Question

YBM’s stock price S is $102 today. — After six months, the stock price can either go up to $115.63212672, or go down to $93.52995844. — Options mature after T = 6 months and have an exercise price of K = $105. — The continuously compounded risk-free interest rate r is 5 percent per year. Given the above data, the hedge ratio and the put option’s value are given by:

Group of answer choices

0.5190 for the hedge ratio and $5.59 for the put option’s value

-0.5190 for the hedge ratio and $5.59 for the put option’s value

0.2523 for the hedge ratio and $4.81 for the put option’s value

-0.2523 for the hedge ratio and $2.35 for the put option’s value

please provide explanation

Answer 1

Current share price = $102

After 6 months, there are 2 possibilities

1. Share price going up to $115.63212672
2. Share price going down to $93.52995844

Exercise Price of Option = $105

Payoff from put option when share price is $115.63212672 after 6 months
Since Share price is more than Exercise Price, put option will lapse and payoff will be 0.

Payoff from put option when share price is $$93.52995844 after 6 months.
Since share price is less than exercise price, put option will exercise in the money.
Payoff from put option = Exercise Price – Share Price
                                             = (105-93.52995844)
                                             = 11.47004156

Assuming probability of up and down move is 50% each.
Expected Payoff from Put Option after 6 months = 0.50 * 21 + 0.50 * 11.47004156
   = $5.73502078

PV at Y0 = $5.73502078/ert
= $5.73502078/e(.05*(6/12)
= $5.73502078/e.025
   = $5.73502078/1.0253
= $ 5.59 (rounded off)

So Value of put option = $5.59

Hedge Ratio is given by the formula

Where

Pu and Pd are value of put option at higher and lower share price respectively and Su and Sd are higher and lower share price respectively.