Q#01. List the six factors affecting stock option prices.
Answer: The six factors affecting stock option prices are the stock price, strike price, riskfree interest rate, volatility, time to maturity, and dividends Q#02. What is a lower bound for the price of a four-month call option on a non-diidendpa!ing stock when the stock price is "2$ the strike price is "2%$ and the risk-free interest rate is & per annum'
Answer: The lower bound is 28- 2e!" #8x # $$$$ % &$'((
Q#0(. What is a lower bound for the price of a one-month )uropean put option on a nondiidend pa!ing stock when the stock price is "12$ the strike price is "1%$ and the risk-free interest rate is *& per annum'
Answer The lower bound is )e#' '#( x #'#8$$$ * ! &2'+$ Q#0+. ,ie reasons wh! the earl! exercise of an merican merican call option on a non-diidendpa!ing stock is not optimal.
Answer elaying exercise delays the payment of the strike price' This means that the option holder is able to earn interest on the strike price for a longer period of time' elaying exercise also provides insurance against the stock price falling below the strike price by the expiration date' Assume that the option holder has an amount of cash and that interest rates are .ero' /xercising early means that the option holder0s position will be worth 1T at expiration' elaying exercise means that it will be worth max , 1 T 3 at expiration'
Q#0%. What is a lower bound for the price of a two-month )uropean put option on a nondiidend pa!ing stock when the stock price is "%$ the strike price is "*%$ and the risk-free interest rate is %& per annum'
Answer The lower bound is (e -"'"x 24)2 5 8 % &g 6g
Q#0*. he price of a non-diidend pa!ing stock is "1/ and the price of a three-month )u ropean call option on the stock with a strike price of "20 is "1. he risk- free rate is +& per annum. What is the price of a three-month )uropean put option with a strike price of "20'
Answer c % ), T * #'2, 1o % )+, * 2#, and r % #'#6' 7rom put-call parity p * c e !rT - 1o p % ) 2#e9" #6 x # 2 - )+ % )'8# so that the /uropean put price is &)'8# Q#07.
What is a lower bound for the price of a six-month call option on a non-diidend pa!ing stock when the stock price is "0$ the strike price is "%$ and the risk- free interest rate is 10& per annum'
Answer The lower bound is 80- 75e“ ° 1 x 0 5 = $8.66
Q#08.
What is a lower bound for the price of a two-month European put option on a non¬ dividend-paying stock when the stock price is $58, the strike price is $5, and the risk- free interest rate is 5! per annum"
Answer The lower bound is 65e -°.°5x 2/12 _ 58 = $ !
Q#09.
four-month )uropean call option on a diidend-pa!ing stock is currentl! selling for "%. he stock price is "*+$ the strike price is "*0$ and a diidend of "0.0 is expected in one month. he risk- free interest rate is 12& per annum for all maturities. What opportunities are there for an arbitrageur'
Answer The present value of the strike price is (#e!#)2x 64)2 % &'(' The present value of the dividend is #'8#e!" )2x)4)2 % #'+' ;ecause < (6 - '(- #'+ An arbitrageur should buy the option and short the stock' This generates (6 * % &+' The arbitrageur invests &#'+ of this at )2= for one month to pay the dividend of &#'8# in one month' The remaining &8'2) is invested for four months at )2=, >egardless of what happens a profit will materiali.e' ?f the stock price declines below &(# in four months, the arbitrageur loses the & spent on the option but gains on the short position' The arbitrageur shorts when the stock price is &(6, has to pay dividends with a present value of &#'+, and closes out the short position when the stock price is &(# or less' ;ecause &'( is the present value o f &(#, the short position generates at least (6 * '( * #'+ % &'( in present value terms' The present value of the arbitrageur0s gain is therefore at least '( * '## * &#'(' ?f the stock price is above &(# at the expiration of the option, the option is exercised' The arbitrageur buys the stock for &(# in four months and closes out the short position' The present value of the &(# paid for the stock is &'( and as before the dividend has a present value of &#'+' The gain from the short position and the exercise of the option is "here#ore ex%"l& 6!- 57.65 ' 0.7( = $5.56. The rbi"reur)s in in *resen" +lue "er,s is ex%"l& 5.56 ' 5.00 = $0.56.
Q#10. A one-month European put option on a non-dividend-paying stock is
currently selling for $2.50.. The stock price is $47, the strike price is $50, and the risk- free interest rate is 6 per annum, !hat opportunities are there for an ar"itrageur#
Answer The present value of the strike price is #e @ "#(x)4)2 * 6+'' 2' < 6+'- 6'## An arbitrageur should borrow &6+'# at (= for one month, buy the stock, and buy the put option' This generates a profit in all circumstances' ?f the stock price is above &# in one month, the option expires worthless, but the stock can be sold for at least &#' A sum of &# received in one month has a present value of &6+' today' The strategy therefore generates profit with a present value of at least &#'2' ?f the stock price is below &# in one month the put option is exercised and the stock owned is sold for exactly &# or &6+' in present value terms3' The trading strategy therefore generates a profit of exactly &#'2 in present value terms
Q#11. #he price of a European call that epires in si months and has a strike price of $%&
is $'( #he underlying stock price is $'), and a dividend of $&(5& is epected in two months and again in Eve months( #he term structure is Eat $ with all risk- free interest rates being *&!( What is the price of a European put option that epires in si months and has a strike price of $%&"
Answer: c e !rT % p 1# p * c e!rT - 1o ?n this case
p % 2 $#e !" ) x (4)2 #'e !" ) x 24)2 #'e@ " lx 4)23 - 2+ % 2') ?n other words the put price is &2')'
Q#12.
#he price of an +merican call on a non-dividend-paying stock is $( #he stock price is $%*, the strike price is $%&, and the epiration date is in three months( #he risk- free interest rate is 8!( erive upper and lower bounds for the price of an +merican put on the same stock with the same strike price and epiration date(
1o - < B - C < 1# - e !rT ?n this case $)- $# < 6- C < $)- $#e@ " #8x# 2 or )'## < 6'##- C < )'+ or 2'6) < C < $'## Dpper and lower bounds for the price of an American put are therefore &2'6) and &$'##'
