Post on 11-Jan-2016
description
23/4/21 1
Martingales and Measures
Chapter 27
23/4/21 2
Derivatives Dependent on a Single Underlying Variable
dzdtd
dzdtd
dzsdtmd
f
f
f
f
Suppose .f and f prices t with and
ononly dependent sderivative twoImagine
process thefollows that security) tradeda of
price y thenecessaril(not , variable,aConsider
222
2
111
1
21
23/4/21 3
Forming a Riskless Portfolio
t
)f ff f (=
f )f (f )f (
derivative 2nd theof f
and derivative1st theof f +
of consisting , portfolio riskless a upset can We
21122121
211122
11
22
23/4/21 4
Since the portfolio is riskless: =
This gives:
or 1
1
r t
r r
r r
1 2 2 1 2 1
2
2
Market Price of Risk
This shows that ( – r )/ is the same for all derivatives dependent only on the same underlying variable, and t.
We refer to ( – r )/ as the market price of risk for and denote it by
23/4/21 5
Extension of the Analysisto Several Underlying Variables
f
dzsdt d
1
1
iii
i
n
iii
n
iii
i
r
dzdtdf
m
23/4/21 6
How to measure λ?
For a nontraded securities(i.e.commodity),we can use its future market information to measure λ.
23/4/21 7
Martingales
A martingale is a stochastic process with zero drfit
A martingale has the property that its expected
future value equals its value today
0)( TE
dzd
23/4/21 8
Alternative Worlds
In the traditional risk -neutral world
In a world where the market price of risk
is
df rfdt fdz
df r fdt fdz
( )
23/4/21 9
A Key Result
(证明手写)ion)considerat
under period theduring income no
provide toassumed are and (
pricessecurity derivative all
for martingale a is that shows
lemma s Ito then ,security a
ofy volatilit the toequal set weIf
gf
f
gf
g
f和 g是否必须同一风险源?
设 :在以 g为记账单位的风险中性世界中:
23/4/21 10
1 2 1 21 2 1 2;f f g gdf rfdt fdz fdz dg rgdt gdz gdz
1 1 2 2 1 2
1 2 1 2
1 1 2 2 1 2 1 2
1 2 1 2
1 1 2 2 1
1 2
2 21 2
2 21 2
2 21 2
2
( ) ;
( )
ln ( 1/ 2 1/ 2 )
ln ( 1/ 2 1/ 2 )
ln / ( 1/ 2 1
f g f g f f
g g g g
f g f g f f f f
g g g g
f g f g f
df r fdt fdz fdz
dg r gdt gdz gdz
d f r dt dz dz
d g r dt dz dz
d f g
2 1 2
1 1 2 2
2 2 2
1 2
/ 2 1/ 2 1/ 2 )
( ) ( )
f g g
f g f g
dt
dz dz
f和 g是否必须同一风险源?令
23/4/21 11
1 1 2 2 1 2 1 2
1 1 2 2 1 1 2 2
1 1 2 2
1 1
2
2 2 2 2
2 21 2
1 2
ln / , /
1/ 2 ( )
( 1/ 2 1/ 2 1/ 2 1/ 2 )
( ) ( ) 1/ 2 (( ) ( ) )
( ) ( )
( / ) / ( / ) ( )
x
x x
xf g f g f f g g
x x xf g f g f g f g
x xf g f g
f g
x f g y f g e
dy e dx e dx
e dt
e dz e dz e dt
e dz e dz
d f g f g
2 21 2( )f gdz dz 是鞅过程
23/4/21 12
Forward Risk Neutrality
We 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.
If Eg denotes a world that is FRN wrt g
f
gE
f
ggT
T
0
0
23/4/21 13
Aleternative Choices for the Numeraire Security g
Money Market Account Zero-coupon bond price Annuity factor
23/4/21 14
Money Market Accountas the Numeraire
The money market account is an account that starts at $1 and is always invested at the short-term risk-free interest rate
The process for the value of the account is
dg=rgdt This has zero volatility. Using the money market
account as the numeraire leads to the traditional risk-neutral world
23/4/21 15
Money Market Accountcontinued
worldneutral-risk ltraditiona
in the nsexpectatio denotes ˆ where
)(ˆˆ
becomes
equation the,= and 1= Since
0
0
0
0
0
0
E
feEfeEf
g
fE
g
f
egg
TTr
Trdt
T
Tg
rdtT
T
T
23/4/21 16
Zero-Coupon Bond Maturing at time T as Numeraire
The equation
becomes
where ( , ) is the zero - coupon
bond price and denotes expectations
in a world that is FRN wrt the bond price
f
gE
f
g
f P T E f
P T
E
gT
T
T T
T
0
0
0 0
0
( , ) [ ]
23/4/21 17
Forward Prices
Consider an variable S that is not an interest rate. A forward contract on S with maturity T is defined as a contract that pays off ST-K at time T. Define f as the value of this forward contract. We have
f0 equals 0 if F=K,
So, F=ET(fT)
F is the forward price.
0 (0, )[ ( ) ]T Tf P T E S K
23/4/21 18
利率 T2时刻到期债券 T1交割的远期价格 F= P(t, T2)/P(t, T
1) 远期价格 F又可写为
2112 ,1
1
TTtRTTF
,-+=
S求终值
23/4/21 19
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
)],,([),,0(:
),(:
)],(),([1
:
]),(
),(),([
1),,(
),(
),(
)],,()(1[
1
211221
2
2112
2
21
1221
1
2
2112
TTTRETTRthen
TtPgand
TtPTtPTT
fSetting
TtP
TtPTtP
TTTTtR
TtP
TtP
TTtRTT
23/4/21 20
Annuity Factor as the Numeraire-1
Let s(t) is the forward swap rate of a swap starting at the time T0, with payment dates at times T1, T2,…,TN. Then the value of the fixed side of the swap is
1
1 10
[( , ( ) ( )N
i i ii
T T t P t T s t A t
-
)s =
23/4/21 21
Annuity Factor as the Numeraire-2
If we add $1 at time TN, the floating side of the swap is worth $1 at time T0. So, the value of the floating side is: P(t,T0)-P(t, TN)
Equating the values of the fixed and floating side we obstain:
0( , ) ( , )
( )( )
NP t T P t Ts t
A t
23/4/21 22
Annuity Factor as the Numeraire-3
0
0
0
( , ) ( , ), ( )
( ) [ ( )]
For any security,
(0)( )
N
A
TA
Let
f P t T P t T g A t
Then
S t E S T
ff A E
A T
23/4/21 23
Extension to Several Independent Factors
In a risk - neutral world
For other worlds that are internally consistent
df t r t f t dt t f t dz
dg t r t g t dt t g t dz
df t r t t f t dt t f t dz
dg t r t t
f i ii
m
gi ii
m
i f ii
m
f i ii
m
i gii
m
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
1
1
1 1
1
g t dt t g t dzgi ii
m
( ) ( ) ( )1
23/4/21 24
Extension to Several Independent Factorscontinued
We define a world that is FRN wrt
where
As in the one-factor case, is a
martingale and the rest of the results hold.
i gi
g
f g
23/4/21 25
Applications
Valuation of a European call option when interest rates are stochastic
Valuation of an option to exchange one asset for another
23/4/21 26
Valuation of a European call option when interest rates are stochastic
Assume ST is lognormal then:
The result is the same as BS except r replaced by R.
)]0,[max()]0,[max(),0(
),0(:
XSEeXSETPc
eTPDefine
TTRT
TT
RT
)()(
)(
)()()()]0,[max(
210
0
21
dNXedNSc
eSSE
dXNdNSEXSE
RT
RTTT
TTTT
-
23/4/21 27
Valuation of an option to exchange one asset(U) for another(V) Choose U as the numeraire, and set f as the value
of the option so that fT=max(VT-UT,0), so,
)()(
)(
)]()()([
]0,1[max(])0,max(
[
20100
0
0
210
000
dNUdNVf
U
VE
U
V
dNdNU
VEU
U
VEU
U
UVEUf
T
TU
T
TU
T
TU
T
TTU
23/4/21 28
Change of Numeraire
qv
qghqv
vhg
q
vqv
and between n correlatio theis
and, ofy volatilit theis ,, of
y volatilit theis whereby increases
variablea ofdrift the, to from
security numeraire thechange When we
23/4/21 29
证明 当记帐单位从 g变为 h时, V的偏移率增加了
, , ,1
q,i , ,
, ,1
, ,
, ,1
/ , ITO
n
v h i g i v ii
h i g i
n
v q i v ii
v i i q i i
i
v q
n
v q q i v ii
q h g
dv v dt dq q dt
dz
vq E dvdq E dv dq
= -
定义 从 引理可知, = - ,故
=
,
由于 不相关,且从 的定义有:
dt= -
=