Question

The FXI (iShares China Large-Cap ETF) has a current spot price of $40.

Assume the risk-free rate is 5%. Over the next year, the ETF could either move up to $64 (+60%) or move down to $25 (-37.5%). Consider a one-year call with a $45 strike (out of the money).

Calculate the risk-neutral probability (q), write the formula relating premium to the risk neutral probability, and solve it to calculate the premium on the call.

Answer 1

Ans) The information given is summerize as below

Spot price or market price = $ 40

Risk free rate of return = 5%

Rise in price up to $ 64 (+60%)

Fall in price up to -37.5% to $ 25

Strike price / Excercise price = $45

We know that if the stock rises than it is worth $ 24 ( 64-40)

Whe it falls it is worth 0.

Since the investor is risk neutral , hence the expected return is same as the risk-free return because he is indifferent to risk. Hence the probability of rise and fall are calculated as below:

Expected return / risk free return in case of risk neutal investor = Probability of rise* rise percent +[ (1-probability of rise )*fall percent]

Putting in formula we get,

0.05 = Probability of rise *0.60 +[ ( 1-probability of rise )*-0.375

0.05 = 0.60 Probability of rise -0.375+0.375 Probability

0.05 = 0.975 Probability of Rise  -0.375

0.05+0.375 = 0.975 Probability of Rise

0.425 = 0.975 Probability of Rise

    0.425 /0.975 = Probability of Rise

0.435 = Probaility of Rise

Probability of fall = 1-0.435 = 0.565

The call premium of price of an optio is calculated as below :

The premium on call = Expected future value of the option / (1+r)^n -------eq 1

Expected future value = Probability of rise * rise value +probability of fall * fall value

Expected Future value = 0.435*24 +0.565 *0

Expected Future value = 10.44

Premium on call as per above ------eq 1, we get

Premium on call = 10.44/(1+0.05)

= 10.44/1.05 = $ 9.94