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Lecture 8

Lecture 8.docx

6 Pages
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Department
Administrative Studies
Course Code
ADMS 4503
Professor
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 12.94 = 7.06
Since its overpriced, we :
At time T
Arbitrage
St K
St K
- Short put
+20
- (K ST)
ST 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 todays profit is 2.94
So its underpriced.
At time T
Arbitrage
St K
- Long put
-10
(K 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 ST) + 0 + ST = 0 - (ST 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 doesn’t pay any dividend or bonds that don’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 – 12.94 = 7.06 Since it’s overpriced, we : At time T Arbitrage St≤ K S t K - Short put +20 - (K – 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 today’s profit is 2.94 So it’s underpriced. At time T Arbitrage St≤ K St≥ K - Long put -10 (K – ST) 0 - Short call +40 0 - (ST– 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 – T ) + 0 + T = 0 - (T – 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 doesn’t pay any dividend or bonds that don’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 It’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 – S --0--------------------------------------------------------------------------------------
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