Question

Consider a stock currently trading at 25 that can go up or down by 15 percent per period. The risk-free rate is 10 percent.

Use one-period binomial model.

A. Determine the two possible stock prices for the next period.

B. Determine the intrinsic values at expiration of a European call with an exercise price of 25.

C. Find the value of the option today.

D. Construct a hedge by combining a position in stock with a position in the call. Calculate the hedge ratio and show that the return on the hedge portfolio is the risk-free rate regardless of the outcome, assuming that the call trading at the price obtained in part c.

E. Determine the rate of return from a risk-free hedge if the call is trading at 3.50 when the hedge is initiated. F. How can an investor use listed options to protect an equity portfolio from price risk?

Answer 1

Given

Stock trading price S0 = 25

Percentage change during period = 15%

Risk free rate rf = 10%

A.

If stock goes up by 15% u = 1+ 0.15 = 1.15

Stock price Su= S0*u = 25*(1.15) = $28.75

If stock goes down by 15% d = 1- 0.15 = 0.85

Stock price Sd= S0*d = 25*(0.85) =$21.25

Two possible stock prices are Su= $28.75 and Sd = $21.25

B.

Exercise price K = 25

Intrinsic values are

Cu = max (0, Su –K)

Cu= max(0, 1.15(25) – 25)

Cu= max(0, 3.75)

Cu= 3.75

Cd = max((0, Sd –K)

Cd= max(0, 0.85(25) – 25)

Cd= max(0, -3.75)

Cd= 0

C.

p = [(1 + r) – d]/ (u – d)

=[ (1+ 0.1) – 0.85]/ (1.15 – 0.85) = 0.83

1 – p = 1 - 0.8333 = 0.17

C = {(p *Cu) + [(1 – p) *Cd]} /(1 + r) = [(0.83 * 3.75) + (0.17 * 0)] / 1.10

C = 2.83

The value of option = $2.83

D.

Hedge h = = (Cu – Cd) ÷ (Su– Sd)

h = (Cu – Cd) / (Su– Sd)

=(3.75 – 0) / (28.75 – 21.25) = 0.50

So, hedge will be buy 500 shares and sell 1000 calls

Hedge ratio = 0.5

Current value of hedge portfolio = 500*25 – 1000 *2.83 = 9670

If stock goes up at $28.75

Value = 500*28.75 – 1000 (28.75-25) = 10,625

If stock goes down to $21.25

Value = 500 *21.25 – 1000* 0 = 10,625

Return on hedge portfolio = (10,625/ 9670) – 1= 0.098759

Hedge portfolio is to buy 500 shares and sell 1000 calls

Hedge ratio = 0.5

Return on hedge portfolio = 0.1