Can someone advise if below answer is correct: Question: A stock price is currently $60. Over each of the next two six-month periods, it is expected to go up by 6% or down by 6%. The risk-free interest rate is 5% per year with semi-annual compounding.

Part I. Use the two-step binomial tree model to calculate the value of a one-year European put option with an exercise price of $61. (5 Marks)

Part II. Discuss how you can hedge risk when you initially write the put option? (4 Marks)

Part III. Assume six months have passed, discuss how you can hedge risk when you realize that the stock price is $63.6?

Answer:

PART I

The standard deviation, which is 50%, has been provided to us.

This is demonstrated by the fact that nature exists in either an upstate or a downstate.

It has the potential to increase or decrease by 6%, respectively.

The current price: $60 The standard deviation: 50% Interest rates (risk-free): 5%

Discounting periods: 2(12 months/6 months). (a) Assuming an upstate, the value will be computed as 0.5 x 6% = 3%

Since the discounting periods are two: effective rate is 3%/2 = 1.5% 60 x (1.015) ^2 =$61.8135 x 1.05 =$64.904175

(b) On the lower side, still the 3% valuation rate works. 60 x [(1.015) ^2 - 1] 60 x 0.030225

1.8135 Value: 60 1.8135. =58.1865 x 1.05 =61.095825

PART II

Value of put when stock price is $63.6

=(0.7083*0+0.2917*1.216)/1.025 = $0.346016

Value of put when stock price is $56.4

=(0.7083*1.216+0.2917*7.984)/1.025 =$3.112195

At t= 0, for a replicating portfolio of long position in shares and $B invested at risk free rate

*63.6+$B*1.025=0.346016

*56.4+$B*1.025=3.112195

Solving, = -0.384192 and B = 24.17619

Thus, a portfolio with short position in 0.384192 shares of stock and long position of $24.17619 in riskfree bonds will have the same payoff as a long put.

Thus if one writes a put option, one can hedge it by short position in 0.384192 shares of stock and long position of $24.17619 in riskfree bonds

PART III

If after 6 months the stock price is $63.6

At t= 1(after 6 months), for a replicating portfolio of long position in shares and $B invested at risk free rate

*67.416+$B*1.025=0

*59.784+$B*1.025=1.216

Solving, = -0.15933 and B = 10.4793

Thus, a portfolio with short position in 0.15933 shares of stock and long position of $10.4793 in riskfree bonds will have the same payoff as a long put.

Thus if one writes a put option, one can hedge it by short position in 0.15933 shares of stock and long position of $10.4793 in riskfree bonds