Help in solving the following questions.

1 An investor claims to be able to value an unusual derivative on a non-dividend-paying share using the pricing formula: where S, denotes the price of the share at time t. (1) Derive formulae for the delta and gamma of the derivative, based on the pricing formula above. (ii) For each of the following scenarios, calculate the number of shares that must be purchased or sold along with a short holding in one derivative, in order to achieve a delta-hedged portfolio: (a) the current share price is 1 (b) the current share price is 3. (ili) Explain which of the scenarios in (ii) is likely to involve more portfolio management in the near future if the investor is determined to maintain a delta-hedged portfolio. 2 (i) Define the delta, gamma and theta of an option. [3] yle (Wi) Describe, using a numerical example, the concept of delta hedging. [6] [Total 9] 3 Give definitions of the 'Greeks' that could be used as an aid to management in each of the yle following situations. State also the desired ranges for their numerical values and define any notation you use. (a) A hedge fund manager wishes to establish a delta-neutral position that would not need frequent rebalancing. (b) A derivatives trader is concerned that a change in the distribution of the daily price movements of particular shares might affect the values of the options held on those shares. (c) The trustee of a pension fund that purchased a large number of options last year as a means of hedging is concerned about changes in the value of the fund as the options approach their expiry date. [6] 4 A call option has a price of 20.15p and a delta of 0.558 at time t . Determine the hedging portfolio of shares and cash for this option at time t , given that the price of the underlying share, St = 240p .8 0 Explain the difference between a recombining and a non-recombining binomial tree. [2] (ii) A researcher is using a two-step binomial tree to determine the value of a 6-month European put option on a non-dividend-paying share. The put option has a strike price of 450. During the first 3 months it is assumed that the share price of 400 will either increase by 10% or decrease by 5% and that the continuously compounded risk-free rate (per 3 months) is 0.01. During the following 3 months it is assumed that the share price will either increase by 20% or decrease by 10% and that the continuously compounded risk-free rate is 0.015 (per 3 months). Calculate the value of the put option. [6] (mii) The researcher is considering subdividing the option term into months. Explain the advantages and disadvantages of this modification of the model and suggest an alternative model based on months that might be more efficient numerically. [5] [Total 13]