SOLVE FI ME!

1. Suppose that at time t a portfolio (t , t) is held, where t represents the number of

units of a stock, with price St , held at time t and t is the number of units of a cash

bond, with price Bt , held at time t. The processes and are previsible.

Let V(t) = t

St + t

Bt be the value of the portfolio at time t.

(i) Explain what it means for this portfolio to be self-financing. [2]

Consider a stock paying a continuous dividend at a rate and denote its price at any

time t by St

Let Ct and Pt be the price at time t of a European call option and European put option

respectively, written on the stock S, each with strike price K and maturity T t.

The instantaneous risk-free rate is denoted by r.

(ii) Prove put-call parity in this context by constructing two self-financing

portfolios whose value must be equal by the principle of no arbitrage.

2.

(i) Define the terms: (a) par swap. (b) extendable swap. (c) step-up swap. 14] (ii) Explain why a pension fund might want to purchase a par swap. [2] XYZ pension scheme enters into a three year interest rate swap contract under which the pension scheme will receive a fixed rate payment stream. The pension scheme is required to pay a floating rate payment stream in return. The following information is available regarding the swap and likely payments: . Term 3 years . Notional value of swap f100m (the notional amount of the swap is not exchanged) . Payments are made in arrears semi-annually. . There are 360 days in a swap year. Period No. of days in Semi-annual forward period interest rate 180 1.52% 180 1.63% 180 1.75% OutWN- 180 1.91% 180 2.02% 180 2.14% (iii) Calculate the following, showing all your workings, using the above information: (a) present value of the floating rate payments [3] (b ) the annualised fixed rate of the swap [3] The pension fund trustees agree to enter into the interest rate swap on these terms. The transaction costs on the swap have the impact of reducing the annualised fixed rate of the swap by 0.02%. (iv) Calculate, showing all your workings, the impact on the value of the swap from transaction costs. [3] (v) Explain the impact on the value of the swap if interest rates fall during the term of the swap contract. [2]Let us consider that we need to sample from a discrete random variable Y with distribution function: W/- 13 w/ - W (i) Set out a direct method of sampling from Y. [2] Consider now another random variable X with distribution: W/ N of w (ii) Set out a direct method of sampling from X. [2] (iii) (a) Explain how you can apply the acceptance-rejection method to sample X by rejecting/accepting samples from Y. (b) Calculate how many samples from Y on average are needed to generate one sample from X. [6](i) Explain why insurance companies make use of run-off triangles. [2] (ii) The run-off triangle below shows incremental claims incurred on a portfolio of general insurance policies. Development Year Policy Year 0 2 2011 4,657 3.440 931 572 2012 6,089 5,275 1,381 2013 5,623 4,799 2014 7.224 Calculate the outstanding claims reserve for this portfolio using the basic chain ladder method. [7]