Question

The price of Ervin Corp. stock will either be $64 or $82 at the end of the year. Call options are available with one year until expiration. Continuously compounded T-bills currently yield 4.59 %. Suppose the current price of Ervin stock is $73.

What is the value of the call if the strike price is $55 per share?

Answer 1

Given the following data,

Current price of Ervin stock = $73

Ervin corp stock will either move up at $82 or move down at $64, i.e.,

Upward move U = High price/Current price = 82/73 = 1.1233

Downward move D = Lower price/ Current price = 64/73 = 0.8767

Risk neutral probability for the upward move is given by the following formula,

P = (e^(risk free rate*time period)-D)/(U-D)

Where

Risk free rate = 4.59% = 0.0459

time period = 1 year

D = 0.8767

U = 1.1233

Substituting these values in the above formula,

P = (e^(0.0459*1)-0.8767)/(1.1233-0.8767)

P = (e^(0.0459)-0.8767)/(0.2466)

P = (1.04697-0.8767)/(0.2466)

P = (0.1703)/(0.2466)

P = 0.6904

Now

call option payoff at high price = MAX(High price-strike price,0)

= MAX(82-55,0)

= MAX(27,0)

= 27

call option payoff at low price = MAX(Low price-strike price,0)

= MAX(64-55,0)

= MAX(9,0)

= 9

Price of the call option is given by the formula,

Price of call option = e^(-r*t)*(P*Payoff H+(1-P)*Payoff L)

Where

r = risk free rate = 4.59% = 0.049

t = 1

P = 0.6904

Payoff at High price = 27

Payoff at low price = 9

Substituting these,

e^(-0.049*1)*(0.6904*27+(1-0.6904)*9)

e^(-0.049)*(18.6408+(0.3096)*9)

e^(-0.049)*(18.6408+2.7864)

0.9522*21.4272

20.4

Thus the value of the call is 20.4