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Hull: Options, Futures, and Other Derivatives, Ninth Edition Chapter 13: Binomial Trees Multiple Choice Test Bank: Questions with Answers 1. The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. An investor sells call options with a strike price of $32. Which of the following hedges the position? A. Buy 0.6 shares for each call option sold B. Buy 0.4 shares for each call option sold C. Sh
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  Hull: Options, Futures, and Other Derivatives, Ninth EditionChapter 13: Binomial TreesMultiple Choice Test Ban: !uestions ith #ns ers  1.The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $6. !ssume the risk-free rate is ero. !n investor sells call options #ith a strike price of $3. hich of the follo#ing hedges the position%!.&uy 0.6 shares for each call option sold&.&uy 0.' shares for each call option sold(.)hort 0.6 shares for each call option sold*.)hort 0.6 shares for each call option sold !ns#er+ & The value of the option #ill ,e either $' or ero. f ∆  is the position in thestock #e reuire 36 ∆ /'6 ∆ so that ∆ 0.'. it follo#s that 0.' shares should ,e purchased for each option sold..The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $6. !ssume the risk-free rate is ero. hat is the risk-neutral pro,a,ility of that the stock price #ill ,e $36% !.0.6&.0.(.0.'*.0.3 !ns#er+ ( The formula for the risk-neutral pro,a,ility of an up movement is d ud e p rT  −−= n this case   u =36/30 or  1.2 and   d  =26/30 =0.8667. !lso   r  =0 and   T  =0.5.  The formula gives    p =(1-0.8667/(1.2-0.8667) =0.4. 3.The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $6. !ssume the risk-free rate is ero. !n investor sells call options #ith a strike price of $3. hat is the value of each call option%!.$1.6&.$.0(.$.'*.$3.0 !ns#er+ !   The formula for the risk-neutral pro,a,ility of an up movement is d ud e p rT  −−= n this case   u =36/30 or   1.2  and   d  =26/30 =0.8667. !lso   r  =0 and   T  =0.5.  The formula gives    p =(1-0.8667/(1.2-0.8667) =0.4.  The payo2 from the call option is   $'   if there is an up movement and   $0   if there is a do#n movement. The value of the option is therefore   0.4×4 +0.6×0= $1.6 . e do not do any discounting ,ecause the interest rate is ero.4'.The current price of a non-dividend-paying stock is $'0. Over the next year it is expected to rise to $' or fall to $35. !n investor ,uys put options #ith a strike price of $'1. hich of the follo#ing is necessary to hedge the position%!.&uy 0. shares for each option purchased&.)ell 0. shares for each option purchased(.&uy 0. shares for each option purchased*.)ell 0. shares for each option purchased !ns#er+ ( The payo2 from the put option is ero if there is an up movement and ' if there is a do#n movement. )uppose that the investor ,uys one put optionand ,uys   ∆   shares. f there is  an up movement the value of the portfolio is   ∆ × ' . f there is a do#n movement it is #orth ∆ × 357' .  These are eual #hen  37 ∆ + '' ∆  or ∆ = 0. .  The investor should therefore ,uy 0. shares foreach option purchased. .The current price of a non-dividend-paying stock is $'0. Over the next year it is expected to rise to $' or fall to $35. !n investor ,uys put options #ith a strike price of $'1. hat is the value of each option% The risk-free interest rate is 8 per annum #ith continuous compounding.!.$3.93&.$.93(.$1.93*.$0.93 !ns#er+ * The formula for the risk-neutral pro,a,ility of an up movement is d ud e p rT  −−= n this case   r  = 0.0 , T  = 1 , u ':'01.0 and   d  35:'00.9 so that    p 0.56   and the value of the option is   0.56;070.';'4 e -0.02×1 0.936.hich of the follo#ing descri,es ho# !merican options can ,e valued using a  ,inomial tree%!.(heck #hether early exercise is optimal at all nodes #here the option is in-the-money &.(heck #hether early exercise is optimal at the <nal nodes(.(heck #hether early exercise is optimal at the penultimate nodes and the <nal nodes*.=one of the a,ove!ns#er+ ! >or an !merican option #e must check #hether exercising is ,etter than not exercising at each node #here the option is in the money. t is clearlynot #orth exercising #hen the option is out of the money45.n a ,inomial tree created to value an option on a stock? the expected return on stock is!.@ero&.The return reuired ,y the market(.The risk-free rate*.t is impossi,le to kno# #ithout more information !ns#er+ ( The expected return on the stock on the tree is the risk-free rate. This is an application of risk-neutral valuation..n a ,inomial tree created to value an option on a stock? #hat is the expected return on the option%!.@ero&.The return reuired ,y the market(.The risk-free rate*.t is impossi,le to kno# #ithout more information !ns#er+ ( The expected return on the option on the tree is the risk-free rate. This is an application of risk-neutral valuation. The expected return on all assets in a risk-neutral #orld is the risk-free rate.9.! stock is expected to return 108 #hen the risk-free rate is '8. hat is the correct discount rate to use for the expected payo2 on an option in the real #orld%!.'8&.108(.Aore than 108*.t could ,e more or less than 108 !ns#er+ *   The correct ans#er is *. There is no easy #ay of determining the correct discount rate for an optionBs expected payo2 in the real #orld. >or a call option the correct discount rate in the real #orld is often uite high and fora put option it is often uite lo# even negative4. The example in the text illustrates this. 10.hich of the follo#ing is true for a call option on a stock #orth $0!.!s a stockBs expected return increases the price of the option increases&.!s a stockBs expected return increases the price of the option decreases(.!s a stockBs expected return increases the price of the option might increase or decrease*.!s a stockBs expected return increases the price of the option on the stock stays the same!ns#er+ * The option price #hen expressed in terms of the underlying stock price is independent of the return on the stock. To put this another #ay? everything relevant a,out the expected return is incorporated in the stockprice.11.hich of the follo#ing are =OT true!.Cisk-neutral valuation and no-ar,itrage arguments give the same option prices&.Cisk-neutral valuation involves assuming that the expected return is the risk-free rate and then discounting expected payo2s at the risk-free rate(.! hedge set up to value an option does not need to ,e changed*.!ll of the a,ove!ns#er+ (  The hedge set up to value an option needs to ,e changed as time passes. ! and & are true.1.The current price of a non-dividend paying stock is $30. Dse a t#o-step tree tovalue a European call option on the stock #ith a strike price of $3 that expires in 6 months. Each step is 3 months? the risk free rate is 8 per annum#ith continuous compounding. hat is the option price #hen u  1.1 and d  0.9. !.$1.9&.$1.'9(.$1.69*.$1.9 !ns#er+ & The pro,a,ility of an up movement is
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