ACT370 - Final Exam, 2008 Questions 1 and 2 relate to the following ...

ACT370 - Final Exam, 2008Questions 1 and 2 relate to the following information. An "exchange call option" gives theowner of the option the right to give up one share of Stock A in exchange for receiving oneshare of Stock B. Stock A currently has a price of $56 and Stock B has a current price of $52.The continuously compounded risk-free rate of interest is 5% and the price of a one-yearEuropean exchange call option is $7.1. Suppose that neither Stock A nor Stock B pays any dividends. Find the price of a Europeanexchange put option expiring in one year which gives the owner the right to give up one shareof Stock B in exchange for receiving one share of Stock A.A) $3 B) $5 C) $7 D) $9 E) $112. Suppose that Stock A pays continuous dividends at a rate of 3% and Stock B will pay asingle dividend in 6 months of $ H. Find the value of H if the one year European exchange putoption has a current price of $10.A) Less than .5 B) At least .5 but less than .75 C) At least .75 but less than 1D) At least 1 but less than 1.25 E) At least 1.25

3. You are given call option prices of $12 and $8 for European options with strike prices of$50 and $60, respectively. The options expire at the same time and are on a non-dividendpaying stock. According to convexity of option prices, how many of the following prices arefeasible prices for a call option on the same stock and expiring at the same time with a strikeprice of $53? Feasible means that there are no arbitrage opportunities.I. $10 II. $10.50 III. $11 IV. $11.50A) None B) One C) Two D) Three E) All

Questions 4 and 5 relate to the following information. In a binomial branching model, the stockprice at time 0 is 120. At time 1, the stock price will be either 144 or 100. The annual effectiverisk free rate of interest from time 0 to time 1 is 3œÞ"! .4. A call option on the stock with exercise price 110 expires at time 1. Find the number ofshares of stock in the replicating portfolio at time 0.A) Less than .2 B) At least .2 but less than .4 C) At least .4 but less than .6D) At least .6 but less than .8 E) At least .85. Another stock is available that has a value of 100 at time 0. The stock price at time 1 is eitherF (when the first stock is 100) or "Þ#F (when the first stock is 144). Find the value of F thatresults in no arbitrage opportunities.A) Less than 80 B) At least 80 but less than 85 C) At least 85 but less than 90D) At least 90 but less than 95 E) At least 95

Questions 6 to 8 relate to the following information. A 3-period binomial tree is constructed.The continuously compounded risk-free rate of interest per period is < , and you are given that

9. A call option on a non-dividend paying stock has a strike price of 100 and expires in one year.You are given the following:- current stock price is 100- current option price is 19.384 based on the Black-Scholes option pricing model- continuously compounded risk free interest rate is 8%- the call delta is .6554For the same stock, find the price of a similar call option with a strike price of 110 using theBlack-Scholes option pricing model.A) Less than 12 B) At least 12, but less than 13 C) At least 13, but less than 14D) At least 14, but less than 15 E) At least 15

10. A call option on a non-dividend paying stock has a current price of 60 and expires in oneyear. You are given the following, based on Black-Scholes option pricing:- the option > is .018817- the volatility of the underlying stock is 5 œÞ%&- the option Vega for a .01 change in 5 is .182$"$- ." !Find the value of the call option ? using Black-Scholes option pricing.A) Less than .50 B) At least .50, but less than .55 C) At least .55, but less than .60D) At least .60, but less than .65 E) At least .65

11. W " , the price of a stock at time 1, has a lognormal distribution under the risk neutral measureand under the physical measure. You are given the following:- under the physical measure IÒ68 W" Ó œ %Þ$*'##( and IÒW"Ó œ )&Þ*(#%- under the risk neutral measure IÒ68 W"Ó œ %Þ$'*##(.Find IÒW"Ó under the risk neutral measure.A) Less than 80 B) At least 80, but less than 81 C) At least 81, but less than 82D) At least 82, but less than 83 E) At least 83

12. You are given the following information regarding European call options on a non-dividendpaying stock with a current price of 60 and a volatility of 5 œÞ%& . The continuouslycompounded risk free rate is 6%.Call Option 1 Call Option 2Strike Price 62 65Time to Expiry (years) .25 .5Option Price 4.9101 6.3348Delta .5133 .5007Gamma .0295 .0209A market maker has a long position in one European put option with strike price 62 expiring in.25 years. The market maker wishes to construct a Delta-Gamma hedge using shares of stock andput options with strike price 65 expiring in .5 years. How many shares of stock are needed in thehedging portfolio?A) Short at least 1 share B) Short between 0 and 1 shares C) No shares neededD) Long between 0 and 1 shares E) Long at least one share

ACT370, 2008 Final Exam Solutions1. According to put-call parityTprepaid forward price of asset to be receivedœGprepaid forward price of asset be delivered .Therefore T&#œ(&' , so that T œ"" . Answer: EÞ!#&Þ!$2. "! &# H/ œ ( &'/ p H œ Þ'( Þ Answer: B"#GÐ&$ÑGÐ&$Ñ)3. We must have&$&! '!&$p GÐ&$Ñ Ÿ "!Þ)! .Two prices are feasible. Answer: C4. ? œ 0Ð?Ñ0Ð.ÑW ?W .! !$%!"%%"!!œ œ Þ(($ . Answer: D‡ "Þ""!!)"Þ#F: FÐ": Ñ"" "Þ""#!5. : œ œ . For no arbitrage, we must have œ "!! ."%% "!!"#! "#!Solving for F results in F œ *'Þ!$ . Answer: E‡ ‡‡ "Þ!&Þ*6. : œ"Þ"Þ*œ Þ(& .Þ(&Þ#& "The derivative value at node 4 is"Þ!&œ"Þ!&.Þ(&The derivative value at node 5 is"Þ!&.The derivative value at node 2 is Þ(&† " Þ#&† Þ(&"Þ!& "Þ!&œThe derivative value at node 3 is Þ(&†The derivative value at node 1 isÞ(&"Þ!&"Þ!&Þ(&†"Þ!&œÐ"Þ!&Ñ#ÐÞ(&ÑÐ"Þ!&ÑÐÞ(&ÑÐ"Þ#&Ñ ÐÞ(&Ñ#Þ#&†Ð"Þ!&Ñ# Ð"Þ!&Ñ##ÐÞ(&ÑÐ"Þ#&ÑÐ"Þ!&Ñ#œ Þ(#* . Answer: D7. Suppose there are Bunits of the option with strike 120 and C units of the option with strike100. At node 7 the combination has value "$Þ"B $$Þ"C œ ".At node 8 the combination has value )Þ*C œ " ."Solving for B and C results in C œ)Þ*œ Þ""#% (long) units of option with strike 100,and B œ Þ#!(' (short) units of option with strike 120. Answer: A

8. The option would not be exercised at any node other than nodes 4 or 7 since the stock pricewould be less than 100 at any other node. The prepaid forward price at time 0 is "!! HÐ"Þ!&Ñ. #H #The stock price at node 4 is Ò"!! ÓÐ"Þ"Ñ H œ "#" Þ!*(&!'H .Ð"Þ!&ÑHÐ"Þ!&ÑThe stock price at node 7 is Ò"!! ÓÐ"Þ"Ñ œ "$$Þ"! "Þ#!(#&'H .##The option value at node 7 is #$Þ"! "Þ#!(#&'H , and the option value at node 8 is 0.The exercise value of the option at node 4 is "" Þ!*(&!'H .The binomial tree backward induction value of the option at node 4 isÐÞ(&ÑÐ#$Þ"!"Þ#!(#&'HÑ"Þ!&œ "'Þ& Þ)'#$#'H .Exercise at node 4 will be optimal if "" Þ!*(&!'H "'Þ& Þ)'#$#'H ,or equivalently, if H (Þ"* ÞAnswer: D$9. ? œ RÐ. " Ñ œ Þ'&&% p ." œ Þ% (from the standard normal table).For the option with a strike price of 100, and X œ "ß < œ Þ!) , we have

12. The put options have the same > as the call options, and ? T œ ? G " .Þ!#*&The hedging portfolio should be short byÞ!#!* œ "Þ%"" put options with a strike price of 65expiring in .5 years. The ? of the long put with strike 62 is Þ&"$$ " œ Þ%)'( .The ? of the 1.411 short puts with strike 65 is "Þ%""ÐÞ&!!( "Ñ œ Þ(!%& .To match the ? of the long put with strike 65, the hedging portfolio should go shortÞ(!%& ÐÞ%)'(Ñ œ Þ#"() shares. Answer: B