Finance Derivatives
Doris Chong
modelindependentoptionvaluation.pptxModel-Independent Option Valuation
Dr. Kurt Smith
Overview
Introduction
Boundaries
Vertical Spread
Butterfly Spread
Calendar Spread
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Finance (Derivative Securities)
Introduction
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Introduction
Model-independent means value relationships between different options on the same underlier that must hold to prevent arbitrage. These relationships must hold for every option pricing model.
The absence of vertical spread, butterfly spread and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier.
Option buyers have the right, not the obligation, to exercise the option at expiry (European) or anytime up to and including expiry (American). Expiry payoff diagrams for options can be obtained via simple rotations about the x- and y-axis.
•
•
•
Model-Independent Option Valuation; Dr. Kurt Smith
The focus in this lecture is on a single underlier with zero intermediate cash flows (e.g., no dividends). For simplicity, interest rates are assumed to be zero unless stated otherwise.
•
Finance (Derivative Securities)
Introduction
Model-Independent Option Valuation; Dr. Kurt Smith
Expiry Payoff
ST
K
European Call
Expiry payoff = MAX(ST – K, 0)=(ST – K)+
100
120
70
ST=120, K=100; then (ST – K)+=20
ST=70, K=100; then (ST – K)+=0
Examples:
Expiry Payoff
ST
K
55
110
30
European Put
Expiry payoff = MAX(K – ST, 0)=(K – ST)+
ST=110, K=55; then (K – ST)+=0
ST=30, K=55; then (K – ST )+=25
Examples:
Finance (Derivative Securities)
Introduction
Model-Independent Option Valuation; Dr. Kurt Smith
Long Call
Long Put
Short Put
Short Call
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Contract maturity (Tj).
Strike price (Ki).
Spot price (S0).
American exercise.
For European Call options Ci,j and European Put options Pi,j.
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Contract maturity (Tj):
At expiry:
Before expiry:
•
•
At the limit:
•
t=0
T
S0
Ki
T=0
S0,Ki
t=0
T=∞
S0
Ki
Call Price
S0
Ki
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Strike price (Ki):
Zero:
At the limit:
•
•
Call Price
S0
Ki
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Spot price (S0):
Zero:
•
Call Price
S0
Ki
•
At the limit:
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
American exercise (Amex.):
•
An American option has all of the features of a European option PLUS the ability to exercise early if it is in the buyer’s interest. Therefore, an American option cannot be worth less than a European option.
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Why is the value of a Call option non-negative (Ci,j ≥ 0) whereas the value of a forward contract Ft can be negative?
A Call option expiry payoff (ST - Ki)+ ≥ 0. Since there is no possibility of loss at T, the option value at t Ci,j ≥ 0. In contrast, the expiry payoff of a forward contract (ST - ft;S,T) is positive, negative, or zero.
Expiry Payoff
ST
Ki
Call Option
Expiry Payoff
ST
Forward Contract
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
European Put-Call parity for an underlier with no interim cash flows (e.g., no dividends): a forward contract and a synthetic forward contract created by options must have the same value.
Expiry Payoff
ST
f(t;S,T)=K
Buy at K thru long Call if ST > K
Buy at K thru short Put if ST < K
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Put options:
Maturity:
Strike:
•
•
Spot:
•
American:
•
Put Price
S0
Ki
Ki
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Bull spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0.
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
In general:
Example:
That is, cannot pay a negative amount today for a future payoff that at worst is zero.
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
| How? | Now t | Payoff at Expiry T | ||
| Portfolio | t | ST < 50 | 50 ≤ ST ≤ 55 | ST > 55 |
| Buy K2 = 55 | 12 | 0 | 0 | ST – 55 |
| Sell K1 = 50 | -18 | 0 | -(ST – 50) | -(ST – 50) |
| Sub-Total | -6 | 0 | 50 – ST | -5 |
| Lend Cash | 6 | ≥ 6 | ≥ 6 | ≥ 6 |
| Total | 0 | > 0 | > 0 | > 0 |
Therefore, pay zero today to get a guaranteed positive payoff in the future (Type 3 arbitrage violation). The trader will do this as many times as possible to pay a multiple of zero today to earn a multiple of a positive amount in the future.
Example: Let C(K1=50)=$18 and C(K2=55)=$12. Is there an arbitrage? If so, how would you exploit it?
Sell the bull spread. Why?
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Bear spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0.
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
In general:
Example:
That is, cannot pay a negative amount today for a future payoff that at worst is zero.
Finance (Derivative Securities)
Butterfly Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Butterfly Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
K3
If K2 - K1 = K3 - K2 then C(K1) – 2C(K2) + C(K3) must have a value greater than zero.
In general:
Example:
Butterfly spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0.
That is, cannot pay a negative amount today for a future payoff that at worst is zero.
Finance (Derivative Securities)
Butterfly Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Asymmetric butterflies
Symmetric butterfly
C(K=70)-1.11C(K=72)+0.11C(K=90)
C(K=70)-10C(K=88)+9C(K=90)
C(K=70)-2C(K=80)+C(K=90)
Finance (Derivative Securities)
Butterfly Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Example: Let C(K=70)=$7, C(K=80)=$6 and C(K=90)=$4. Is there an arbitrage? If so, how would you exploit it?
Hence, yes there is an arbitrage. Buy Call(K1=70), sell 2 Call(K2=80), buy Call(K3=90). The trader will receive $1 now [i.e., at t=0 will pay 7-2(6)+4=-$1 ]; and will have zero probability of loss in the future (refer to expiry payoff figure).
Finance (Derivative Securities)
Calendar Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Calendar Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Expiry Payoff
ST
Call Price
S0
Calendar spread: same strikes (Ki), different maturities (Tj). Also r=0 & div=0.
In general:
Example:
That is, cannot pay a negative amount today for a future payoff that at worst is zero.
Finance (Derivative Securities)
Calendar Spread
Model-Independent Option Valuation; Dr. Kurt Smith
Example: the price of a Call option expiring at T1 is $5 and T2 is $4, where T1 < T2. Is there an arbitrage? If so, how would you exploit it?
| Expiry Payoff at T2 | |||||
| ST2 < K | ST2 > K | ||||
| Now | Expiry Payoff at T1 | ||||
| Portfolio | t | ST1 < K | ST1 > K | ST1 < K | ST1 > K |
| Sell C(T1) | -5 | 0 | -(ST2-K) | 0 | -(ST2-K) |
| Buy C(T2) | 4 | 0 | 0 | ST2-K | ST2-K |
| Total | -1 | 0 | K-ST2 | ST2-K | 0 |
The trader receives $1 today (t) for non-negative expiry payoffs at T2 . This is a Type 2 arbitrage violation. Sell near (T1) and buy far (T2) maturity to extract the arbitrage profit.
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Finance (Derivative Securities)
Conclusion
Model-independent means value relationships between different options on the same underlier that must hold to prevent arbitrage. These relationships must hold for every option pricing model.
The absence of vertical spread, butterfly spread and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier.
Option buyers have the right, not the obligation, to exercise the option at expiry (European) or anytime up to and including expiry (American). Expiry payoff diagrams for options can be obtained via simple rotations about the x- and y-axis.
•
•
•
Model-Independent Option Valuation; Dr. Kurt Smith
Vertical Spread:
Butterfly Spread:
Calendar Spread:
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
Vertical Spread
Bull Spread
Bear Spread
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Expiry Payoff
Expiry Payoff
ST
ST
K1
K2
K3
Butterfly Spread
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Call Price
S0
Calendar Spread
,
0
,
iiji
KCSK
æö
ç÷
èø
+
"=-
,0
ij
C
>
,
ij
C
=¥
0
T
=
0
T
>
,
0
,
j
rT
iiji
KCSKe
æö
ç÷
ç÷
èø
+
-
"=-
j
rT
i
Ke
-
T
=¥
,
0
, lim
iij
T
KCS
®¥
"=
(
)
r
i
Ke
-¥
,0
ij
C
=
,
00
,
j
rT
jiji
TCSKeS
æö
ç÷
ç÷
èø
+
-
"=-=
0
K
=
K
=¥
,
, lim0
jij
K
TC
®¥
"=
0
K
=
0,0
ij
CS
=
=
0
0
S
=
0
0
S
=
,
0
,, 0
j
rT
ijiji
KTCSKe
æö
ç÷
ç÷
èø
+
-
"=-=
0
S
=¥
,
0
,,
ijij
KTCS
"=
.
,,
,, 0
Amex
ijijij
KTCC
"³³
,,
ijij
CP
=-
00
j
rT
i
FSKe
-
==-
,,
0
j
rT
ijiji
CPSKe
-
-=-
,,
0
j
rT
ijiji
PCSKe
-
\=-+
.
,,
,, 0
Amex
ijijij
KTPP
"³³
,
, lim
jij
K
TPK
®¥
"=
,
0
,,
jj
rTrT
ijijii
KTPKeSKe
æö
ç÷
ç÷
èø
+
--
"=-=
,
,, 0
ijij
KTP
"=
,
0
,
iiji
KPKS
æö
ç÷
èø
+
"=-
,
0
,
j
rT
iiji
KPKeS
æö
ç÷
ç÷
èø
+
-
"=-
,
, lim0
iij
T
KP
®¥
"=
,
0
, 0
j
rT
jiji
TPKeS
æö
ç÷
ç÷
èø
+
-
"=-=
12
0
CKCK
æöæö
ç÷ç÷
èøèø
-³
,
1,
0
ij
ij
CC
-
-³
,
1,1
iji
iji
CCKK
--
-£-
1221
CKCKKK
æöæö
ç÷ç÷
èøèø
-£-
,
1,1
Since 1812655550: arbitrage.
iji
iji
CCKK
--
>
-=-==-=-
21
0
PKPK
æöæö
ç÷ç÷
èøèø
-³
,
1,
0
ij
ij
PP
-
-³
,
1,1
iji
iji
PPKK
--
-£-
2121
PKPKKK
æöæö
ç÷ç÷
èøèø
-£-
111
,
1,1,
11
0
i
iii
ij
ijij
ii
ii
KKKK
CCC
KKKK
+--
-+
++
--
-+³
--
(
)
(
)
(
)
31
21
123
3232
0
KK
KK
CKCKCK
KKKK
-
-
-+³
--
00.511.522.5507090110130150Expiry PayoffExpiry Spot
K=70, 72, 90
024681012507090110130150Expiry PayoffExpiry Spot
K=70, 80, 90
02468101214161820507090110130150Expiry PayoffExpiry Spot
K=70, 88, 90
(
)
(
)
90708070
7641
90809080
--
-+=-
--
,
,1
0; ,0
ij
ij
CCij
+
-³³
,1
ij
C
+
,
ij
C
,0
ij
C
=
21
0
CTCT
æöæö
ç÷ç÷
èøèø
-³
(
)
(
)
1221
0
CKCKKK
£-£-
(
)
(
)
(
)
31
21
123
3232
0
KK
KK
CKCKCK
KKKK
-
-
-+³
--
(
)
(
)
21
0
CTCT
-³
