# MAT3104 Mathematical Modelling in Financial Economics- Lemuria

### Questions

1. Suppose you are the head of the treasury department of Lemuria an island nation,  which has re- cently been plunged into recession by an unexpected failure of the travel sector which traditionally forms a substantial part of the Lemurian economy. Officials have estimated that within a few weeks unemployment will reach 12 million, 14% of the work-force, unless government action is taken.
Suppose the consumption function of Lemuria is estimated to be C  =  30 + 0.75Y ,  in  billions  of dollars, in which Y represents household income and C is household consumption. Currently there is no income tax, net government expenditure is \$65 billion, net exports are worth \$25 billion, and investment spending is currently \$35 billion.

(a) What is the equilibrium level of consumption (C) and household income (Y ) at the current settings of government spending?
(b) What changes to government spending should be made to increase output by 12%?

Suppose the government decides that any increase in spending should be 50% funded by an income tax. By how much should the government increase (or decrease) spending and taxes, to achieve an increase of 7.5% in aggregate output?

2. (a) Consider a one-step binomial tree model for a share which has the current value of \$18 at time 0 and at time 1 the share price either increases to \$25 or decreases to \$12.
The risk free interest rate during this period is 7% (per unit time).

(i) Find the value of an European call option with an exercise price of \$15 by hand. Remember to show full working.
(ii) Suppose that you have sold 2600 units of the option in part (i). How many shares should you now buy, or sell, to hedge your sale in such a way that you will earn the same income from the option whatever happens to the share?

(b) Consider a two-step binomial tree model for a share which has the current value of \$10 at time 0 and at the end of step the share price either increases by a factor of 1.1 or decreases by the factor of 0.9.
The risk free interest rate during this period is 2% per unit time and each step is 1 unit of time.

(i) Find the value of an American put option with an exercise price of \$11 by hand. Please remember  to  show  full  working and include a diagram of your tree.
Should the option be exercised early? If so at what share price value or values?

3. (a) Sketch a plot for a European call option at expiry, assuming the final value of the share is always \$45, as a function of the strike price which varies between \$0 and \$70.

(b) Consider an option to buy a share, in τ months time, for \$X, assuming the current value of the share is \$S0, the volatility of the share is σ, and the risk-free interest rate is r% per month.
Determine the value of the option as the time to expiry becomes large (i.e. goes to infinity) if σ < √2r.

(c) What is the value of an option to sell a share, in one years time, for \$X, assuming the current value of the share is \$85, X 85 (i.e. X is so much greater than 85 that we can ignore the possibility of the share rising to X), and the risk-free interest rate is 3% per year?
Explain a strategy, involving no trading, which can realise this value without risk. You may assume that short selling of shares is allowed, with no brokerage fee, and money can be borrowed or invested at the risk-free interest rate without risk.

(d) You have just been issued 750 shares which you must continue to hold for six months, and another 350 shares which you must continue to hold for one year.
Assuming you can buy and/or sell put and call options in the shares, at a variety of strike prices and for durations of 6 months and of one year, explain a strategy that you should use to lock in the current value of the shares, i.e. to eliminate any risk from their value falling (or rising)?
Include in your explanation how the strategy will work in the event the share price increases and in the event the share price decreases.

4. (a) Find the solution to stochastic differential equation

dYt = (18 + 6 cos 3t) Yt dt + 6Yt dWt,    Y0 = 7

Please show full working

(b) Find a stochastic differential equation in terms Yt if Yt = loge (3Xt + 4t2) where

dXt = 3tXt dt + 2t2 dWt

Please show full working.

5. Derive the formula for the value of a European option with the payout at expiry of

by using the same method as in Example 4.4 and then rewriting X1 and X2 in present value terms. Give your final answer in terms of the standard cumulative normal distribution N (z).