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Martingales and Measures
Chapter 21
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
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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
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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
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Differential Equation for ƒ
Using Ito’s lemma to obtain expressions for and in terms of m and sThe equation
=r
becomes
2
1 )(
2
222 rf
fssm
f
t
f
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Risk-Neutral Valuation
This analogy shows that we can value ƒ in a risk-neutral world providing the drift rate of is reduced from m to m – s
Note: When is not the price of an investment asset, the risk-neutral valuation argument does not necessarily tell us anything about what would happen with in a risk-neutral world .
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Extension of the Analysisto Several Underlying Variables
f
dzsdt d
1
1
iii
i
n
iii
n
iii
i
r
dzdtdf
m
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Traditional Risk-Neutral Valuation with Several Underlying Variables
A derivative can always be valued as if the would is risk neutral, provided that the expected growth rate of each underlying variable is assumed to be mi-λisi rather than mi.
The volatility of the variables and the coefficient of the correlation between variables are not changed. (CIR,1985).
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How to measure λ?
For a nontraded securities(i.e.commodity),we can use its future market information to measure λ.
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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
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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
( )
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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
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讨论 f 和 g是否必须同一风险源?
推导过程手写。
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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
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Aleternative Choices for the Numeraire Security g
Money Market Account Zero-coupon bond price Annuity factor
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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
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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
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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
( , ) [ ]
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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
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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
,-+=
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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
)],,([),,0(:
),(:
)],(),([1
:
]),(
),(),([
1),,(
),(
),(
)],,()(1[
1
211221
2
2112
2
21
1221
1
2
2112
TTTRETTRthen
TtPgand
TtPTtPTT
fSetting
TtP
TtPTtP
TTTTtR
TtP
TtP
TTtRTT
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Annuity Factor as the Numeraire-1
Let Sn(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 =
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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
tA
TtPTtPtS N
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Annuity Factor as the Numeraire-3
)()0(
security,any For
)]([)(
)(),,(),(
0
0
TA
fEAf
TSEtS
Then
tAgTtPTtPf
Let
TA
A
N
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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
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Extension to Several Independent Factorscontinued
We define a world that is FRN wrt
as world where
As in the one - factor case, is a
martingale and the rest of the results hold.
i
g
f g
gi
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Applications
Valuation of a European call option when interest rates are stochastic
Valuation of an option to exchange one asset for another
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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
-
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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
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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
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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= -
=
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Quantos
Quantos are derivatives where the payoff is defined using variables measured in one currency and paid in another currency
Example: contract providing a payoff of ST – K dollars ($) where S is the Nikkei stock index (a yen number)
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Quantos continued
When we move from the traditional risk
neutral world in currency to the traditional
risk neutral world in currency , the growth
rate of a variable increases by
where is the volatility of , is the volatility
of the exchange rate (units of Y per unit of X), and
is the coefficient of correlation between
the two
Y
X
V
VV S
V S
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Quantos continuedWhen we move from a forward risk
neutral world in currency to a forward
risk neutral world in currency both being
wrt to zero - coupon bonds maturing at time
, the growth rate of a variable increases by
where is the volatility of the forward value of ,
is the volatility of the forward exchange rate
(units of Y per unit of X), and is the coefficient
of correlation between the two
Y
X
T V
VF G
F
G
(
)
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Siegel’s Paradox
An exchange rate (units of currency per untit of
currency ) follows the risk -neutral process
This implies from Ito's lemma that
Given that the process for S has a drift
rate of we expect the process for
to have a drift of
What is going on here?
S Y
X
dS r r Sdt Sdz
d S r r S dt S dz
r r
S r r
Y X S
X Y S S
Y X
X Y
[ ]
( / ) [ ]( / ) ( / )
,
.
1 1 1
1
2
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Siegel’s Paradox(2)
In the process of dS, the numeraire is the money market account in currency Y. In the second equation, the numeraire is also the money market account in currency Y.
To change the numeraire from Y to X,the growth rate of 1/S increase by ρσVσS where V=1/S and ρ is the correlation between S and 1/S.In this case, ρ=-1, and σv =σS.It follows that the change of numeraire causes the growth rate of 1/S to increase – σS
2.