Part C: Properties of options and valuation (47 marks)Problem C1: Properties of options (10 marks)The price of a European call that expires in six m

Part C: Properties of options and valuation (47 marks)Problem C1: Properties of options (10 marks)The price of a European call that expires in six months and has a strike price of $72 is $3.73. Theunderlying stock price is $75, and a dividend of $2 is expected in four months. The risk-free interestrate is 5% per annum (cont. comp.).a. What is the price of a European put option on the same stock that expires in six months andhas a strike price of $72? [1 mark]b. Let us assume some mispricing now. Show in detail the arbitrage strategies and the arbitrageprofit for the following two scenarios for the European put option price in tabular form:[9 marks]1) Scenario 1: The European put price is $2.0.2) Scenario 2: The European put price is $0.4.Note: Presenting the solution in a text form and not in a tabular form will attract a penalty.

Problem C2: Option Valuation and Properties of Options (26 marks)In this question, you need to price options with different valuation approaches and comment on yourresults. You will consider puts and calls on a share with a spot price of $47.8. Strike price is $50. Therisk-free interest rate is 6.14% per annum with continuous compounding.Binomial trees:Furthermore, assume that over each of the next two three-month periods, the share price is expectedto go up by 8% or down by 8%.a. Draw a two-step binomial tree and populate the individual nodes with the share price values ateach node. [1 marks]b. Use the two-step binomial tree from a. to calculate the value of a six-month European call optionusing the no-arbitrage approach. [3 marks]c. Use the two-step binomial tree from a. to calculate the value of a six-month European put optionusing the no-arbitrage approach. [3 marks]d. Show whether the put-call-parity holds for the European call and the European put prices youjust calculated in b. and c. [2 marks]e. Use the two-step binomial tree from a. to calculate the value of a six-month European call optionusing risk-neutral valuation. [2 marks]f. Use the two-step binomial tree from a. to calculate the value of a six-month European put optionusing risk-neutral valuation. [2 marks]g. Verify whether the no-arbitrage approach and the risk-neutral valuations lead to the sameresults. [1 mark]h. Use the two-step binomial tree from a. to calculate the value of a six-month American put option.[2 marks]Notes:1. When you use no-arbitrage arguments, you need to show in detail how to set up the risklessportfolios at the individual nodes of the binomial tree.

Black-Scholes-Merton model:Furthermore, assume that the volatility of the underlying asset is 16% per annum.i. What is the Black-Scholes-Merton price of a six-month European call option? [2 marks]j. Without calculations, show the price of a six-month American call and provide an explanationfor your answer. [1 mark]k. What is the Black-Scholes-Merton price of a six-month European put option? [2 marks]l. Verify whether the put-call parity holds for the option prices you just calculated in i. and k. [1mark]m.Without calculations: What would happen to the option prices you just calculated in i. and k. ifthe interest rate declines to 5.5%? Why? [2 marks]Comparison across models:n. Compare the call option prices you just calculated in e. and i. Compare also the put option pricesyou just calculated in f. and k. Do you expect these prices to be the same? Why/Why not? [2marks]

Problem C3: Binomial Trees Consider a stock which currently sells for $105. Assume that during each two-month period for thenext four months this share price is expected to increase by 5% or decrease by 5% and the risk-freeinterest rate is 4% per annum (cont. comp.).Consider a derivative that has a payoff given by the formula (max[(ST-103),0])2 whereTS is the stockprice in four months.a. Draw a two-step binomial tree and populate the individual nodes with the share price values ateach node. [1 mark]b. If this derivative is of European-style, value the derivative using no-arbitrage arguments. [5marks]c. If this derivative is of European-style, value the derivative using risk-neutral valuation. [2marks]d. Verify whether both approaches lead to the same result. [1 mark]e. If the derivative is of American style, should it be exercised early if the payoff at time t is givenby the formula (max[(St-103),0])2? [2 marks]

Note: When you use no-arbitrage arguments, you need to show in detail how to set up the risklessportfolios at the individual nodes of the binomial tree.