期货期权及其衍生品配套课件Ch27

Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
11
Money Market Account continued
T
Since
g0
= 1 and
gT
=
e
rdt
The process for the value of the account is
dg=rg dt
This has zero volatility. Using the money market account as the numeraire leads to the traditional risk-neutral world where l=0
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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Annuity Factor as the Numeraire
The equation
Βιβλιοθήκη Baiduf0 g0
E g
Suppose that s(t) is the swap rate corresponding to the annuity factor A. Then:
s(t)=EA[s(T)]
Options, Futures, and Other Derivatives, 7th International
d ?1 ƒ1
μ1
dt
σ1
dz
d?2 ƒ2
μ2
dt
σ2
dz
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
2
Forming a Riskless Portfolio
Martingales and Measures
Chapter 27
Options, Futures, and Other Derivatives, 7th International Edition,
Copyright © John C. Hull 2019
1
Derivatives Dependent on a Single Underlying Variable
and E T denotes expectatio ns in a w orld that is FRN w rt the bond price
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
T higsive:sμ1σ2 μ2σ1 rσ2 rσ1
or μ1r μ2 r
σ1
σ2
This shows that (m – r )/s is the same for all derivatives dependent on the same underlying
variable,
We refer to (m – r )/s as the market price of risk for and denote it by l
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Extension to Several Independent Factors
(Page 625)
In the traditiona l risk - neutral world
m
df (t ) r (t ) f (t )dt s f ,i (t ) f (t )dz i i 1
13
Forward Prices
In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price
Options, Futures, and Other Derivatives, 7th International
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Interest Rates
In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate
Consider a variable (not necessaril y the price
of a traded security) that follows the process
d m dt s dz
Imagine two derivative s dependent on
with prices ƒ1 and ƒ2. Suppose
We can set up a riskless portfolio , consisting of
+ σ 2 ƒ2 of the 1st derivative and σ1ƒ1 of the 2nd derivative
(σ 2 ƒ2 ) ƒ1 (σ 1ƒ1 ) ƒ2
= ( μ1σ 2 ƒ1ƒ2 μ 2 σ 1ƒ1ƒ2 ) t
0
,
the
equation
f0 g0
E g
fT gT
becomes
f0
Eˆ
e
T
rdt
0
fT
where Eˆ denotes expectatio ns in the
traditiona l risk - neutral world
Options, Futures, and Other Derivatives, 7th International
fT gT
becomes
f0
A
(0
)
E
A
A
fT (T
)
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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Annuity Factors and Swap Rates
df rfdtσfdz
In a worldwherethemarketpriceof risk is l
df (r ls) f dtsf dz
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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The Equivalent Martingale Measure Result (Page 620-621)
If wesetl equalto thevolatilityof
a securityg, thenIto's lemma showsthat f g is a martingalefor all derivativesecurityprices f
m
dg (t ) r (t ) g (t )dt s g ,i (t ) g (t )dz i i 1
For other worlds that are internally
consistent
df
(t)
r
(t
)
m i 1
l
is
f
,i
(t
)
f
(t )dt
m i 1
s f ,i (t ) f
Alternative Choices for the Numeraire Security g
Money Market Account Zero-coupon bond price Annuity factor
Options, Futures, and Other Derivatives, 7th International
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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Extension of the Analysis to Several Underlying Variables
Edition, Copyright © John C. Hull 2019
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Money Market Account as the Numeraire
The money market account is an account that starts at $1 and is always invested at the shortterm risk-free interest rate
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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Market Price of Risk (Page 616)
Sincteheportfoilsioriskle:ss =rt
Edition, Copyright © John C. Hull 2019
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Martingales (Page 620)
A martingale is a stochastic process with zero drift A variable following a martingale has the property that its expected future value equals its value today
Options, Futures, and Other Derivatives, 7th International
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Forward Risk Neutrality
We will refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g.
Options, Futures, and Other Derivatives, 7th International
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Alternative Worlds
In thetraditional risk-neutralworld
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Zero-Coupon Bond Maturing at time T as Numeraire
The equation
f0 g0
E g
fT gT
becomes
f0 P (0,T )ET [ fT ]
w here P ( 0,T ) is the zero - coupon bond price
(t )dz i
dg
(t)
r
(t
)
m i 1
l
i
s
g
,i
(
t
)
g
(
t
)
dt
m i 1
s g ,i (t ) g (t )dz i
Options, Futures, and Other Derivatives, 7th International
(Equations 27.12 and 27.13, page 619)
If f depends on several underlying
with
d?
ƒ
μ dt
n
σi
i 1
dz i
then
n
μ r λ iσ i i 1
variables
Options, Futures, and Other Derivatives, 7th International
If Eg denotes a world that is FRN wrt g
f0 g0
E
g
f
T
gT
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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