. You are given the following two sets of prices of European options as a function of the strike price, for a stock with S = 100. Assume that all options mature in 6 months,
and that the interest rate (continuously compounding, annualized) is 10%.
(1) p(90) = 4; p(100) = 9 1/8 ; p(110) = 16; p(120) = 25 3/4
(2) p(90) = 2 ¾ ; p(100) = 81/2 ; p(110) = 17; p(120) = 24
For each set of prices, please answer the following questions:
(a) Assume that the stock will not pay any dividend in the next 6 months. Do
these prices satisfy arbitrage restrictions on options values? If yes, prove it. If
not, construct an arbitrage portfolio to realize riskless pro_ts and show how that
portfolio performs whatever the underlying price does
To determine if the given prices satisfy the arbitrage restrictions on options values, we need to check if there are any opportunities for riskless profits (arbitrage) based on the prices.
The arbitrage restrictions on options values state that the prices of options should be consistent with the principle of no-riskless-profit. This means that there should not be any combination of buying and selling options that guarantees a riskless profit without any investment.
To determine if the prices satisfy the arbitrage restrictions, we can use the put-call parity relationship for European options. The put-call parity states:
C - P = S - K * e^(-r*t)
Where:
C = Price of the call option
P = Price of the put option
S = Current stock price
K = Strike price
r = Interest rate
t = Time to maturity
Let's now apply the put-call parity to the given prices in set (1):
p(90) = 4
p(100) = 9 1/8
p(110) = 16
p(120) = 25 3/4
Based on the given prices, we can calculate the corresponding call option prices using the put-call parity formula. Assuming the options are European options with a 6-month maturity and an interest rate of 10% (converted to continuous compounding), we have:
c(90) = p(90) + S - K * e^(-r*t)
c(100) = p(100) + S - K * e^(-r*t)
c(110) = p(110) + S - K * e^(-r*t)
c(120) = p(120) + S - K * e^(-r*t)
Using the given stock price (S = 100), strike prices (K = 90, 100, 110, 120), interest rate (r = 0.10), and time to maturity (t = 0.5), we can calculate the call option prices:
c(90) = 4 + 100 - 90 * e^(-0.10 * 0.5) ≈ 23.71
c(100) = 9 1/8 + 100 - 100 * e^(-0.10 * 0.5) ≈ 10.19
c(110) = 16 + 100 - 110 * e^(-0.10 * 0.5) ≈ 7.83
c(120) = 25 3/4 + 100 - 120 * e^(-0.10 * 0.5) ≈ 8.33
Now let's compare the calculated call option prices with the given call option prices in set (1):
c(90) ≈ 23.71 ≠ p(90)
c(100) ≈ 10.19 ≠ p(100)
c(110) ≈ 7.83 ≠ p(110)
c(120) ≈ 8.33 ≠ p(120)
We can see that the calculated call option prices are not equal to the given put option prices. This implies that there are arbitrage opportunities in this scenario.
To construct an arbitrage portfolio, we can take advantage of the mispricing by simultaneously buying the cheaper option (in this case, the put option) and selling the relatively more expensive option (the call option). By doing this, we can potentially earn a riskless profit.
For example, if we consider the option with a strike price of 90:
- Buy the put option at a price of 4
- Sell the call option at a price of 23.71
This will result in a net cash flow of 23.71 - 4 = 19.71. Regardless of the underlying stock price at expiration, the maximum profit we can earn is 19.71.
By constructing this arbitrage portfolio, we have ensured a riskless profit because the net cash flow at the start of the strategy is positive and no future investment is required. This violates the no-riskless-profit principle, indicating that these prices do not satisfy the arbitrage restrictions.