Question

1. The current price of a stock is $80, and at the end of one year, its price will be either $88 or $72. The annual risk-free rate is 3.0% based on daily compounding. Based on the binominal model, what is the present value for a 1-year call option on this stock with an exercise price of $86.

2. Warren Corporation's stock sells for $40 per share. The company wants to sell some 10-year, semi annual coupon payment bond, at $1,000 par value. Each bond would have 40 warrants attached to it, each exercisable into one share of stock at an exercise price of $45. The firm's straight bonds yield to maturity is 10%. Each warrant is expected to have a market value of $5 that the stock sells for $42. What annual coupon rate must be set on the bonds in order to sell the bonds-with-warrants at par value?

Answer 1

1.)As per Binomial model,

Value of call option = Expected payoff / {(1+r)^t} or (e)^rt, in case of continuous compounding

Where, Expected payoff ={Payoff at high price(HP)* Probability of HP} + {Pay off at low price(LP) * Probability of LP)

''r'' is rate of interest i.e 3% compounded daily, ''t'' is number of time period i.e 1 year

''e'' is matheamatical constant(2.71828) used for calculation in case of continuous compounding

Value of call option = $1.30(refer working note)/(2.71828)^(0.03*1)

Value of call option = $1.30/1.03(as calculated in working note)

Value of call option = $1.26(rounded off to two decimals)

Working note: Calculation of expected payoff

Expected payoff ={Payoff at high price(HP)* Probability of HP} + {Pay off at low price(LP) * Probability of LP)

Where,(a) Pay off at high price = HP - Strike price = $88-$86 =$2

(b) Pay off at low price = LP - Strike price = $72-$86 = NIL, since price is lower than strike price, it will not be exercised and hence no payoff will arise

(c) Probablility of HP = {Spot price*(e)^rt - LP}/(HP-LP)

  Probablility of HP = {$80*(2.71828)^(0.03*1)-$72}/($88-$72)

Probablility of HP = {($80*1.03)-$72}/($88-$72), (2.71828*0.03 = 1.03, using calculator)

Probablility of HP = 0.65

(d) Probability of LP = 1 - Probability of HP

Probability of LP = 1-0.65 = 0.35

Expected payoff = ($2*0.65) + (0*0.35) = $1.3