Question

Homework Problem 18_1

An analysis of HL Corporation suggests that in the next year the price of its stock will be either $50 or $30. The current price is $40. The 1-year riskless rate is 10%.

Consider a call option that expires in one year with an exercise price of $35.

a) The Replicating Portfolio is a portfolio of stocks and bonds that exactly replicates the payoff of the call option under all conditions, in this case the two states.

What portfolio of stocks (number of shares bought or shorted) and bonds (dollars borrowed/shorted or lent/long) that has same payoff if this Call option?

b) What is the fair value of this call? and why?

c) Create a riskless hedge with the call and stock. A riskless hedge is a portfolio of stocks and calls that have the same payoff in all conditions, in this case the two states.

For example, Write 1 calls and buy .75 shares or Buy 1 call and short .75 shares.

d) An arbitrage is a strategy that has all positive cash flows with no investment or negative cash flows.

How would you create an arbitrage if the Call were priced at $15?

e) How would you create an arbitrage if the Option were priced at $5?

Answer 1

the replicative portfolio for a call is made by buying N no, of stocks by borrowing money

let the portfolio be given by

C0= N*S0 +B

where

C0 = current price of the call

N= no. of shares to be bought

S0= current stock price =40$

B = amount to be invested ( if comes out to be positive, then it will be a borrowing)

upside price after one year(Su) =

upside factor(u)= Su/S0 =50/40=1.25

Downside price after one year (Sd)=30

downside factor = 30/40=.75

after 1 year call payoffs will be= Max(0,(St -K))

where St = stock price 1 year hence which may either b4 Su or Sd

K= strike price=35$

Call payoff if stock price goes up (Cu)= Max(0,(50-35))=15

call payoff if stock price goes down = Max(0,(30-35))=0

risk free rate (r) = 10%

future value of B after 1 year = B*1.1 =1.1B

pay off call and the portfolio( i.e. value of stocks and borrowing after 1 year ) should be equal

N*Su +1.1B= Cu

N*Sd+1.1B=Cd

50N+1.1B=15

30N+1.1B=0

solving for N, we get N= .75

plugging N in the equation we get B= (15-.75*50)/1.1= -20.45 it is negetive i.e. borrowing

which means we will buy .75 shares by borrowing 20.45$ @10% for each call

part 2

our original equation of portfolio at time 0 was

N*S0 +B =C0

now we have all the values to find C0

.75*40-20.45=9.55 this is the current price of call

part c

risk less hedge is nothing but the inverse of replicative portfolio

our replicative portfolio was .75*S0-20.45 =C0

so our risk less hedge would be

.75S0-C0=20.45

which means that will buy .75 shares for each call sold in simple words we will buy 3 stocks and sell 4 calls, and why it is called risk less because it is basically replicating the behavior of investing money at risk free rate of 10%. after 1 year this investment will give us 20.45*1.1=22.5 let what does this risk less portfolio gives us after 1 year

if stock price goes up = .75*50-15=22.5

if Stock price goes down = .75*30-0=22.5

see ! it does provide the same values irrespective of stock and call movement, that is why it is called riskless portfolio.

part 4

we have calculated that the theoretical call price should be = 9.55

if the running price is 15 then we can say that call overvalued, and we should go short on this call,and buy stocks by borrowing the$15mount basically we will create a riskless portfolio

our pay off now would be

we get 15$ by selling call now

we paid 0.75*40=30$ for stock

we borrowed 15$ at 10%

our bowwng will mature at 15*1.1 =16.5$

our portfolio will provide

outflow from call=50-35=15

inflow from selling stock = .75*50=37.5$

net inflow from riskless portfolio= 22.5$

outflow to repay borrowing = 16.5$

we gained = 22.5-16.5=6

if the price of the call is 5 then it is undervalued it should be bought and .75 stocks should be sold

we bought a call for $5 cashflow -5

we sold .75 stock for 40 30

we invested money at 10% -25

net investment now 0

after 1 year

call payoff 50-35 15

buying of stock .75*50 -37.5

investment matured at 25*1.1= 27.5

net gain = 5