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Lecture 9

# Lecture 9.docx

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Nabil Tahani

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LECTURE 9
Note: all black lines are initial positions, all red lines are the second step
Homework:
S = 100, K = 100 Div = 4 (Time = 0.5) r = 4% T = 1 cmkt = 40, pmkt = 20
Note that :
C + Ke-rT â‰  S0 â€“ PV(Div) + p
Call put parity is not satisfied , so we can arbitrage
What we do is :
C + Ke-rT = 136.0789
S0 â€“ PV(Div) + p = 116.0792
So we short expensive side and go long on the cheaper side: ( long put + short call + long stock = k at
maturity )
In one year
Long put
-20
K = 100
Short call
+40
Long stock
-100
borrow
+80
80e4%x1 = -83.2649
Sub total
0
dividend
4e4% x 0.5 = 4.08
Total
20.82
We can find a rough estimate of how much we can make from the arbitrage = 136.0789 â€“ 116.0792 =
19.9997
Now, if we discount the value 20.82 , we will get 19.9997
For a butterfly effect:
-c1 â€“c3 + 2c2 < 0
So that c2 < (c1 + c3)/2
Max profit : K2 â€“ K1 â€“ ( c1 + c3 -2c2
Max loss : - (c1 +c3 - 2c2)
Profit
ST â€“ K1
K3 - ST
Cost
- (c1 +c3 - 2c2)
- (c1 +c3 - 2c2)
Pay off = profit â€“cost = 0,
We shall take breakeven point=
X
X = K1 + cost
X = K3 - cost
BE1 = K1 + (c1 +c3 - 2c2)
BE2 = K3 â€“ (c1 +c3 - 2c2)
Players buying such butterfly would be a person who feels that the stock would be somewhat at balance
and constant around K2, it`s a low volatility gain but still in the spread with limited upper and lower
bounds
Players going short on butterfly would be a person who feels that the market is moving.
Combinations
Long call K -c
Long put K p
-------------------------------
-( c + p) which would be less than 0, also note that k would be ``at the money``
ST â‰¤ K
ST â‰¥ K
0
ST â€“K
K â€“ St
0
K â€“ St â€“(c+p)
St â€“ K â€“ (c+p)
K
K â€“ (c+p)
K
BE1 BE2
-(c+p)
Note:
You might not wanna engage in such a combination ( straddle) just when the financial results are
announced.
Person engaging in this take on a highly risky investment .
K1 â€“(c+p)
K1 K2
BE1 K BE2
-(c+p)
You would have to wait longer, but you pay less. ( note: we put the blue line showing the original
strategy as seen in the first diagram, so as to show you that since we chose an out of the money strategy
here, the time period between the BE and K s are longer, you would have to wait longer than usual, but
unlike the first strategy, since these are out of the money, you would pay lesser cost for the call and put
this time)
Also note that the region between the K1 and K2 is flat because , as we are out of the money, we get
nothing.
When you go extreme bearish or extreme bullish- you take on double position in the one you believe
in:
Strip:
Long call K
-c
2 long puts K
-2p
-(c+2P) < 0 , this is debit

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Description
LECTURE 9 Note: all black lines are initial positions, all red lines are the second step Homework: S = 100, K = 100 Div = 4 (Time = 0.5) r = 4% T = 1 c mkt= 40, p mkt= 20 Note that : C + Ke â‰  S â€“ 0V(Div) + p Call put parity is not satisfied , so we can arbitrage What we do is : C + Ke = 136.0789 S0â€“ PV(Div) + p = 116.0792 So we short expensive side and go long on the cheaper side: ( long put + short call + long stock = k at maturity ) In one year Long put -20 K = 100 Short call +40 Long stock -100 4%x1 borrow +80 80e = -83.2649 Sub total 0 4% x 0.5 dividend 4e = 4.08 Total 20.82 We can find a rough estimate of how much we can make from the arbitrage = 136.0789 â€“ 116.0792 = 19.9997 Now, if we discount the value 20.82 , we will get 19.9997 For a butterfly effect: -c1â€“c 3 2c <20 So that c2< (c 1 c 3/2 Max profit : K 2 K 1 ( c 1 c -3c 2 Max loss : - (c +c - 2c ) 1 3 2 Profit STâ€“ K 1 K 3 S T Cost - (1 +c3- 2c 2 - (1 +c3- 2c 2 Pay off = profit â€“cost = 0, X = K1+ cost X = K3- cost We shall take breakeven point= X BE 1 K +1(c +1 - 3c ) 2 BE = K â€“ (c +c - 2c ) 2 3 1 3 2 Players buying such butterfly would be a person who feels that the stock would be somewhat at balance and constant around K , i2`s a low volatility gain but still in the spread with limited upper and lower bounds Players going short on butterfly would be a person who feels that the market is moving. Combinations Long call K -c Long put K p ------------------------------- -( c + p) which would be less than 0, also note that k would be ``at the money`` STâ‰¤ K S T K 0 S TK K â€“ St 0 K â€“ Stâ€“(c+p) S t K â€“ (c+p) K K â€“ (c+p) K BE 1 BE2 -(c+p) Note: You might not wanna engage in such a combination ( straddle) just when the financial results are announced. Person engaging in this take on a highly risky investment . K â€“(c+p) 1 K1 K 2 BE 1 K BE2 -(c+p) You would have to wait longer, but you pay less. ( note: we put the blue line showing the original strategy as seen in the first diagram, so as to show you that since we chose an out of the money strategy here, the time period between the BE and K s are longer, you would have to wait longer than usual, but unlike the first strategy, since these are out of the money, you would pay lesser cost for the call and put this time) Also note that the region between the K an1 K is 2lat because , as we are out of the money, we get nothing. When you go extreme bearish or extreme bullish- you take on double position in the one you believe in: Strip: Long call K -c 2 long puts K -2p -(c+2P) < 0 , this is debit S â‰¤ K S â‰¥ K T T Payoffs 0 S TK 2(K â€“ St) 0 Profit 2(K â€“ St)â€“(c+2p) S t K â€“ (c+2p) 2K 2K â€“ (c+2p) BE 1 K BE 2 -(c+2p) BE 1 K â€“ Â½ (c+2p) BE 2 K + (c + 2p) Formulation : We take the payoff and equate it to zero so that 2K -2S T (c+ 2p) = 0 So that S T K â€“ Â½ ( c+2p) When you take double position on the put, you are making quicker money on the put side than on the call Example of a diagonal spread: Here not only the strike prices are different , but the maturities are also different. ï‚· Long call K 1 80 , T= 6 months ï‚· Short call K = 100, T = 9 months 2 Note that the more the duration, the more curvish it is. So that it looks like this. Payoff for call K1 K2 The 9 months duration this is the initial put pay off had the duration Would make it the payoff for put been 6 mon
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