Some Exotic Options

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FINA 6220Professor Andrew Chen

• Some Exotic Options
• Lecture Note 6

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Outline

• Lookback Options
• Options on Average
• Barrier Options (Knock-out and Knock-in Options)
• Forward Start at-the-money Options
• Chooser Options
• Compound Options

AC 62
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Pricing of Some Exotic Options (Binomial OPM)

• The Basic Model
• n 2
• S 100
• K 100
• u 1.2
• d 0.8333
• T 6-month
• ?t 0.5
• rf 20
• p 0.7415
• 1 p 0.2585

Prices of a stock over a 2-period lattice
144
(Suu)
120
100
100
(Sud)
83.33
69.44
(Sdd)
AC 63
4
Pricing of Some Exotic Options (Binomial OPM)

• At-the-Money Call (Put) Prices

44
(0)
29.52
(0)
0
19.79
(0)
(1.67)
0
0
(7.15)
(30.56)
AC 64
5
Lookback Options

• Call on Minimum (changing K)
• COMin Max ST Min(S0,S1,ST),0
• Call on Maximum (constant K)
• COMax Max Max(S0,S1,ST) - K,0
• Put on Maximum (changing K)
• POMax Max Max(S0,S1,ST) ST,0
• Put on Minimum (constant K)
• POMin Max K Min(S0,S1,ST),0

AC 65
6
Pricing Lookback Call Options on the Minimum
Suu 144
Min 100
Cuu 44
Su 120
Cu 29.52
Sud 100
Min 100
S 100
Cud 0
C 22.42
Sdu 100
Min 83.33
Sd 83.33
Cdu 16.67
Cd 11.18
Sdd 69.44
Min 69.44
Cdd 0
AC 66
7
Pricing Lookback Put Options on the Maximum
Suu 144
Max 144
Puu 0
Su 120
Pu 4.68
Sud 100
Max 120
S 100
Pud 20
P 4.81
Sdu 100
Max 100
Sd 83.33
Pdu 0
Pd 7.15
Sdd 69.44
Max 100
Pdd 30.56
AC 67
8
Options on Average

• Call on Average Strike
• Call (Avg K) Max ST Avg(S0,S1,ST),0
• Put on Average Strike 
• Put (Avg K) Max Avg(S0,S1,ST) ST,0

AC 68
9
Lookback Call Options on the Average Strike
Suu 144
Avg 121.33
Cuu 22.67
Su 120
Cu 15.21
Sud 100
Avg 106.67
S 100
Cud 0
C 11.07
Sdu 100
Avg 94.44
Sd 83.33
Cdu 5.56
Cd 3.73
Sdd 69.44
Avg 84.26
Cdd 0
AC 69
10
Lookback Options on the Average Strike with a
Delayed Start
Suu 144
Avg 132
Cuu 12
Su 120
Cu 8.05
Sud 100
Avg 110
S 100
Cud 0
C 6.71
Sdu 100
Avg 91.67
Sd 83.33
Cdu 8.33
Cd 5.59
Sdd 69.44
Avg 76.39
Cdd 0
AC 610
11
Options on Average Price (Asian Options)

• Asian Call Avg(S0,S1,ST) K,0
• Asian Put K Avg(S0,S1,ST),0

AC 611
12
Pricing Average-Price Call Options
Suu 144
Avg 121.33
Cuu 21.33
Su 120
Cu 15.87
Sud 100
Avg 106.67
S 100
Cud 6.67
C 10.64
Sdu 100
Avg 94.44
Sd 83.33
Cdu 0
Cd 0
Sdd 69.44
Avg 84.26
Cdd 0
AC 612
13
Barrier Options

• Knocked-In Options Options come to life if
  certain price level is reached.
• Up-and-In Options
• Down-and-In Options
• Knocked-Out Options Options are extinguished if
  certain price level is reached.
• Up-and-Out Options
• Down-and-Out Options
• Immediate Rebate Barrier Options
• Baseball Option
• A regular call option that is knocked out
  (worthless) if the stock price falls below a
  barrier on three separate occasions prior to
  expiration.

AC 613
14
Valuing A Barrier Option
Suu 144
Su 120
Cu 18
Sud 100
S 100
C 12.08
Sdu 100
Sd 83.33
Cdu 0
Cd 0
Sdd 69.44
Cdd 0
AC 614
15
Forward Start (Delayed Option)

• (Start at-the-money at t1)

Suu 144
Cuu 24
K 120
Su 120
Sud 100
Cu 16.10
S 100
Cud 0
C 13.42
Sdu 100
K 83.33
Cdu 16.67
Sd 83.33
Cd 11.18
Sdd 69.44
Cdd 0
AC 615
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Forward Start (Delayed Option)

• Cu (0.7415 x 24) x e-0.10 16.10
• Cd (0.7415 x 16.67) x e-0.10 11.18
• C (0.7415 x 16.10 0.2585 x 11.18) x 0.9048
• 13.42

AC 616
17
Chooser (or As You Like It) Options

• (Choose Call or Put at t1)

C100 P100
44 0
Suu 144
Su 120
Cu 29.52
Pu 0
0 0
Sud 100
S 100
C 21.48
Sd 83.33
0 0
Sdu 100
Cd 0
Pd 7.15
0 30.56
Sdd 69.44
AC 617
18
Chooser (or As You Like It) Options

• C(Chooser) 0.7415 x 29.52 0.2585 x 7.15 x
  0.9048
• 21.48
• At the choice date, tc, where t tc T, the
  chooser is
• Chooser (tc)

AC 618
19
Compound Options

• Call on put (COP)
• Call on call (COC)
• Put on call (POC)
• Put on put (POP)
• Terms
• Front fee the price of call (COP)
• Back fee the strike price of call (COP)
• To (t 1) the expiration date of overlying
• Tu (t 2) the expiration date of underlying
• Ko the strike price of overlying (call option)
• Ku the strike price of underlying (put option)

AC 619
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Compound Options Example
Ko 2.00
COP
Ku 100.00
Suu 144
Ko 20.00
COC
Ku 100.00
Su 120
Sud 100
S 100
Sud 100
Sd 83.33
Sdd 69.44
AC 620
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Compound Options Example
Ko 2.00
COP
Ku 100.00
Cuu 44
Ko 20.00
COC
Puu (0)
(9.52)
Ku 100.00
Cu 29.52
Pu (0)
Cud 0
C 19.67
Pud (0)
P (1.67)
Cd 0
Pd (7.15)
(5.15)
Cdd 0
Pdd (30.56)
AC 621
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Compound Options Example
Ko 2.00
COP
Ku 100.00
COC
Ko 20.00
Ku 100.00
COP (0.2585 x 5.15) x 0.9048 1.20
COC (0.7415 x 9.52) x 0.9048 6.39
AC 622
23
Binomial Compound Options Formulas

• (COC) p x Max(Cu-Ko,0) (1-p) x
  Max(Cd-Ko,0)e-r(?t)
• where
• Cu p x Cuu (1-p) x Cud e-r(?t)
• Cuu Max(Suu-Ku,0) and Cdd Max(Sud-Ku,0
• (COP) p x Max(Pu-Ko,0) (1-p) x
  Max(Pd-Ko,0)e-r(?t)

AC 623
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Binomial Compound Options Formulas

• (POC) p x Max(Ko-Cu,0) (1-p) x
  Max(Ko-Cd,0)e-r(?t)
• (POP) p x Max(Ko-Pu,0) (1-p) x
  Max(Ko-Pd,0)e-r(?t)

AC 624
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The Compound Option Call on call (COC)

• One-year Zero Coupon Bond
• Current Value of the firm
• V 100
• Promised Payments
• H 120
• ?t 0.5
• u 1.2
• d 0.833
• r 5
• p 0.523
• 1 p 0.477

AC 625
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The Compound Option Call on call (COC)
One-year Zero Coupon Bond
144
120
100
100
100
83.33
69.44
AC 626
27
The Compound Option Call on call (COC)
One-year Zero Coupon Bond
Cuu 24
Cu 12.24
Cud 0
6.24
Cd 0
Cdd 0
VE 6.24 and VD 93.76
AC 627
28
One-year Coupon Bond

• Face value 100
• Semi-annual coupon 10 K0
• Shareholder owns a compound option (COC)

AC 628
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One-year Coupon Bond
Cuu 34
Cu 17.34
3.75
Cud 0
(COC)
Cd 0
Cdd 0
AC 629
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One-year Coupon Bond

• COC p x Max(Cu-Ko,0) (1-p) x
  Max(Cd-Ko,0)e-r(?t)
• VE 3.75 and VD 96.25

AC 630