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Consider a European call option with price c, written on an underlying non-dividend- paying security with price S, at current time /. (1) State whether each of the following changes in underlying factors would increase or reduce the price of this option: (a) a fall in the price of the underlying security (b) an increase in the strike price of the option (c) an increase in the volatility of the underlying security price (d) a fall in the risk-free rate of interest [You should assume that each change occurs on a standalone basis, i.c. all other factors are unchanged.] [2] (ii) Explain each of your statements in part (i). [4] Consider a European put option with price p, written on the same underlying security. with the same strike price K and the same maturity ' as the call option described above. The continuously compounded risk-free rate of interest is r. (iii) Write down a formula that relates the values of c, and p,. [1] The call option has value 10.50 at time /= 0, and the put option has value $1.00. Both options are written on a security with current value So = $5, and both options have strike price $6.00 and maturity 7 =3 years. (iv) Determine the continuously compounded risk-free rate r. [2](i) Write down an expression for the price of a derivative in a Black-Scholes market in terms of an expectation under the risk-neutral measure, defining any additional notation that you use. [3] Consider an option on a non-dividend-paying stock when the stock price is $50, the exercise price is $49, the continuously compounded risk-free rate of interest is 5% per annum, the volatility is 25% per annum, and the time to maturity is six months. (ii) Calculate the price of the option using the Black-Scholes formula, if the option is a European call. [4] (iii) Determine the price of the option if it is an American call. [1] (iv) Calculate the price of the option if it is a European put. [2] (v) Determine how the prices of the contracts in parts (ii) to (iv) would change in the case of a dividend-paying underlying stock. [Note that you do not have to perform any further calculations.] [3] [Total 13]The current price of a non-dividend-paying share is $7 and its volatility is thought to be 40% per annum. The continuously compounded risk-free interest rate is 5% per annum. A European call option on this share has a strike price of $6.50 and term to maturity of one year. Calculate the price of this call option, assuming that the Black-Scholes model applies. [4] The market price for the option is actually $2. (ii) Show that the volatility of the share implied by the true market price of the option is 60% per annum, to the nearest 1% per annum. [6] [Total 10]