Long Put, Strike E₁ = £20 and cost P₁ = £0.5.

(ii) Long Call, Strike E₂ = £30 and cost C₂ = £1.0.

Draw the profit/loss diagram for this trading strategy and determine under what conditions a profit is made ? [3]

(c) Consider the binomial tree method of pricing an option. In each step, the share price (S) may either increase by a multiple u (> 1) or decrease by d (< 1). Assuming that p is the probability of an up move, construct a binomial tree with δt = 1 month to calculate the price of a 3 month European Put option if S(t = 0) = £40, E = £64, r = 2% and σ = 25%. For the calculation you may assume

3. 金融数学代考

(a) For the case where V = V (S), so that the financial derivative is independent of time, the Black-Scholes Equation reduces to an ordinary differential equation (ODE). Solve this ODE to obtain its two possible solutions. [4]

(b) (i) A financial derivative pays out an amount bS^ⁿ at time T, where S is the value of a share, b a constant and n a positive integer. Assuming that,

V (S, t) = A(t)S^ⁿ

is a solution of the Black-Scholes equation, determine the ordinary differential equation that A(t) satisfies. [3]

(ii) State the boundary condition that A(t) must satisfy to produce the stated pay out of the financial derivative at t = T. [1]

(iii) Solve the ordinary differential equation for A(t), to find an expression for the value of the financial derivative V at time t before expiry, T. [4]