Question 1: Check for the Call Put parity for 3 different couples of options: at the money, in and out of the money. Comment.
-qT -rT Call Put Parity: C - P = S?- K?000
CCall option price 0:
PPut option price 0:
S: Index price 0
K: Strike price
q: Dividend yield
r: Risk free rate
We base our analysis on European options. In the case of American options, we may
-qT -qT -rTalso check the modified call-put parity inequality: S?? K ? C - P ?S?- K? 0 000
Method 1:
From the current index price 3874.54 and future price 3855.75 (mid price as of 19 Nov 2010), risk free rate of 0.256% (LIBOR as of 22 Oct 2010), we can back out dividend yield q:
(r-q)*TF = S? 00
(0.256% - q)*(28/360)3855.75 = 3874.54?
= > q = 0.065063 = 6.5063%
Then we plug r and q into equations below to test whether Call Put Parity holds for the 3 couples of options we choose. Currently the underlying index price is 3874.54, hence for at the money options we = > At the Money Option, we choose Call & Put options with strike 3875:
-q*(28/360)-r*(28/360) 71.1 – 90.4 = 3874.54? - 3875? (1)
Equation (1) is valid, which means the Call Put Parity holds for ATM options.
Another two couples of options:
Strike at 3800:
-q*(28/360)-r*(28/360)127 – 59.4 = 3874.54? - 3800? (2)
-q*(28/360)-r*(28/360)LHS = 127-59.4=67.6 ; RHS=3874.54? - 3800? = 55.74
Both sides of Equation (2) are not equal, which means the Call Put Parity does not hold. In reality, Call Put Parity may fail to hold due to transaction costs, unequal borrowing and lending rates, market inefficiencies, low liquidity in certain options, margin requirements, taxes and other imperfections. Thus, option prices may exhibit significant deviation from put-call parity without there being any arbitrage opportunity, or economically significant deviations. In our case, the bid-offer spread for call option is
quite large, which is 105-127, leading to higher costs for option investors
Strike at 3900:
-q*(28/360)-r*(28/360)58.9 – 103.7 = 3874.54? - 3900? (3)
Equation (3) is valid, which means the Call Put Parity holds.
Method 2:
Make use of Euribor 0.82% as of 22 Oct 2010 as the risk free rate. Use Excel Solver to get optimal dividend yield q. Set the bid-offer spreads for Call and Put prices as constraint, LHS equals RHS for all the 3 pairs of options.
Sample 1: Taking options at strike 3875 (ATM)
-qT -rT -0.0709*28/365-0.0082*28/365C = S?- K?+P = 3874.54 * e – 3875 * e + 90.4 00 0
= 71.36
Bid price for call is 70.0 (No arbitrage as profit negative). No profit can be made by creating a synthetic Call and selling it in the market (Loss = 70.0 – 71.36 = -1.36).
-qT -rT -0.0709*28/365-0.0082*28/365-C= -S?+ K?-P= -3874.54 * e + 3875 * e – 88.6 0 0 0
= -69.56
Offer price for call is 71.1 (No arbitrage as profit negative). No profit can be made by selling a synthetic Call and buying it in the market (Loss = 69.56 – 71.1 = -1.54).
-qT -rT -0.0709*28/365-0.0082*28/365P=C-S?+ K?= 71.1 - 3874.54 * e + 3875 * e 0 0 0
= 90.14
Bid price for put is 88.6 (No arbitrage as profit negative). No profit can be made by creating a synthetic Put and selling it in the market (Loss = 88.6 – 90.14 = -1.54).
-qT -rT -0.0709*28/365-0.0082*28/365-P=-C+S?- K?= -70 + 3874.54 * e - 3875 * e 0 0 0
= -89.04
Offer price for Put is 90.4 (No arbitrage as profit negative). No profit can be made by selling a synthetic Put and buying it in the market (Loss = 89.04 – 90.4 = -1.36).
Put Call Parity holds true for ATM option.
Sample 2: Taking options at strike 3850 (ITM for calls)
-qT -rT -0.0709*28/365-0.0082*28/365C = S?- K?+P = 3874.54 * e – 3850 * e + 78.8 00 0
= 84.74
Bid price for call is 83.4 (No arbitrage as profit negative). No profit can be made by creating a synthetic Call and selling it in the market (Loss = 83.4 – 84.74 = -1.34).
-qT -rT -0.0709*28/365-0.0082*28/365-C= -S?+ K?-P= -3874.54 * e + 3850 * e – 77.4 0 0 0
= -83.34
Offer price for call is 84.7 (No arbitrage as profit negative). No profit can be made by selling a synthetic Call and buying it in the market (Loss = 83.34 – 84.7 = -1.36).
-qT -rT -0.0709*28/365-0.0082*28/365P=C-S?+ K?= 84.7 - 3874.54 * e + 3850 * e 0 0 0
= 78.76
Bid price for put is 77.4 (No arbitrage as profit negative). No profit can be made by creating a synthetic Put and selling it in the market (Loss = 77.4 – 78.76 = -1.36).
-qT -rT -0.0709*28/365-0.0082*28/365-P=-C+S?- K?= -83.4 + 3874.54 * e - 3850 * e 0 0 0
= -77.46
Offer price for Put is 78.8 (No arbitrage as profit negative). No profit can be made by selling a synthetic Put and buying it in the market (Loss = 77.46 – 78.8 = -1.34).
Put Call Parity holds true for OTM (on call) option.
Sample 3: Taking options at strike 3900 (OTM for calls)
-qT -rT -0.0709*28/365-0.0082*28/365C = S?- K?+P = 3874.54 * e – 3900 * e + 103.7 00 0
= 59.67
Bid price for call is 57.9 (No arbitrage as profit negative). No profit can be made by creating a synthetic Call and selling it in the market (Loss = 57.9 – 59.67 = -1.77).
-qT -rT -0.0709*28/365-0.0082*28/365-C= -S?+ K?-P= -3874.54 * e + 3900 * e – 101 0 0 0
= -56.97
Offer price for call is 58.9 (No arbitrage as profit negative). No profit can be made by selling a synthetic Call and buying it in the market (Loss = 56.97 – 58.9 = -1.93).
-qT -rT -0.0709*28/365-0.0082*28/365P=C-S?+ K?= 58.9 - 3874.54 * e + 3900 * e 0 0 0
= 102.93
Bid price for put is 101 (No arbitrage as profit negative). No profit can be made by creating a synthetic Put and selling it in the market (Loss = 101 – 102.93 = -1.93).
-qT -rT -0.0709*28/365-0.0082*28/365-P=-C+S?- K?= -57.9 + 3874.54 * e - 3900 * e 0 0 0
= -101.93
Offer price for Put is 103.7 (No arbitrage as profit negative). No profit can be made by selling a synthetic Put and buying it in the market (Loss = 101.93 – 103.7 = -1.77).
Put Call Parity holds true for ITM (on call) option.
Conclusion is Put Call Parity hold true in market as shown by lack of arbitrage opportunities.
Question 2: Build a butterfly strategy with calls and one with puts that makes money if the market stays stable up to the maturity of the options.
The maturity of the options is 19 Nov 10, and the CAC 40 Index Futures Price for 19 Nov is 3855.5. If the market stays stable up to the maturity, in order to build up the butterfly strategy, we should pick the options which would be sold with strike price close to 3855.5.
Strategy with Calls:
Buy 2 Call options, one with strike K1, the other one with strike K3. Sell 2 Call option with strike K2.
K1 < K2 < K3 and K2 should be close to the underlying price 3855.5. Hence we buy Call option with strike 3825 at price 99.8. Buy Call option with strike 3875 at price 71.1. Sell 2 Call options with strike 3850 at price 83.4.
The total cost is 99.8 + 71.1 – 83.4 x 2 = 4.1
If the market is stable, at maturity date, the index price is 3855.5, the profit is (3855.5 – 3825) – (3855.5 – 3850) x 2 – 4.1 = 15.4
This is a gain of (15.4-4.1)/4.1=275.61% on the original amount invested. Strategy with Puts:
Buy 2 Put options, one with strike K1, the other one with strike K3. Sell 2 Put options with strike K2.
1 < K2 < K3 and K2 should be close to the underlying price 3855.5. K
Hence we buy Put option with strike 3800 at price 59.4. Buy Put option with strike 3900 at price 103.7. Sell 2 Put options with strike 3850 at price 77.4.
The total cost is 59.4 + 103.7 – 77.4 x 2 = 8.3
If the market is stable, at maturity date, the index price is 3855.5, the profit is (3900 – 3855.5) – 8.3 = 36.2
This is a gain of (36.2-8.3)/8.3=336.14%.
Question 3: What is the dividend yield that the market expects from the stock index for November?
From the current index price 3874.54 and future price 3855.75 (mid price as of 19 Nov 2010), risk free rate of 0.82% (EURIBOR as of 22 Oct 2010), we can back out dividend yield q:
(r-q)*TF = S? 00
(0.82% - q)*(28/365)3855.75 = 3874.54?
= > q = 0.07157 = 7.157%