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＝ －

＝