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 call price of $6. Then the arbitrage profit that you can make today by trading this call and related securities is:

Group of answer choices

$1.22

$0.81

$0

$0.25

please provide explantion

Answer 1

Answer: 0.81

Current Stock Price (So) = $ 102

Risk free Rate (r) = 5% per annum continously compounded

Expected Price in 6 Months

S(upward) = $ 115.63212672

S(downward) = $ 93.52995844

Exercise Price = $ 105

Risk Neutralisation Model:

Fair Future Price = So * e^rt

= $ 102* e^0.05*(6/12)

= $ 102* e^0.025

=102*1.02532

= $ 104.58

Let the Probability of attaining Upward price at the time of Expiry = "P"

Then,

($ 115.63212672 * P) + ($ 93.52995844 * (1 - P)) = $ 104.58

($ 115.63212672 - $ 93.52995844 P) = $ 104.58 - $ 93.52995844

22.1022 P = $ 11.05

P = 0.5 (approx)

Therefore P(Downward) = 1- 0.5

P(Downward) = 0.5

Therefore, Price of Call Option =

= [(0.50 * ($ 115.63212672 - 105)) + (0.5 * 0)] / e^0.05*(6/12)

= $ 5.32 / 1.02532

Fair Price of Call Option = $ 5.19

Traders Quote of Call Option = $ 6

Traders Quote is Overpriced.

Arbitrage Profit = $ 6 - $ 5.19

Arbitrage Profit = $ 0.81 (approx)