23/4/21 1

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

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

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

23/4/21 6

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 .

23/4/21 7

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 8

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).

23/4/21 9

How to measure λ?

For a nontraded securities(i.e.commodity),we can use its future market information to measure λ.

23/4/21 10

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 11

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 12

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

23/4/21 13

讨论 f 和 g是否必须同一风险源？

推导过程手写。

23/4/21 14

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 15

Aleternative Choices for the Numeraire Security g

Money Market Account Zero-coupon bond price Annuity factor

23/4/21 16

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 17

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 18

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 19

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 20

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

，－＋＝

23/4/21 21

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 22

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 ＝

23/4/21 23

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

23/4/21 24

Annuity Factor as the Numeraire-3

)()0(

security,any For

)]([)(

)(),,(),(

0

0

TA

fEAf

TSEtS

Then

tAgTtPTtPf

Let

TA

A

N

23/4/21 25

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 26

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

23/4/21 27

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 28

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 29

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 30

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 31

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＝ －

＝

23/4/21 32

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)

23/4/21 33

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

23/4/21 34

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

(

)

23/4/21 35

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

23/4/21 36

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.