Question

Brigham, E. F., & Ehrhardt, M. C. (2017). Financial management: theory and practice. Australia: South-Western.

Chapter 8 Mini Case P. 370

d. Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 that expires in 6 months. The risk-free rate is 6%.

(1) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?

(2) Suppose you write one call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (i.e., a hedge portfolio)? What is the payoff of the portfolio?

(3) What is the present value of the hedge portfolio? What is the value of the call option?

(4) What is a replicating portfolio? What is arbitrage?

Answer 1

Ans.

1.

Information available in question

Current stock price, P =	$27.00
Risk-free rate, Rf =	6%
Strike price, X =	$25.00
Up factor for stock price, u =	1.41
Down factor for stock price, d =	0.71
Years to expiration, t =	0.50

Calculation of Payoffs

Strike price: X =	$25.00
Current stock price: P =	$27.00
Up factor for stock price: u =	1.41
Down factor for stock price: d =	0.71
Up option payoff: Cu = MAX[0,P(u)-X] =	$13.07
Down option payoff: Cd =MAX[0,P(d)-X] =	$0.00

Calculation of Ending Values of the stock Price

current Stock Price P = $27 Ending up" Stock Price = P(u) = $38.07 1 Option payots: Cu = Max [0, P(u) -x] Cu = $13.07 Ending"down" Stock Price - Pld) = $ 1917 Option payoff: Cd= max [0, Pld)- x) cd= $0.00

2.

Number of shares buy to create a portfolio with a riskless payoff is calculated as

Ns = Cu - Cd / P(u-d) = 13.07 - 0 / 27(1.41-0.71) = 13.07 / 18.90 = 0.69153

The Hedge Portfolio with Riskless Payoffs

Strike price: X =	$25.00
Current stock price: P =	$27.00
Up factor for stock price: u =	1.41
Down factor for stock price: d =	0.71
Up option payoff: Cu = MAX[0,P(u)-X] =	$13.07
Down option payoff: Cd =MAX[0,P(d)-X] =	$0.00
Number of shares of stock in portfolio: Ns = (Cu - Cd) / P(u-d) =	0.69153

Current stock Price Stock Price =P() = $38.07 7 Portfolio's stock payoff - P(U) (Ns) - $26-33 Less: option's payoff Cu = 3 = $13.07 Portfolio's net payoff = P(uNs)- Cu = $13.26 P - $27 V Stock Price = P(d) = $19.17 Portfolio's Stock payoff= P(a) (N5) = $13-24 Less: option's payoff: cd = $0.00 Portfolio's net payoff = P(d) (NS)-Cd = $13.26

3.

The present value of the riskless payoff discounted at the risk-free rate ( assuming daily compounding) is:

Pv of payoff = Payoff / (1+ Rf/365)^365*t = $ 13.2567/1.03045 = $ 12.865

The current value of hedge portfolio is determine by deducting call value (Vc) from the stock value (Ns*P). Here, this portfolio has riskless payoff so its value is equal to to the present value of the riskless payoff discounted at the risk-free rate.

So, the value of call option is calculated as

Vc = Ns (P) - Present value of riskless payoff

Vc = 0.69153 ($27) - $ 12.865 = $ 5.81

4.

If we borrow an amount equal to the present value of the riskless payoff and buy Ns shares of stock then the payoffs of this portfolio replicate the payoffs of the call option.

Ns =	0.69153
Amount borrowed = PV of riskless payoff =	$12.86
Repayment of riskless payoff =	$13.26

Payoff if stock is up

Stock price =	$38.07
Value of stock in portfolio =	$26.33
Less repayment of borrowing =	$13.26
Net payoff of portfolio =	$13.07

Payoff if stock is down

Stock price =	$19.17
Value of stock in portfolio =	$13.26
Less repayment of borrowing =	$13.26
Net payoff of portfolio =	$0.00

So ,we observe that these are the same payoffs as that of the option payoffs.