Real Option Valuation vs. DCF Valuation - An application to a North
Model-Independent Option Valuation Dr. Kurt Smith.
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Transcript of Model-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 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
Expi
ry P
ayoff
STK
European Call
Expiry payoff = MAX(ST – K, 0)=(ST – K)+
100 12070
ST=120, K=100; then (ST – K)+=20
ST=70, K=100; then (ST – K)+=0
Examples:
Expi
ry P
ayoff
STK55 11030
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 CallLong Put
Short Put Short Call
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: , 0, i i j iK C S K
Before expiry:
0T
0T , 0, jrTi i j iK C S K e
•
•
At the limit:• T , 0, limi i jTK C S
t=0 T
S0 Ki
T=0
S0,Ki
t=0 T=∞
S0 Ki
Call
Pric
e
S0KijrT
iK e r
iK e
, 0i jC , 0i jC ,i jC
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Strike price (Ki):
Zero: , 0 0, jrTj i j iT C S K e S
At the limit:
0K
K
•
• ,, lim 0j i jKT C
Call
Pric
e
S0Ki0K
, 0i jC 0, 0i jC S
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Spot price (S0):
Zero: 0 0S •
Call
Pric
e
S0Ki0 0S
, 0, , 0jrTi j i j iK T C S K e
• At the limit: 0S , 0, , i j i jK T C S
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
American exercise (Amex.):
• ., ,, , 0Amex
i j i j i jK T C C
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.
Expi
ry P
ayoff
STKi
Call Option
Expi
ry P
ayoff
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.
Expi
ry P
ayoff
ST
f(t;S,T)=K
Buy at K thru long Call if ST > K
Buy at K thru short Put if ST < K , ,i j i jC P
0 0jrT
iF S K e , , 0
jrTi j i j iC P S K e
, , 0jrT
i j i j iP C S K e
Finance (Derivative Securities)
Boundaries
Model-Independent Option Valuation; Dr. Kurt Smith
Put options:
., ,, , 0Amex
i j i j i jK T P P
Maturity:
Strike:
0T
0T
•
•
Spot:•
T
, 0, i i j iK P K S
, 0, jrTi i j iK P K e S
,, lim 0i i jTK P
, 0, 0jrTj i j iT P K e S
0K
K ,, limj i jKT P K
0 0S , 0, , j jrT rT
i j i j i iK T P K e S K e
0S ,, , 0i j i jK T P
American:•Pu
t Pric
e
S0Ki
Ki
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
1 2 0C K C K
Bull spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0.
,1, 0i ji jC C
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2
In general:
,1, 1i j ii j iC C K K
Example:
1 2 2 1C K C K K K
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?
,1, 1Since 18 12 6 5 55 50 : arbitrage.i j ii j iC C K K
Sell the bull spread. Why?
Finance (Derivative Securities)
Vertical Spread
Model-Independent Option Valuation; Dr. Kurt Smith
2 1 0P K P K
Bear spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0.
, 1, 0i j i jP P
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2
In general:
, 1, 1i j ii j iP P K K
Example:
2 1 2 1P K P K K K
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
1 1 1,1, 1,
1 10ii i i
i ji j i ji ii i
K K K KC C C
K K K K
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2 K3
3 1 2 11 2 3
3 2 3 20
K K K KC K C K C K
K K K K
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
0
0.5
1
1.5
2
2.5
50 70 90 110 130 150
Expi
ry P
ayoff
Expiry Spot
K=70, 72, 90
0
2
4
6
8
10
12
50 70 90 110 130 150
Expi
ry P
ayoff
Expiry Spot
K=70, 80, 90
02468
101214161820
50 70 90 110 130 150
Expi
ry P
ayoff
Expiry Spot
K=70, 88, 90
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?
1 1 1,1, 1,
1 10ii i i
i ji j i ji ii i
K K K KC C C
K K K K
3 1 2 11 2 3
3 2 3 20
K K K KC K C K C K
K K K K
90 70 80 707 6 4 190 80 90 80
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).
0
2
4
6
8
10
12
50 70 90 110 130 150
Expi
ry P
ayoff
Expiry Spot
K=70, 80, 90
Finance (Derivative Securities)
Calendar Spread
Model-Independent Option Valuation; Dr. Kurt Smith
,, 1 0; , 0i ji jC C i j
Expi
ry P
ayoff
ST
Call
Pric
e
S0
, 1i jC ,i jC
, 0i jC
Calendar spread: same strikes (Ki), different maturities (Tj). Also r=0 & div=0.
In general:
Example:
2 1 0C T C T
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-ST2ST2
-K 0The 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 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:
1 2 2 10 C K C K K K
3 1 2 11 2 3
3 2 3 20
K K K KC K C K C K
K K K K
2 1 0C T C T
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2
Vertical Spread
Bull Spread Bear Spread
1 2 0C K C K
1 2 2 1C K C K K K
2 1 0P K P K
2 1 2 1P K P K K K
Finance (Derivative Securities)
Conclusion
Model-Independent Option Valuation; Dr. Kurt Smith
Expi
ry P
ayoff
Expi
ry P
ayoff
ST
STK1 K2 K3
3 1 2 11 2 3
3 2 3 20
K K K KC K C K C K
K K K K
Butterfly Spread