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, suppose that a trader quotes a put price of $5. Then the arbitrage profit that you can make today by trading this call and related securities is:

Group of answer choices

$0.33

$0.59

$1.54

$0

please provide explanation

Answer 1

$0.59 is the arbitrage profit that you can make today.

The stock has equal probability (=0.5) of moving up or down

Consider a case in which the an arbitrageur by the out option with strike price =$105 for $5

Up-front payment = $5

If the stock moves to $115.63212672, the put option remains out-of-money and hence expires worthless

If the stock moves to $93.52995844, the put-option pays off ($105-$93.52995844) = $11.47 at maturity

Hence, expected payoff by the put option at maturity = Probability of up-move * Payoff at up-move +Probability of down-move * Payoff at down-move

Expected payoff by the put option at maturity = 0.5*0 + 0.5*11.47 = $5.735

The risk-free rate is 0.05 and time to maturity is 0.5 years

Present value of the Expected payoff by the put option at maturity = $5.735 * e^(-0.05*0.5)

Present value of the Expected payoff by the put option at maturity = $5.59

Up-front payment for ourchasing the put option = $5

Arbitrage profit today = $5.59 - $5 = $0.59