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Lecture 8

# Lecture 8.docx

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Nabil Tahani

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Put call parity
c + Ke-rT = p + S0
Example :
S0 = 200, K = 180 r = 4% T = 1 year c = 40
a. What is p
We get p = c + Ke-rT + S0
P = 12.94
b. Assume pmkt = 20, now there is a chance of arbitrage.
Without any complicated calculations, the profit is 20 â€“ 12.94 = 7.06
Since itâ€™s overpriced, we :
At time T
Arbitrage
St â‰¤ K
St â‰¥ K
- Short put
+20
- (K â€“ ST)
ST â€“ k
- Long call
-40
0
-ST
- Short stock
+200
-ST
-k
- Invest
-180
+180 e4%
+180 e4%
-
180 e4% - 180
180 e4% - 180
Which is = 7.35
Note that the pay off at the end of everything : short put , long call and short stock = -k
- 7.35e-4% = 7.06
c. Assume pmkt = 10 so that todayâ€™s profit is 2.94
So itâ€™s underpriced.
At time T
Arbitrage
St â‰¤ K
- Long put
-10
(K â€“ ST)
- Short call
+40
0
- Long stock
-200
+ ST
- borrow
+170
-170e4%
= 3.06
Note that the pay off at the end of everything : long put, short call, long stock, the pay off is K
: (K â€“ ST) + 0 + ST = 0 - (ST â€“ K) + ST = K
- so that 3.06e-4% = 2.94
Homework:
S = 100, K = 100 Div = 4 (Time = 0.5) r = 4% T = 1 cmkt = 40, pmkt = 20
d. Assume the stock price from the market S0 = 220 ( fair price = 200)
This means its over priced.
At time T
Short put
+12.94
-K
Long call
-40
Short stock
+220
invest
192.94
192.94e4%
= 20.81
Note that because no one is there in the market to trade with me at 200 since the market price is 220,
we need to create a synthetic long position by short put long call and investment ( which is again a long
position in the T bill)
American options
American call Vs European call
Assumption : we assume that the asset does not pay any income during the life of the options.
For E.g. Stock that doesnâ€™t pay any dividend or bonds that donâ€™t pay any interest.
Result : American call must never be excercised before the maturity date.
Scenarios:
0------------------------------ t -------------------------------------------T
Exercise date : pay k , receive stock
Now we keep stock beyond T
If K = 100, and price goes up to 300
You excercising at t , you make 200.
But the person excercising at T or beyond , is making more than 200, because the 100 paid is less
than the 100 you paid earlier.
When price falls to 10, again you lose 90 because you excercised it at t ,
While the person at time T would not exercise at all and therefore, makes 0 profit/ loss

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Description
Put call parity -rT c + Ke = p + S 0 Example : S0= 200, K = 180 r = 4% T = 1 year c = 40 a. What is p We get p = c + Ke + ST 0 P = 12.94 b. Assume p mkt= 20, now there is a chance of arbitrage. Without any complicated calculations, the profit is 20 â€“ 12.94 = 7.06 Since itâ€™s overpriced, we : At time T Arbitrage Stâ‰¤ K S t K - Short put +20 - (K â€“ ST) S T k - Long call -40 0 -ST - Short stock +200 -ST -k 4% 4% - Invest -180 +180 e +180 e - 180 e - 180 180 e - 180 Which is = 7.35 Note that the pay off at the end of everything : short put , long call and short stock = -k -4% - 7.35e = 7.06 c. Assume p mkt= 10 so that todayâ€™s profit is 2.94 So itâ€™s underpriced. At time T Arbitrage Stâ‰¤ K Stâ‰¥ K - Long put -10 (K â€“ ST) 0 - Short call +40 0 - (STâ€“ K) - Long stock -200 + S +S T 4% T 4% - borrow +170 -170e -170e = 3.06 = 3.06 Note that the pay off at the end of everything : long put, short call, long stock, the pay off is K : (K â€“ T ) + 0 + T = 0 - (T â€“ K) + T = K - so that 3.06e -4%= 2.94 Homework: S = 100, K = 100 Div = 4 (Time = 0.5) r = 4% T = 1 c mkt= 40, p mkt= 20 d. Assume the stock price from the market S = 200 ( fair price = 200) This means its over priced. At time T Short put +12.94 -K Long call -40 Short stock +220 4% invest 192.94 192.94e = 20.81 Note that because no one is there in the market to trade with me at 200 since the market price is 220, we need to create a synthetic long position by short put long call and investment ( which is again a long position in the T bill) American options American call Vs European call Assumption : we assume that the asset does not pay any income during the life of the options. For E.g. Stock that doesnâ€™t pay any dividend or bonds that donâ€™t pay any interest. Result : American call must never be excercised before the maturity date. Scenarios: 0------------------------------ t -------------------------------------------T Exercise date : pay k , receive stock Now we keep stock beyond T If K = 100, and price goes up to 300 You excercising at t , you make 200. But the person excercising at T or beyond , is making more than 200, because the 100 paid is less than the 100 you paid earlier. When price falls to 10, again you lose 90 because you excercised it at t , While the person at time T would not exercise at all and therefore, makes 0 profit/ loss The intrinsic value S T K < C t Itâ€™s not optimal to exercise instead sell the call. American put Intrinsic (K) value American put price K = 100 Note that the American put price can go on the line of K , but not below ( following the arbitrage rule). However the European put price has to below American put price, but it can go below the K line and this would not go against the arbitrage rule. So that European put price -rT Ke â€“ S --0--------------------------------------------------------------------------------------
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