Answer Happy

Question 3 (Marks: 50) For the stock that corresponds to your Group, select an option expiry date among the available ones in the market so that T> 0.5 years. Given the spot price of the stock (So), select three strike prices K₁, K₂ and K3 near the money such that K₁ < K₂ < K3. a) Pricing Call and Put options with strike price K₂ and maturity T (in detail calculations). The time horizon should be divided into 5 intervals of duration 4t, each. Consider that the stock price may increase by 1 and decrease by da on each time interval. The volatility of the stock price is the historical volatility found in Question 2, part (c). - Calculate analytically the p* probability Draw the binomial tree for the stock price evolution Provide and explain the formula of calculating the probabilities of reaching each node Price the European Call and Put Options (show the required trees; explain) Price the American Call and Put Options (show the required trees; explain) Comment on the difference of the premiums b) Pricing Call and Put options with strike prices K₁, K₂ and K, and maturity T. The time horizon should be divided into 15 intervals now of duration 4t, each. Consider that the stock price may increase by u, and decrease by u, on each time interval. The volatility of the stock price is again the historical volatility found in Question 2, part (c). Calculate analytically the p" probability Price the European Call and Put Options of strike price K₁7. Price the European Call and Put Options of strike price K₂. Price the European Call and Put Options of strike price K3. Compare the European Call and Put premiums for strike price K₂ of (a) and (b) Explain the difference between the calculated premiums and those in the market. c) Given the premiums c; and p; (i = 1,2,3) that correspond to the options with strike prices K₁, K₂ and K3 that have been already calculated, built the following trading strategies and discuss their suitability: Option (K₂) and underlying asset. i) Covered Call and ii) Protective Put Multiple options (K₁, K3) of the same type. i) Bull Spread with Calls and ii) Bear Spread with Puts Combinations. i) Strip (K₂) and Butterfly Spread (K₁, K₂, K3) Note: For the risk-free rate you can use the United States 52 Week Bill Yield®.

Question 1 (Marks: 20) A Credit Default Swaps (CDS)¹ is a contract where one party (credit protection buyer) pays the other one (credit protection seller) a fixed periodic coupon for the life of the contract on a specified reference asset. The party paying the premium is effectively buying insurance against specific credit events, such as default, bankruptcy or failure-to-pay or debt restructuring. If such a credit event occurs, the party receiving the premium makes a payment to the protection buyer, and the swap then terminates. Consider now that party A wishes to get covered from a potential loss of the face value (VA) of an asset in case of a credit event. Hence, party A decides to purchase today (to = 0) some protection from party B that lasts until some specified maturity date T. To pay for this protection, party A makes a regular stream of payments to party B. The size of these payments is a fixed percentage of the face value of the asset being insured and it is based on the yearly contractual spread W₁y, which represents the percentage used to determine the payments' amount over one year. The payments are made every 3 months until maturity of the contract or until a credit event occurs, whichever occurs first. Assume that the credit event occurs as the first event of a Poisson counting process and hence default time is exponentially distributed with parameter λ. Denote the short rate with r. The aim is to value the premium leg, i.e. to write a mathematical expression for this stream of payments taking into account both the appropriate discounting and the probabilities of default events. a) Illustrate the problem with a sketch representing the various payments occurring over the considered time period. Make sure you include the time at which the payments are made and the size of each undiscounted payments. b) Express the discounting factor at time t, where i E {0,1,..., N}. c) Express the probability that a credit event occurs before time t, (P) and the survival probability at time t₁, i.e. the probability that no credit event has occurred before time t (PND). d) Using the above, write down the full expression for the premium leg. e) Using the values that correspond to your Group, calculate the premium leg and price the CDS.